id sid tid token lemma pos cana-5942 1 1 communications communication NOUN cana-5942 1 2 on on ADP cana-5942 1 3 applied apply VERB cana-5942 1 4 nonlinear nonlinear ADJ cana-5942 1 5 analysis analysis NOUN cana-5942 1 6 issn issn NOUN cana-5942 1 7 : : PUNCT cana-5942 1 8 1074 1074 NUM cana-5942 1 9 - - PUNCT cana-5942 1 10 133x 133x NUM cana-5942 1 11 vol vol NOUN cana-5942 1 12 31 31 NUM cana-5942 1 13 no no NOUN cana-5942 1 14 . . PUNCT cana-5942 2 1 8s 8s PROPN cana-5942 2 2 ( ( PUNCT cana-5942 2 3 2024 2024 NUM cana-5942 2 4 ) ) PUNCT cana-5942 2 5 1128 1128 NUM cana-5942 2 6 https://internationalpubls.com https://internationalpubls.com X cana-5942 2 7 reflection reflection NOUN cana-5942 2 8 at at ADP cana-5942 2 9 the the DET cana-5942 2 10 free free ADJ cana-5942 2 11 surface surface NOUN cana-5942 2 12 of of ADP cana-5942 2 13 generalized generalized ADJ cana-5942 2 14 thermoelastic thermoelastic ADJ cana-5942 2 15 medium medium NOUN cana-5942 2 16 with with ADP cana-5942 2 17 voids voids NOUN cana-5942 2 18 under under ADP cana-5942 2 19 three three NUM cana-5942 2 20 phase phase NOUN cana-5942 2 21 lag lag NOUN cana-5942 2 22 effect effect NOUN cana-5942 2 23 neelam neelam PROPN cana-5942 2 24 hooda1 hooda1 PROPN cana-5942 2 25 and and CCONJ cana-5942 2 26 renu renu PROPN cana-5942 2 27 yadav yadav PROPN cana-5942 2 28 * * PUNCT cana-5942 2 29 ( ( PUNCT cana-5942 2 30 corresponding correspond VERB cana-5942 2 31 author author NOUN cana-5942 2 32 ) ) PUNCT cana-5942 2 33 1assistant 1assistant NUM cana-5942 2 34 professor professor NOUN cana-5942 2 35 of of ADP cana-5942 2 36 mathematics mathematics PROPN cana-5942 2 37 govt govt NOUN cana-5942 2 38 . . PUNCT cana-5942 3 1 college college NOUN cana-5942 3 2 for for ADP cana-5942 3 3 women woman NOUN cana-5942 3 4 , , PUNCT cana-5942 3 5 lakhan lakhan PROPN cana-5942 3 6 majra majra NOUN cana-5942 3 7 , , PUNCT cana-5942 3 8 rohtak rohtak PROPN cana-5942 3 9 , , PUNCT cana-5942 3 10 haryana haryana PROPN cana-5942 3 11 * * PUNCT cana-5942 3 12 assistant assistant PROPN cana-5942 3 13 professor professor NOUN cana-5942 3 14 of of ADP cana-5942 3 15 mathematics mathematics PROPN cana-5942 3 16 f. f. PROPN cana-5942 3 17 g. g. PROPN cana-5942 3 18 m. m. PROPN cana-5942 3 19 govt govt PROPN cana-5942 3 20 . . PUNCT cana-5942 4 1 college college NOUN cana-5942 4 2 adampur adampur PROPN cana-5942 4 3 , , PUNCT cana-5942 4 4 hisar hisar PROPN cana-5942 4 5 , , PUNCT cana-5942 4 6 haryana haryana PROPN cana-5942 4 7 email email NOUN cana-5942 4 8 : : PUNCT cana-5942 5 1 renumath87@gmail.com renumath87@gmail.com X cana-5942 5 2 article article NOUN cana-5942 5 3 history history NOUN cana-5942 5 4 : : PUNCT cana-5942 5 5 received receive VERB cana-5942 5 6 : : PUNCT cana-5942 5 7 02 02 NUM cana-5942 5 8 - - SYM cana-5942 5 9 10 10 NUM cana-5942 5 10 - - PUNCT cana-5942 5 11 2024 2024 NUM cana-5942 5 12 revised revise VERB cana-5942 5 13 : : PUNCT cana-5942 5 14 25 25 NUM cana-5942 5 15 - - SYM cana-5942 5 16 11 11 NUM cana-5942 5 17 - - PUNCT cana-5942 5 18 2024 2024 NUM cana-5942 5 19 accepted accept VERB cana-5942 5 20 : : PUNCT cana-5942 5 21 26 26 NUM cana-5942 5 22 - - SYM cana-5942 5 23 12 12 NUM cana-5942 5 24 - - PUNCT cana-5942 5 25 2024 2024 NUM cana-5942 5 26 abstract abstract NOUN cana-5942 5 27 : : PUNCT cana-5942 5 28 in in ADP cana-5942 5 29 this this DET cana-5942 5 30 paper paper NOUN cana-5942 5 31 , , PUNCT cana-5942 5 32 propagation propagation NOUN cana-5942 5 33 of of ADP cana-5942 5 34 plane plane NOUN cana-5942 5 35 waves wave NOUN cana-5942 5 36 in in ADP cana-5942 5 37 an an DET cana-5942 5 38 isotropic isotropic NOUN cana-5942 5 39 , , PUNCT cana-5942 5 40 homogeneous homogeneous ADJ cana-5942 5 41 , , PUNCT cana-5942 5 42 generalized generalized ADJ cana-5942 5 43 magneto magneto NOUN cana-5942 5 44 - - PUNCT cana-5942 5 45 thermo thermo NOUN cana-5942 5 46 - - PUNCT cana-5942 5 47 viscoelastic viscoelastic NOUN cana-5942 5 48 semi semi ADJ cana-5942 5 49 - - ADJ cana-5942 5 50 infinite infinite ADJ cana-5942 5 51 medium medium NOUN cana-5942 5 52 ( ( PUNCT cana-5942 5 53 having have VERB cana-5942 5 54 stressfree stressfree ADJ cana-5942 5 55 insulated insulated ADJ cana-5942 5 56 boundary boundary NOUN cana-5942 5 57 ) ) PUNCT cana-5942 5 58 containing contain VERB cana-5942 5 59 a a DET cana-5942 5 60 distribution distribution NOUN cana-5942 5 61 of of ADP cana-5942 5 62 vacuous vacuous ADJ cana-5942 5 63 pores pore NOUN cana-5942 5 64 ( ( PUNCT cana-5942 5 65 voids void NOUN cana-5942 5 66 ) ) PUNCT cana-5942 5 67 has have AUX cana-5942 5 68 been be AUX cana-5942 5 69 investigated investigate VERB cana-5942 5 70 . . PUNCT cana-5942 6 1 basic basic ADJ cana-5942 6 2 governing governing NOUN cana-5942 6 3 equations equation NOUN cana-5942 6 4 are be AUX cana-5942 6 5 based base VERB cana-5942 6 6 on on ADP cana-5942 6 7 the the DET cana-5942 6 8 linear linear PROPN cana-5942 6 9 theory theory NOUN cana-5942 6 10 of of ADP cana-5942 6 11 thermoelasticity thermoelasticity NOUN cana-5942 6 12 , , PUNCT cana-5942 6 13 namely namely ADV cana-5942 6 14 , , PUNCT cana-5942 6 15 three three NUM cana-5942 6 16 - - PUNCT cana-5942 6 17 phase phase NOUN cana-5942 6 18 - - PUNCT cana-5942 6 19 lag lag NOUN cana-5942 6 20 ( ( PUNCT cana-5942 6 21 3pl 3pl ADJ cana-5942 6 22 ) ) PUNCT cana-5942 6 23 thermoelasticity thermoelasticity NOUN cana-5942 6 24 to to PART cana-5942 6 25 account account VERB cana-5942 6 26 for for ADP cana-5942 6 27 finite finite ADJ cana-5942 6 28 velocity velocity NOUN cana-5942 6 29 of of ADP cana-5942 6 30 the the DET cana-5942 6 31 temperature temperature NOUN cana-5942 6 32 . . PUNCT cana-5942 7 1 it it PRON cana-5942 7 2 is be AUX cana-5942 7 3 found find VERB cana-5942 7 4 that that SCONJ cana-5942 7 5 three three NUM cana-5942 7 6 compressional compressional ADJ cana-5942 7 7 waves wave NOUN cana-5942 7 8 and and CCONJ cana-5942 7 9 a a DET cana-5942 7 10 shear shear NOUN cana-5942 7 11 vertical vertical ADJ cana-5942 7 12 ( ( PUNCT cana-5942 7 13 sv sv NOUN cana-5942 7 14 ) ) PUNCT cana-5942 7 15 wave wave NOUN cana-5942 7 16 can can AUX cana-5942 7 17 exist exist VERB cana-5942 7 18 in in ADP cana-5942 7 19 the the DET cana-5942 7 20 medium medium NOUN cana-5942 7 21 adopted adopt VERB cana-5942 7 22 . . PUNCT cana-5942 8 1 reflection reflection NOUN cana-5942 8 2 phenomena phenomenon NOUN cana-5942 8 3 of of ADP cana-5942 8 4 compressional compressional ADJ cana-5942 8 5 and and CCONJ cana-5942 8 6 shear shear ADJ cana-5942 8 7 waves wave NOUN cana-5942 8 8 from from ADP cana-5942 8 9 the the DET cana-5942 8 10 free free ADJ cana-5942 8 11 surface surface NOUN cana-5942 8 12 of of ADP cana-5942 8 13 the the DET cana-5942 8 14 medium medium NOUN cana-5942 8 15 is be AUX cana-5942 8 16 considered consider VERB cana-5942 8 17 . . PUNCT cana-5942 9 1 the the DET cana-5942 9 2 formulae formulae NOUN cana-5942 9 3 for for ADP cana-5942 9 4 reflection reflection NOUN cana-5942 9 5 coefficients coefficient NOUN cana-5942 9 6 of of ADP cana-5942 9 7 various various ADJ cana-5942 9 8 reflected reflect VERB cana-5942 9 9 waves wave NOUN cana-5942 9 10 have have AUX cana-5942 9 11 been be AUX cana-5942 9 12 obtained obtain VERB cana-5942 9 13 in in ADP cana-5942 9 14 the the DET cana-5942 9 15 closed closed ADJ cana-5942 9 16 form form NOUN cana-5942 9 17 . . PUNCT cana-5942 10 1 the the DET cana-5942 10 2 numerical numerical ADJ cana-5942 10 3 values value NOUN cana-5942 10 4 of of ADP cana-5942 10 5 the the DET cana-5942 10 6 modulus modulus NOUN cana-5942 10 7 of of ADP cana-5942 10 8 reflection reflection NOUN cana-5942 10 9 coefficients coefficient NOUN cana-5942 10 10 are be AUX cana-5942 10 11 presented present VERB cana-5942 10 12 graphically graphically ADV cana-5942 10 13 . . PUNCT cana-5942 11 1 keywords keyword NOUN cana-5942 11 2 : : PUNCT cana-5942 11 3 three three NUM cana-5942 11 4 - - PUNCT cana-5942 11 5 phase phase NOUN cana-5942 11 6 - - PUNCT cana-5942 11 7 lag lag NOUN cana-5942 11 8 thermoelasticity thermoelasticity NOUN cana-5942 11 9 , , PUNCT cana-5942 11 10 reflection reflection NOUN cana-5942 11 11 , , PUNCT cana-5942 11 12 voids voids NOUN cana-5942 11 13 . . PUNCT cana-5942 12 1 introduction introduction NOUN cana-5942 12 2 concept concept NOUN cana-5942 12 3 of of ADP cana-5942 12 4 generalized generalized ADJ cana-5942 12 5 thermoelasticity thermoelasticity NOUN cana-5942 12 6 has have AUX cana-5942 12 7 drawn draw VERB cana-5942 12 8 the the DET cana-5942 12 9 attention attention NOUN cana-5942 12 10 of of ADP cana-5942 12 11 many many ADJ cana-5942 12 12 researchers researcher NOUN cana-5942 12 13 during during ADP cana-5942 12 14 the the DET cana-5942 12 15 last last ADJ cana-5942 12 16 decades decade NOUN cana-5942 12 17 . . PUNCT cana-5942 13 1 series series NOUN cana-5942 13 2 of of ADP cana-5942 13 3 generalized generalized ADJ cana-5942 13 4 theories theory NOUN cana-5942 13 5 of of ADP cana-5942 13 6 thermoelasticity thermoelasticity NOUN cana-5942 13 7 has have AUX cana-5942 13 8 been be AUX cana-5942 13 9 developed develop VERB cana-5942 13 10 to to PART cana-5942 13 11 overcome overcome VERB cana-5942 13 12 the the DET cana-5942 13 13 shortcomings shortcoming NOUN cana-5942 13 14 inherent inherent ADJ cana-5942 13 15 in in ADP cana-5942 13 16 the the DET cana-5942 13 17 classical classical ADJ cana-5942 13 18 coupled couple VERB cana-5942 13 19 dynamical dynamical ADJ cana-5942 13 20 theory theory NOUN cana-5942 13 21 of of ADP cana-5942 13 22 elasticity elasticity NOUN cana-5942 13 23 . . PUNCT cana-5942 14 1 these these DET cana-5942 14 2 theories theory NOUN cana-5942 14 3 are be AUX cana-5942 14 4 characterized characterize VERB cana-5942 14 5 by by ADP cana-5942 14 6 the the DET cana-5942 14 7 finite finite ADJ cana-5942 14 8 speed speed NOUN cana-5942 14 9 of of ADP cana-5942 14 10 propagation propagation NOUN cana-5942 14 11 of of ADP cana-5942 14 12 thermal thermal ADJ cana-5942 14 13 disturbance disturbance NOUN cana-5942 14 14 . . PUNCT cana-5942 15 1 generalized generalize VERB cana-5942 15 2 thermoelastic thermoelastic ADJ cana-5942 15 3 model model NOUN cana-5942 15 4 known know VERB cana-5942 15 5 as as ADP cana-5942 15 6 dual dual ADJ cana-5942 15 7 - - PUNCT cana-5942 15 8 phase phase NOUN cana-5942 15 9 - - PUNCT cana-5942 15 10 lag lag NOUN cana-5942 15 11 model model NOUN cana-5942 15 12 was be AUX cana-5942 15 13 developed develop VERB cana-5942 15 14 by by ADP cana-5942 15 15 tzou tzou NOUN cana-5942 15 16 ( ( PUNCT cana-5942 15 17 1995 1995 NUM cana-5942 15 18 ) ) PUNCT cana-5942 15 19 by by ADP cana-5942 15 20 considering consider VERB cana-5942 15 21 micro micro ADJ cana-5942 15 22 - - ADJ cana-5942 15 23 structural structural ADJ cana-5942 15 24 effects effect NOUN cana-5942 15 25 into into ADP cana-5942 15 26 the the DET cana-5942 15 27 delayed delay VERB cana-5942 15 28 response response NOUN cana-5942 15 29 in in ADP cana-5942 15 30 time time NOUN cana-5942 15 31 at at ADP cana-5942 15 32 the the DET cana-5942 15 33 macroscopic macroscopic ADJ cana-5942 15 34 level level NOUN cana-5942 15 35 by by ADP cana-5942 15 36 taking take VERB cana-5942 15 37 into into ADP cana-5942 15 38 account account NOUN cana-5942 15 39 that that SCONJ cana-5942 15 40 the the DET cana-5942 15 41 increase increase NOUN cana-5942 15 42 in in ADP cana-5942 15 43 the the DET cana-5942 15 44 lattice lattice NOUN cana-5942 15 45 temperature temperature NOUN cana-5942 15 46 is be AUX cana-5942 15 47 delayed delay VERB cana-5942 15 48 due due ADJ cana-5942 15 49 to to ADP cana-5942 15 50 phonon phonon NOUN cana-5942 15 51 - - PUNCT cana-5942 15 52 electron electron NOUN cana-5942 15 53 interactions interaction NOUN cana-5942 15 54 on on ADP cana-5942 15 55 the the DET cana-5942 15 56 macroscopic macroscopic ADJ cana-5942 15 57 level level NOUN cana-5942 15 58 . . PUNCT cana-5942 16 1 tzou tzou NOUN cana-5942 16 2 introduced introduce VERB cana-5942 16 3 two two NUM cana-5942 16 4 different different ADJ cana-5942 16 5 phase phase NOUN cana-5942 16 6 lags lag VERB cana-5942 16 7 , , PUNCT cana-5942 16 8 one one NUM cana-5942 16 9 for for ADP cana-5942 16 10 the the DET cana-5942 16 11 heat heat NOUN cana-5942 16 12 flux flux NOUN cana-5942 16 13 vector,πœπ‘ž vector,πœπ‘ž NOUN cana-5942 16 14 and and CCONJ cana-5942 16 15 the the DET cana-5942 16 16 other other ADJ cana-5942 16 17 for for ADP cana-5942 16 18 the the DET cana-5942 16 19 temperature temperature NOUN cana-5942 16 20 gradient,πœπ‘‡. gradient,πœπ‘‡. NOUN cana-5942 16 21 for for ADP cana-5942 16 22 this this DET cana-5942 16 23 model model NOUN cana-5942 16 24 , , PUNCT cana-5942 16 25 classical classical ADJ cana-5942 16 26 fourier fourier NOUN cana-5942 16 27 ’s ’s PART cana-5942 16 28 law law NOUN cana-5942 16 29 is be AUX cana-5942 16 30 thus thus ADV cana-5942 16 31 modified modify VERB cana-5942 16 32 to to ADP cana-5942 16 33 οΏ½ οΏ½ PROPN cana-5942 16 34 βƒ— βƒ— NOUN cana-5942 16 35 οΏ½ οΏ½ NOUN cana-5942 16 36 (𝑝 (𝑝 PROPN cana-5942 16 37 , , PUNCT cana-5942 16 38 𝑑 𝑑 PROPN cana-5942 16 39 + + NOUN cana-5942 16 40 πœπ‘ž πœπ‘ž X cana-5942 16 41 ) ) PUNCT cana-5942 16 42 = = PUNCT cana-5942 16 43 βˆ’πΎ[ βˆ’πΎ[ PROPN cana-5942 16 44 οΏ½ οΏ½ PROPN cana-5942 16 45 βƒ—βƒ— βƒ—βƒ— PROPN cana-5942 16 46 οΏ½ οΏ½ PROPN cana-5942 16 47 𝑇(𝑝 𝑇(𝑝 PROPN cana-5942 16 48 , , PUNCT cana-5942 16 49 𝑑 𝑑 PROPN cana-5942 16 50 + + X cana-5942 16 51 πœπ‘‡ πœπ‘‡ NOUN cana-5942 16 52 ) ) PUNCT cana-5942 16 53 ] ] PUNCT cana-5942 16 54 . . PUNCT cana-5942 17 1 the the DET cana-5942 17 2 delay delay NOUN cana-5942 17 3 time time NOUN cana-5942 17 4 πœπ‘ž πœπ‘ž NOUN cana-5942 17 5 is be AUX cana-5942 17 6 interpreted interpret VERB cana-5942 17 7 as as ADP cana-5942 17 8 the the DET cana-5942 17 9 relaxation relaxation NOUN cana-5942 17 10 time time NOUN cana-5942 17 11 due due ADJ cana-5942 17 12 to to ADP cana-5942 17 13 fast fast ADJ cana-5942 17 14 transient transient ADJ cana-5942 17 15 effects effect NOUN cana-5942 17 16 of of ADP cana-5942 17 17 thermal thermal ADJ cana-5942 17 18 inertia inertia NOUN cana-5942 17 19 . . PUNCT cana-5942 18 1 the the DET cana-5942 18 2 other other ADJ cana-5942 18 3 delay delay NOUN cana-5942 18 4 time time NOUN cana-5942 18 5 πœπ‘‡is πœπ‘‡is PROPN cana-5942 18 6 interpreted interpret VERB cana-5942 18 7 as as ADP cana-5942 18 8 that that PRON cana-5942 18 9 caused cause VERB cana-5942 18 10 by by ADP cana-5942 18 11 the the DET cana-5942 18 12 microstructural microstructural ADJ cana-5942 18 13 interactions interaction NOUN cana-5942 18 14 . . PUNCT cana-5942 19 1 stability stability NOUN cana-5942 19 2 of of ADP cana-5942 19 3 the the DET cana-5942 19 4 dual dual ADJ cana-5942 19 5 - - PUNCT cana-5942 19 6 phase phase NOUN cana-5942 19 7 - - PUNCT cana-5942 19 8 lag lag NOUN cana-5942 19 9 heat heat NOUN cana-5942 19 10 conduction conduction NOUN cana-5942 19 11 is be AUX cana-5942 19 12 discussed discuss VERB cana-5942 19 13 by by ADP cana-5942 19 14 quintanilla quintanilla NOUN cana-5942 19 15 and and CCONJ cana-5942 19 16 racke racke ADJ cana-5942 19 17 ( ( PUNCT cana-5942 19 18 2006 2006 NUM cana-5942 19 19 ) ) PUNCT cana-5942 19 20 . . PUNCT cana-5942 20 1 sixth sixth ADJ cana-5942 20 2 and and CCONJ cana-5942 20 3 the the DET cana-5942 20 4 most most ADV cana-5942 20 5 recent recent ADJ cana-5942 20 6 development development NOUN cana-5942 20 7 in in ADP cana-5942 20 8 generalized generalized ADJ cana-5942 20 9 thermoelasticity thermoelasticity NOUN cana-5942 20 10 is be AUX cana-5942 20 11 three three NUM cana-5942 20 12 - - PUNCT cana-5942 20 13 phase phase NOUN cana-5942 20 14 - - PUNCT cana-5942 20 15 lag lag NOUN cana-5942 20 16 thermoelastic thermoelastic ADJ cana-5942 20 17 model model NOUN cana-5942 20 18 . . PUNCT cana-5942 21 1 roychoudhuri roychoudhuri PROPN cana-5942 21 2 ( ( PUNCT cana-5942 21 3 2007 2007 NUM cana-5942 21 4 ) ) PUNCT cana-5942 21 5 established establish VERB cana-5942 21 6 this this DET cana-5942 21 7 model model NOUN cana-5942 21 8 by by ADP cana-5942 21 9 modifying modify VERB cana-5942 21 10 heat heat NOUN cana-5942 21 11 conduction conduction NOUN cana-5942 21 12 law law NOUN cana-5942 21 13 to to ADP cana-5942 21 14 the the DET cana-5942 21 15 form form NOUN cana-5942 21 16 οΏ½ οΏ½ NOUN cana-5942 21 17 βƒ— βƒ— NOUN cana-5942 21 18 οΏ½ οΏ½ NOUN cana-5942 21 19 (𝑝 (𝑝 PROPN cana-5942 21 20 , , PUNCT cana-5942 21 21 𝑑 𝑑 PROPN cana-5942 21 22 + + NOUN cana-5942 21 23 πœπ‘ž πœπ‘ž X cana-5942 21 24 ) ) PUNCT cana-5942 21 25 = = PUNCT cana-5942 21 26 βˆ’[𝐾 βˆ’[𝐾 NOUN cana-5942 21 27 οΏ½ οΏ½ NOUN cana-5942 21 28 βƒ—βƒ— βƒ—βƒ— PROPN cana-5942 21 29 οΏ½ οΏ½ PROPN cana-5942 21 30 𝑇(𝑝 𝑇(𝑝 PROPN cana-5942 21 31 , , PUNCT cana-5942 21 32 𝑑 𝑑 PROPN cana-5942 21 33 + + X cana-5942 21 34 πœπ‘‡ πœπ‘‡ NOUN cana-5942 21 35 ) ) PUNCT cana-5942 22 1 + + CCONJ cana-5942 22 2 πΎβˆ— πΎβˆ— NUM cana-5942 22 3 οΏ½ οΏ½ PROPN cana-5942 22 4 βƒ—βƒ— βƒ—βƒ— PROPN cana-5942 22 5 οΏ½ οΏ½ PROPN cana-5942 22 6 𝑣(𝑝 𝑣(𝑝 PROPN cana-5942 22 7 , , PUNCT cana-5942 22 8 𝑑 𝑑 X cana-5942 22 9 + + X cana-5942 22 10 𝜏𝜐 𝜏𝜐 ADJ cana-5942 22 11 ) ) PUNCT cana-5942 22 12 ] ] PUNCT cana-5942 22 13 . . PUNCT cana-5942 23 1 here𝜏𝜐 here𝜏𝜐 PROPN cana-5942 23 2 , , PUNCT cana-5942 23 3 the the DET cana-5942 23 4 delay delay NOUN cana-5942 23 5 time time NOUN cana-5942 23 6 in in ADP cana-5942 23 7 thermal thermal ADJ cana-5942 23 8 displacement displacement NOUN cana-5942 23 9 gradient gradient NOUN cana-5942 23 10 is be AUX cana-5942 23 11 also also ADV cana-5942 23 12 introduced introduce VERB cana-5942 23 13 in in ADP cana-5942 23 14 addition addition NOUN cana-5942 23 15 to to ADP cana-5942 23 16 πœπ‘ž πœπ‘ž NOUN cana-5942 23 17 and and CCONJ cana-5942 23 18 πœπ‘‡. πœπ‘‡. PROPN cana-5942 23 19 stability stability NOUN cana-5942 23 20 of of ADP cana-5942 23 21 three three NUM cana-5942 23 22 - - PUNCT cana-5942 23 23 phase phase NOUN cana-5942 23 24 - - PUNCT cana-5942 23 25 lag lag NOUN cana-5942 23 26 heat heat NOUN cana-5942 23 27 conduction conduction NOUN cana-5942 23 28 equation equation NOUN cana-5942 23 29 and and CCONJ cana-5942 23 30 the the DET cana-5942 23 31 relations relation NOUN cana-5942 23 32 among among ADP cana-5942 23 33 the the DET cana-5942 23 34 three three NUM cana-5942 23 35 material material NOUN cana-5942 23 36 communications communication NOUN cana-5942 23 37 on on ADP cana-5942 23 38 applied apply VERB cana-5942 23 39 nonlinear nonlinear ADJ cana-5942 23 40 analysis analysis NOUN cana-5942 23 41 issn issn NOUN cana-5942 23 42 : : PUNCT cana-5942 23 43 1074 1074 NUM cana-5942 23 44 - - PUNCT cana-5942 23 45 133x 133x NUM cana-5942 23 46 vol vol NOUN cana-5942 23 47 31 31 NUM cana-5942 23 48 no no NOUN cana-5942 23 49 . . PUNCT cana-5942 24 1 8s 8s PROPN cana-5942 24 2 ( ( PUNCT cana-5942 24 3 2024 2024 NUM cana-5942 24 4 ) ) PUNCT cana-5942 24 5 1129 1129 NUM cana-5942 24 6 https://internationalpubls.com https://internationalpubls.com X cana-5942 24 7 parameters parameter NOUN cana-5942 24 8 are be AUX cana-5942 24 9 discussed discuss VERB cana-5942 24 10 by by ADP cana-5942 24 11 quintanilla quintanilla NOUN cana-5942 24 12 and and CCONJ cana-5942 24 13 racke racke ADJ cana-5942 24 14 ( ( PUNCT cana-5942 24 15 2008 2008 NUM cana-5942 24 16 ) ) PUNCT cana-5942 24 17 . . PUNCT cana-5942 25 1 mukhopdhyay mukhopdhyay NOUN cana-5942 25 2 and and CCONJ cana-5942 25 3 kumar kumar PROPN cana-5942 25 4 ( ( PUNCT cana-5942 25 5 2010 2010 NUM cana-5942 25 6 ) ) PUNCT cana-5942 25 7 analysed analyse VERB cana-5942 25 8 the the DET cana-5942 25 9 effects effect NOUN cana-5942 25 10 of of ADP cana-5942 25 11 phase phase NOUN cana-5942 25 12 lags lag VERB cana-5942 25 13 on on ADP cana-5942 25 14 wave wave NOUN cana-5942 25 15 propagation propagation NOUN cana-5942 25 16 in in ADP cana-5942 25 17 a a DET cana-5942 25 18 thick thick ADJ cana-5942 25 19 plate plate NOUN cana-5942 25 20 under under ADP cana-5942 25 21 axisymmetric axisymmetric ADJ cana-5942 25 22 temperature temperature NOUN cana-5942 25 23 distribution distribution NOUN cana-5942 25 24 . . PUNCT cana-5942 26 1 a a DET cana-5942 26 2 magneto magneto ADJ cana-5942 26 3 - - PUNCT cana-5942 26 4 thermoelastic thermoelastic ADJ cana-5942 26 5 problem problem NOUN cana-5942 26 6 in in ADP cana-5942 26 7 an an DET cana-5942 26 8 unbounded unbounded ADJ cana-5942 26 9 , , PUNCT cana-5942 26 10 perfectly perfectly ADV cana-5942 26 11 conducting conduct VERB cana-5942 26 12 medium medium NOUN cana-5942 26 13 with with ADP cana-5942 26 14 three three NUM cana-5942 26 15 phase phase NOUN cana-5942 26 16 lags lag NOUN cana-5942 26 17 has have AUX cana-5942 26 18 been be AUX cana-5942 26 19 discussed discuss VERB cana-5942 26 20 by by ADP cana-5942 26 21 das das PROPN cana-5942 26 22 and and CCONJ cana-5942 26 23 kanoria kanoria PROPN cana-5942 26 24 ( ( PUNCT cana-5942 26 25 2012 2012 NUM cana-5942 26 26 ) ) PUNCT cana-5942 26 27 . . PUNCT cana-5942 27 1 wave wave NOUN cana-5942 27 2 propagation propagation NOUN cana-5942 27 3 in in ADP cana-5942 27 4 a a DET cana-5942 27 5 porous porous ADJ cana-5942 27 6 media medium NOUN cana-5942 27 7 has have VERB cana-5942 27 8 its its PRON cana-5942 27 9 significance significance NOUN cana-5942 27 10 in in ADP cana-5942 27 11 diversified diversified ADJ cana-5942 27 12 fields field NOUN cana-5942 27 13 of of ADP cana-5942 27 14 science science NOUN cana-5942 27 15 and and CCONJ cana-5942 27 16 engineering engineering NOUN cana-5942 27 17 . . PUNCT cana-5942 28 1 the the DET cana-5942 28 2 general general ADJ cana-5942 28 3 non non ADJ cana-5942 28 4 - - ADJ cana-5942 28 5 linear linear ADJ cana-5942 28 6 theory theory NOUN cana-5942 28 7 of of ADP cana-5942 28 8 elastic elastic ADJ cana-5942 28 9 materials material NOUN cana-5942 28 10 with with ADP cana-5942 28 11 voids voids NOUN cana-5942 28 12 has have AUX cana-5942 28 13 been be AUX cana-5942 28 14 formulated formulate VERB cana-5942 28 15 by by ADP cana-5942 28 16 nunziato nunziato PROPN cana-5942 28 17 and and CCONJ cana-5942 28 18 cowin cowin PROPN cana-5942 28 19 ( ( PUNCT cana-5942 28 20 1979 1979 NUM cana-5942 28 21 ) ) PUNCT cana-5942 28 22 . . PUNCT cana-5942 29 1 linearized linearize VERB cana-5942 29 2 version version NOUN cana-5942 29 3 of of ADP cana-5942 29 4 the the DET cana-5942 29 5 said say VERB cana-5942 29 6 theory theory NOUN cana-5942 29 7 is be AUX cana-5942 29 8 developed develop VERB cana-5942 29 9 by by ADP cana-5942 29 10 cowin cowin NOUN cana-5942 29 11 and and CCONJ cana-5942 29 12 nunziato nunziato PROPN cana-5942 29 13 ( ( PUNCT cana-5942 29 14 1983 1983 NUM cana-5942 29 15 ) ) PUNCT cana-5942 29 16 , , PUNCT cana-5942 29 17 where where SCONJ cana-5942 29 18 void void ADJ cana-5942 29 19 volume volume NOUN cana-5942 29 20 has have AUX cana-5942 29 21 been be AUX cana-5942 29 22 included include VERB cana-5942 29 23 as as ADP cana-5942 29 24 an an DET cana-5942 29 25 additional additional ADJ cana-5942 29 26 kinematic kinematic ADJ cana-5942 29 27 variable variable NOUN cana-5942 29 28 . . PUNCT cana-5942 30 1 this this DET cana-5942 30 2 theory theory NOUN cana-5942 30 3 reduces reduce VERB cana-5942 30 4 to to ADP cana-5942 30 5 the the DET cana-5942 30 6 classical classical ADJ cana-5942 30 7 theory theory NOUN cana-5942 30 8 of of ADP cana-5942 30 9 elasticity elasticity NOUN cana-5942 30 10 in in ADP cana-5942 30 11 the the DET cana-5942 30 12 limiting limit VERB cana-5942 30 13 case case NOUN cana-5942 30 14 when when SCONJ cana-5942 30 15 the the DET cana-5942 30 16 void void ADJ cana-5942 30 17 volume volume NOUN cana-5942 30 18 vanishes vanish VERB cana-5942 30 19 . . PUNCT cana-5942 31 1 iesan iesan PROPN cana-5942 31 2 ( ( PUNCT cana-5942 31 3 1986 1986 NUM cana-5942 31 4 ) ) PUNCT cana-5942 31 5 established establish VERB cana-5942 31 6 a a DET cana-5942 31 7 theory theory NOUN cana-5942 31 8 of of ADP cana-5942 31 9 linear linear ADJ cana-5942 31 10 thermoelastic thermoelastic ADJ cana-5942 31 11 materials material NOUN cana-5942 31 12 with with ADP cana-5942 31 13 voids voids NOUN cana-5942 31 14 singh singh PROPN cana-5942 31 15 and and CCONJ cana-5942 31 16 tomar tomar PROPN cana-5942 31 17 ( ( PUNCT cana-5942 31 18 2007 2007 NUM cana-5942 31 19 ) ) PUNCT cana-5942 31 20 investigated investigate VERB cana-5942 31 21 the the DET cana-5942 31 22 propagation propagation NOUN cana-5942 31 23 of of ADP cana-5942 31 24 plane plane NOUN cana-5942 31 25 waves wave NOUN cana-5942 31 26 in in ADP cana-5942 31 27 a a DET cana-5942 31 28 thermoelastic thermoelastic ADJ cana-5942 31 29 material material NOUN cana-5942 31 30 with with ADP cana-5942 31 31 voids voids NOUN cana-5942 31 32 . . PUNCT cana-5942 32 1 many many ADJ cana-5942 32 2 problems problem NOUN cana-5942 32 3 of of ADP cana-5942 32 4 waves wave NOUN cana-5942 32 5 and and CCONJ cana-5942 32 6 vibrations vibration NOUN cana-5942 32 7 are be AUX cana-5942 32 8 studied study VERB cana-5942 32 9 by by ADP cana-5942 32 10 several several ADJ cana-5942 32 11 researchers researcher NOUN cana-5942 32 12 in in ADP cana-5942 32 13 an an DET cana-5942 32 14 elastic elastic ADJ cana-5942 32 15 material material NOUN cana-5942 32 16 with with ADP cana-5942 32 17 voids voids NOUN cana-5942 32 18 . . PUNCT cana-5942 33 1 some some PRON cana-5942 33 2 of of ADP cana-5942 33 3 them they PRON cana-5942 33 4 are be AUX cana-5942 33 5 abo abo NOUN cana-5942 33 6 - - PUNCT cana-5942 33 7 dahab dahab NOUN cana-5942 33 8 et et PROPN cana-5942 33 9 al al PROPN cana-5942 33 10 . . PROPN cana-5942 34 1 ( ( PUNCT cana-5942 34 2 2013 2013 NUM cana-5942 34 3 ) ) PUNCT cana-5942 34 4 , , PUNCT cana-5942 34 5 sharma sharma PROPN cana-5942 34 6 and and CCONJ cana-5942 34 7 kumar kumar PROPN cana-5942 34 8 ( ( PUNCT cana-5942 34 9 2013 2013 NUM cana-5942 34 10 ) ) PUNCT cana-5942 34 11 , , PUNCT cana-5942 34 12 malik malik PROPN cana-5942 34 13 et et PROPN cana-5942 34 14 . . PUNCT cana-5942 34 15 al.(2022 al.(2022 PROPN cana-5942 34 16 ) ) PUNCT cana-5942 34 17 , , PUNCT cana-5942 34 18 and and CCONJ cana-5942 34 19 kundu kundu NOUN cana-5942 34 20 et et PROPN cana-5942 34 21 al al PROPN cana-5942 34 22 . . PROPN cana-5942 34 23 ( ( PUNCT cana-5942 34 24 2022 2022 NUM cana-5942 34 25 ) ) PUNCT cana-5942 34 26 . . PUNCT cana-5942 35 1 the the DET cana-5942 35 2 two two NUM cana-5942 35 3 - - PUNCT cana-5942 35 4 dimensional dimensional ADJ cana-5942 35 5 deformation deformation NOUN cana-5942 35 6 of of ADP cana-5942 35 7 a a DET cana-5942 35 8 generalized generalize VERB cana-5942 35 9 thermoelastic thermoelastic ADJ cana-5942 35 10 half half ADJ cana-5942 35 11 - - PUNCT cana-5942 35 12 space space NOUN cana-5942 35 13 with with ADP cana-5942 35 14 voids voids NOUN cana-5942 35 15 and and CCONJ cana-5942 35 16 microtemperatures microtemperature NOUN cana-5942 35 17 under under ADP cana-5942 35 18 the the DET cana-5942 35 19 action action NOUN cana-5942 35 20 of of ADP cana-5942 35 21 a a DET cana-5942 35 22 mechanical mechanical ADJ cana-5942 35 23 force force NOUN cana-5942 35 24 was be AUX cana-5942 35 25 investigated investigate VERB cana-5942 35 26 by by ADP cana-5942 35 27 l. l. PROPN cana-5942 35 28 rani rani PROPN cana-5942 35 29 ( ( PUNCT cana-5942 35 30 2023 2023 NUM cana-5942 35 31 ) ) PUNCT cana-5942 35 32 . . PUNCT cana-5942 36 1 in in ADP cana-5942 36 2 this this DET cana-5942 36 3 paper paper NOUN cana-5942 36 4 , , PUNCT cana-5942 36 5 we we PRON cana-5942 36 6 have have AUX cana-5942 36 7 studied study VERB cana-5942 36 8 reflection reflection NOUN cana-5942 36 9 of of ADP cana-5942 36 10 plane plane NOUN cana-5942 36 11 waves wave NOUN cana-5942 36 12 at at ADP cana-5942 36 13 the the DET cana-5942 36 14 free free ADJ cana-5942 36 15 surface surface NOUN cana-5942 36 16 of of ADP cana-5942 36 17 generalized generalized ADJ cana-5942 36 18 thermoelastic thermoelastic ADJ cana-5942 36 19 medium medium NOUN cana-5942 36 20 with with ADP cana-5942 36 21 voids voids NOUN cana-5942 36 22 . . PUNCT cana-5942 37 1 the the DET cana-5942 37 2 study study NOUN cana-5942 37 3 is be AUX cana-5942 37 4 in in ADP cana-5942 37 5 the the DET cana-5942 37 6 context context NOUN cana-5942 37 7 of of ADP cana-5942 37 8 three three NUM cana-5942 37 9 phase phase NOUN cana-5942 37 10 lag lag NOUN cana-5942 37 11 thermoelasticity thermoelasticity NOUN cana-5942 37 12 . . PUNCT cana-5942 38 1 it it PRON cana-5942 38 2 is be AUX cana-5942 38 3 found find VERB cana-5942 38 4 that that SCONJ cana-5942 38 5 there there PRON cana-5942 38 6 exist exist VERB cana-5942 38 7 three three NUM cana-5942 38 8 sets set NOUN cana-5942 38 9 of of ADP cana-5942 38 10 coupled couple VERB cana-5942 38 11 dilatational dilatational ADJ cana-5942 38 12 waves wave NOUN cana-5942 38 13 and and CCONJ cana-5942 38 14 one one NUM cana-5942 38 15 set set NOUN cana-5942 38 16 of of ADP cana-5942 38 17 coupled couple VERB cana-5942 38 18 transversal transversal ADJ cana-5942 38 19 waves wave NOUN cana-5942 38 20 . . PUNCT cana-5942 39 1 the the DET cana-5942 39 2 reflection reflection NOUN cana-5942 39 3 coefficients coefficient NOUN cana-5942 39 4 of of ADP cana-5942 39 5 various various ADJ cana-5942 39 6 reflected reflect VERB cana-5942 39 7 waves wave NOUN cana-5942 39 8 are be AUX cana-5942 39 9 computed compute VERB cana-5942 39 10 numerically numerically ADV cana-5942 39 11 for for ADP cana-5942 39 12 a a DET cana-5942 39 13 specific specific ADJ cana-5942 39 14 model model NOUN cana-5942 39 15 and and CCONJ cana-5942 39 16 their their PRON cana-5942 39 17 variations variation NOUN cana-5942 39 18 with with ADP cana-5942 39 19 angle angle NOUN cana-5942 39 20 of of ADP cana-5942 39 21 incidence incidence NOUN cana-5942 39 22 are be AUX cana-5942 39 23 presented present VERB cana-5942 39 24 graphically graphically ADV cana-5942 39 25 . . PUNCT cana-5942 40 1 the the DET cana-5942 40 2 effect effect NOUN cana-5942 40 3 of of ADP cana-5942 40 4 viscosity viscosity NOUN cana-5942 40 5 on on ADP cana-5942 40 6 reflection reflection NOUN cana-5942 40 7 coefficients coefficient NOUN cana-5942 40 8 is be AUX cana-5942 40 9 demonstrated demonstrate VERB cana-5942 40 10 graphically graphically ADV cana-5942 40 11 . . PUNCT cana-5942 41 1 nomenclature nomenclature ADJ cana-5942 41 2 πœπ‘–π‘— πœπ‘–π‘— PROPN cana-5942 41 3 components component NOUN cana-5942 41 4 of of ADP cana-5942 41 5 stress stress NOUN cana-5942 41 6 tensor tensor NOUN cana-5942 41 7 πœ† πœ† NOUN cana-5942 41 8 , , PUNCT cana-5942 41 9 πœ‡ πœ‡ PRON cana-5942 41 10 lame lame NOUN cana-5942 41 11 ’s ’s PART cana-5942 41 12 constants constant NOUN cana-5942 41 13 𝛽 𝛽 PROPN cana-5942 41 14 ( ( PUNCT cana-5942 41 15 3πœ† 3πœ† NOUN cana-5942 41 16 + + CCONJ cana-5942 41 17 2πœ‡)𝛼𝑑 2πœ‡)𝛼𝑑 NUM cana-5942 42 1 𝛼𝑑 𝛼𝑑 ADP cana-5942 42 2 coefficient coefficient NOUN cana-5942 42 3 of of ADP cana-5942 42 4 linear linear ADJ cana-5942 42 5 thermal thermal ADJ cana-5942 42 6 expansion expansion NOUN cana-5942 42 7 𝑐𝑒 𝑐𝑒 PRON cana-5942 42 8 specific specific ADJ cana-5942 42 9 heat heat NOUN cana-5942 42 10 at at ADP cana-5942 42 11 constant constant ADJ cana-5942 42 12 strain strain NOUN cana-5942 42 13 𝐾 𝐾 PROPN cana-5942 42 14 thermal thermal ADJ cana-5942 42 15 conductivity conductivity NOUN cana-5942 42 16 πΎβˆ— πΎβˆ— PUNCT cana-5942 43 1 𝑐𝑒(πœ†+2πœ‡ 𝑐𝑒(πœ†+2πœ‡ X cana-5942 43 2 ) ) PUNCT cana-5942 43 3 4 4 NUM cana-5942 43 4 , , PUNCT cana-5942 43 5 material material NOUN cana-5942 43 6 constant constant ADJ cana-5942 43 7 𝑇 𝑇 PROPN cana-5942 43 8 absolute absolute ADJ cana-5942 43 9 temperature temperature NOUN cana-5942 43 10 𝑇0 𝑇0 NOUN cana-5942 43 11 reference reference NOUN cana-5942 43 12 temperature temperature NOUN cana-5942 43 13 𝛩 𝛩 NOUN cana-5942 43 14 temperature temperature NOUN cana-5942 43 15 deviation deviation NOUN cana-5942 43 16 from from ADP cana-5942 43 17 the the DET cana-5942 43 18 reference reference NOUN cana-5942 43 19 temperature temperature NOUN cana-5942 43 20 𝛩 𝛩 NOUN cana-5942 43 21 = = SYM cana-5942 43 22 𝑇 𝑇 PROPN cana-5942 43 23 βˆ’ βˆ’ PROPN cana-5942 43 24 𝑇0 𝑇0 NOUN cana-5942 43 25 , , PUNCT cana-5942 43 26 | | ADV cana-5942 43 27 𝛩 𝛩 PRON cana-5942 43 28 𝑇0 𝑇0 VERB cana-5942 43 29 | | ADV cana-5942 43 30 β‰ͺ β‰ͺ ADJ cana-5942 43 31 1 1 NUM cana-5942 43 32 communications communication NOUN cana-5942 43 33 on on ADP cana-5942 43 34 applied apply VERB cana-5942 43 35 nonlinear nonlinear ADJ cana-5942 43 36 analysis analysis NOUN cana-5942 43 37 issn issn NOUN cana-5942 43 38 : : PUNCT cana-5942 43 39 1074 1074 NUM cana-5942 43 40 - - PUNCT cana-5942 43 41 133x 133x NUM cana-5942 43 42 vol vol NOUN cana-5942 43 43 31 31 NUM cana-5942 43 44 no no NOUN cana-5942 43 45 . . PUNCT cana-5942 44 1 8s 8s PROPN cana-5942 44 2 ( ( PUNCT cana-5942 44 3 2024 2024 NUM cana-5942 44 4 ) ) PUNCT cana-5942 44 5 1130 1130 NUM cana-5942 44 6 https://internationalpubls.com https://internationalpubls.com X cana-5942 44 7 𝑒𝑖 𝑒𝑖 PRON cana-5942 44 8 components component NOUN cana-5942 44 9 of of ADP cana-5942 44 10 displacement displacement ADJ cana-5942 44 11 vector vector NOUN cana-5942 44 12 𝜌 𝜌 ADP cana-5942 44 13 density density NOUN cana-5942 44 14 of of ADP cana-5942 44 15 the the DET cana-5942 44 16 medium medium ADJ cana-5942 44 17 𝑒𝑖𝑗 𝑒𝑖𝑗 PROPN cana-5942 44 18 components component NOUN cana-5942 44 19 of of ADP cana-5942 44 20 strain strain NOUN cana-5942 44 21 tensor tensor NOUN cana-5942 44 22 π‘’π‘˜π‘˜ π‘’π‘˜π‘˜ NOUN cana-5942 44 23 e e NOUN cana-5942 44 24 , , PUNCT cana-5942 44 25 cubical cubical ADJ cana-5942 44 26 dilatation dilatation NOUN cana-5942 44 27 πœ™ πœ™ NOUN cana-5942 44 28 change change NOUN cana-5942 44 29 in in ADP cana-5942 44 30 volume volume NOUN cana-5942 44 31 fraction fraction NOUN cana-5942 44 32 field field NOUN cana-5942 44 33 𝛿𝑖𝑗 𝛿𝑖𝑗 PROPN cana-5942 44 34 kronecker kronecker PROPN cana-5942 44 35 delta delta PROPN cana-5942 44 36 function function VERB cana-5942 44 37 components component NOUN cana-5942 44 38 of of ADP cana-5942 44 39 equilibrated equilibrated ADJ cana-5942 44 40 stress stress NOUN cana-5942 44 41 vector vector NOUN cana-5942 44 42 components component NOUN cana-5942 44 43 of of ADP cana-5942 44 44 heat heat NOUN cana-5942 44 45 flux flux PROPN cana-5942 44 46 vector vector PROPN cana-5942 44 47 𝛼 𝛼 PROPN cana-5942 44 48 , , PUNCT cana-5942 44 49 𝑏 𝑏 NOUN cana-5942 44 50 , , PUNCT cana-5942 44 51 πœ‰1 πœ‰1 PROPN cana-5942 44 52 void void ADJ cana-5942 44 53 material material NOUN cana-5942 44 54 parameters parameter NOUN cana-5942 44 55 π‘š π‘š ADP cana-5942 44 56 thermo thermo NOUN cana-5942 44 57 - - PUNCT cana-5942 44 58 void void ADJ cana-5942 44 59 coefficient coefficient NOUN cana-5942 44 60 πœ’ πœ’ NUM cana-5942 44 61 equilibrated equilibrate VERB cana-5942 44 62 inertia inertia NOUN cana-5942 44 63 t t PROPN cana-5942 44 64 time time NOUN cana-5942 44 65 variable variable ADJ cana-5942 44 66 basic basic ADJ cana-5942 44 67 equations equation NOUN cana-5942 44 68 and and CCONJ cana-5942 44 69 problem problem NOUN cana-5942 44 70 formulation formulation NOUN cana-5942 44 71 in in ADP cana-5942 44 72 this this DET cana-5942 44 73 section section NOUN cana-5942 44 74 , , PUNCT cana-5942 44 75 a a DET cana-5942 44 76 resumΓ© resumΓ© NOUN cana-5942 44 77 of of ADP cana-5942 44 78 the the DET cana-5942 44 79 basic basic ADJ cana-5942 44 80 equations equation NOUN cana-5942 44 81 has have AUX cana-5942 44 82 been be AUX cana-5942 44 83 presented present VERB cana-5942 44 84 for for ADP cana-5942 44 85 the the DET cana-5942 44 86 analysis analysis NOUN cana-5942 44 87 of of ADP cana-5942 44 88 wave wave NOUN cana-5942 44 89 propagation propagation NOUN cana-5942 44 90 in in ADP cana-5942 44 91 a a DET cana-5942 44 92 generalized generalized ADJ cana-5942 44 93 thermo thermo NOUN cana-5942 44 94 - - PUNCT cana-5942 44 95 viscoelastic viscoelastic NOUN cana-5942 44 96 medium medium NOUN cana-5942 44 97 containing contain VERB cana-5942 44 98 voids voids NOUN cana-5942 44 99 . . PUNCT cana-5942 45 1 the the DET cana-5942 45 2 constitutive constitutive ADJ cana-5942 45 3 relation relation NOUN cana-5942 45 4 is be AUX cana-5942 45 5 πœπ‘–π‘— πœπ‘–π‘— PROPN cana-5942 45 6 = = SYM cana-5942 45 7 πœ†π›Ώπ‘–π‘—π‘’π‘˜,π‘˜ πœ†π›Ώπ‘–π‘—π‘’π‘˜,π‘˜ PUNCT cana-5942 45 8 + + CCONJ cana-5942 45 9 πœ‡(𝑒𝑖,𝑗 πœ‡(𝑒𝑖,𝑗 NOUN cana-5942 45 10 + + CCONJ cana-5942 45 11 𝑒𝑗,𝑖 𝑒𝑗,𝑖 NOUN cana-5942 45 12 ) ) PUNCT cana-5942 45 13 + + CCONJ cana-5942 45 14 π‘πœ™π›Ώπ‘–π‘— π‘πœ™π›Ώπ‘–π‘— NOUN cana-5942 45 15 βˆ’ βˆ’ PROPN cana-5942 45 16 𝛽𝛩𝛿𝑖𝑗 𝛽𝛩𝛿𝑖𝑗 PROPN cana-5942 45 17 , , PUNCT cana-5942 45 18 ( ( PUNCT cana-5942 45 19 1 1 X cana-5942 45 20 ) ) PUNCT cana-5942 45 21 the the DET cana-5942 45 22 balance balance NOUN cana-5942 45 23 of of ADP cana-5942 45 24 linear linear ADJ cana-5942 45 25 momentum momentum NOUN cana-5942 45 26 in in ADP cana-5942 45 27 the the DET cana-5942 45 28 absence absence NOUN cana-5942 45 29 of of ADP cana-5942 45 30 body body NOUN cana-5942 45 31 forces force NOUN cana-5942 45 32 may may AUX cana-5942 45 33 be be AUX cana-5942 45 34 written write VERB cana-5942 45 35 as as ADP cana-5942 45 36 𝜌 𝜌 X cana-5942 45 37 οΏ½ οΏ½ NOUN cana-5942 45 38 ̈ ̈ X cana-5942 45 39 οΏ½ οΏ½ NOUN cana-5942 45 40 𝑖 𝑖 SYM cana-5942 45 41 = = NOUN cana-5942 45 42 πœπ‘—π‘–,𝑗 πœπ‘—π‘–,𝑗 NOUN cana-5942 45 43 ( ( PUNCT cana-5942 45 44 2 2 NUM cana-5942 45 45 ) ) PUNCT cana-5942 45 46 again again ADV cana-5942 45 47 , , PUNCT cana-5942 45 48 the the DET cana-5942 45 49 volume volume NOUN cana-5942 45 50 fraction fraction NOUN cana-5942 45 51 field field NOUN cana-5942 45 52 πœ™ πœ™ PROPN cana-5942 45 53 satisfies satisfy VERB cana-5942 45 54 the the DET cana-5942 45 55 following follow VERB cana-5942 45 56 equation equation NOUN cana-5942 45 57 [ [ X cana-5942 45 58 iesan iesan ADJ cana-5942 45 59 ( ( PUNCT cana-5942 45 60 1986 1986 NUM cana-5942 45 61 ) ) PUNCT cana-5942 45 62 ] ] PUNCT cana-5942 45 63 𝛼𝛻2πœ™ 𝛼𝛻2πœ™ NUM cana-5942 46 1 βˆ’ βˆ’ PROPN cana-5942 46 2 𝑏(𝛻 𝑏(𝛻 NUM cana-5942 46 3 β‹… β‹… PROPN cana-5942 46 4 οΏ½ οΏ½ PROPN cana-5942 46 5 βƒ—βƒ— βƒ—βƒ— PROPN cana-5942 46 6 οΏ½ οΏ½ PROPN cana-5942 46 7 ) ) PUNCT cana-5942 46 8 βˆ’ βˆ’ PROPN cana-5942 46 9 πœ‰1πœ™ πœ‰1πœ™ NOUN cana-5942 46 10 + + CCONJ cana-5942 46 11 π‘šπ›© π‘šπ›© NOUN cana-5942 46 12 = = SYM cana-5942 46 13 πœŒπœ’ πœŒπœ’ ADP cana-5942 46 14 οΏ½ οΏ½ PROPN cana-5942 46 15 ̈ ̈ X cana-5942 46 16 οΏ½ οΏ½ PROPN cana-5942 46 17 ( ( PUNCT cana-5942 46 18 3 3 NUM cana-5942 46 19 ) ) PUNCT cana-5942 46 20 the the DET cana-5942 46 21 heat heat NOUN cana-5942 46 22 equation equation NOUN cana-5942 46 23 corresponding correspond VERB cana-5942 46 24 to to ADP cana-5942 46 25 generalized generalized ADJ cana-5942 46 26 thermoelasticity thermoelasticity NOUN cana-5942 46 27 theory theory NOUN cana-5942 46 28 with with ADP cana-5942 46 29 three three NUM cana-5942 46 30 phase phase NOUN cana-5942 46 31 lags lag VERB cana-5942 46 32 [ [ PUNCT cana-5942 46 33 roychoudhuri roychoudhuri PROPN cana-5942 46 34 ( ( PUNCT cana-5942 46 35 2008 2008 NUM cana-5942 46 36 ) ) PUNCT cana-5942 46 37 ] ] PUNCT cana-5942 46 38 is be AUX cana-5942 46 39 [ [ X cana-5942 46 40 πΎβˆ— πΎβˆ— NUM cana-5942 46 41 ( ( PUNCT cana-5942 46 42 1 1 NUM cana-5942 46 43 + + NUM cana-5942 46 44 𝜏𝜐 𝜏𝜐 INTJ cana-5942 46 45 πœ• πœ• PROPN cana-5942 46 46 πœ•π‘‘ πœ•π‘‘ PROPN cana-5942 46 47 ) ) PUNCT cana-5942 47 1 + + CCONJ cana-5942 47 2 𝐾 𝐾 PROPN cana-5942 47 3 πœ• πœ• PROPN cana-5942 47 4 πœ•π‘‘ πœ•π‘‘ NOUN cana-5942 47 5 ( ( PUNCT cana-5942 47 6 1 1 NUM cana-5942 47 7 + + NUM cana-5942 47 8 πœπ‘‡ πœπ‘‡ NOUN cana-5942 47 9 πœ• πœ• PROPN cana-5942 47 10 πœ•π‘‘ πœ•π‘‘ PROPN cana-5942 47 11 ) ) PUNCT cana-5942 47 12 ] ] PUNCT cana-5942 48 1 𝛻2𝛩 𝛻2𝛩 PROPN cana-5942 48 2 = = SYM cana-5942 48 3 ( ( PUNCT cana-5942 48 4 1 1 NUM cana-5942 48 5 + + NUM cana-5942 48 6 πœπ‘ž πœπ‘ž NOUN cana-5942 48 7 + + NUM cana-5942 48 8 πœπ‘ž πœπ‘ž NOUN cana-5942 48 9 2 2 NUM cana-5942 48 10 2 2 NUM cana-5942 48 11 πœ•2 πœ•2 NOUN cana-5942 48 12 πœ•π‘‘2 πœ•π‘‘2 NOUN cana-5942 48 13 ) ) PUNCT cana-5942 48 14 ( ( PUNCT cana-5942 48 15 πœŒπ‘π‘’ πœŒπ‘π‘’ PROPN cana-5942 48 16 οΏ½ οΏ½ PROPN cana-5942 48 17 ̈ ̈ ADJ cana-5942 48 18 οΏ½ οΏ½ PROPN cana-5942 48 19 + + CCONJ cana-5942 48 20 𝛽𝑇0 𝛽𝑇0 PROPN cana-5942 48 21 οΏ½ οΏ½ PROPN cana-5942 48 22 ̈ ̈ NUM cana-5942 48 23 οΏ½ οΏ½ PROPN cana-5942 48 24 + + CCONJ cana-5942 48 25 π‘šπ‘‡0 π‘šπ‘‡0 PROPN cana-5942 48 26 οΏ½ οΏ½ PROPN cana-5942 48 27 ̈ ̈ X cana-5942 48 28 οΏ½ οΏ½ PROPN cana-5942 48 29 ) ) PUNCT cana-5942 48 30 ( ( PUNCT cana-5942 48 31 4 4 X cana-5942 48 32 ) ) PUNCT cana-5942 48 33 in in ADP cana-5942 48 34 the the DET cana-5942 48 35 above above NOUN cana-5942 48 36 , , PUNCT cana-5942 48 37 the the DET cana-5942 48 38 usual usual ADJ cana-5942 48 39 summation summation NOUN cana-5942 48 40 convention convention NOUN cana-5942 48 41 on on ADP cana-5942 48 42 repeated repeat VERB cana-5942 48 43 indices index NOUN cana-5942 48 44 has have AUX cana-5942 48 45 been be AUX cana-5942 48 46 followed follow VERB cana-5942 48 47 . . PUNCT cana-5942 49 1 indices index NOUN cana-5942 49 2 following follow VERB cana-5942 49 3 comma comma PROPN cana-5942 49 4 indicate indicate VERB cana-5942 49 5 partial partial ADJ cana-5942 49 6 derivative derivative NOUN cana-5942 49 7 with with ADP cana-5942 49 8 respect respect NOUN cana-5942 49 9 to to ADP cana-5942 49 10 those those DET cana-5942 49 11 indices index NOUN cana-5942 49 12 . . PUNCT cana-5942 50 1 we we PRON cana-5942 50 2 consider consider VERB cana-5942 50 3 the the DET cana-5942 50 4 propagation propagation NOUN cana-5942 50 5 of of ADP cana-5942 50 6 plane plane NOUN cana-5942 50 7 waves wave NOUN cana-5942 50 8 in in ADP cana-5942 50 9 a a DET cana-5942 50 10 homogeneous homogeneous ADJ cana-5942 50 11 , , PUNCT cana-5942 50 12 isotropic isotropic NOUN cana-5942 50 13 , , PUNCT cana-5942 50 14 generalized generalized ADJ cana-5942 50 15 semi semi ADJ cana-5942 50 16 - - ADJ cana-5942 50 17 infinite infinite ADJ cana-5942 50 18 solid solid ADJ cana-5942 50 19 occupying occupy VERB cana-5942 50 20 the the DET cana-5942 50 21 region region NOUN cana-5942 50 22 𝑧 𝑧 PRON cana-5942 50 23 β‰₯ β‰₯ NOUN cana-5942 50 24 0 0 NUM cana-5942 50 25 . . PUNCT cana-5942 51 1 solid solid ADJ cana-5942 51 2 is be AUX cana-5942 51 3 assumed assume VERB cana-5942 51 4 to to PART cana-5942 51 5 be be AUX cana-5942 51 6 composed compose VERB cana-5942 51 7 of of ADP cana-5942 51 8 material material NOUN cana-5942 51 9 possessing possess VERB cana-5942 51 10 voids voids NOUN cana-5942 51 11 . . PUNCT cana-5942 52 1 the the DET cana-5942 52 2 surface surface NOUN cana-5942 52 3 𝑧 𝑧 PROPN cana-5942 52 4 = = SYM cana-5942 52 5 0 0 NUM cana-5942 52 6 is be AUX cana-5942 52 7 assumed assume VERB cana-5942 52 8 to to PART cana-5942 52 9 be be AUX cana-5942 52 10 unstressed unstressed ADJ cana-5942 52 11 , , PUNCT cana-5942 52 12 unstrained unstrained ADJ cana-5942 52 13 , , PUNCT cana-5942 52 14 thermally thermally ADV cana-5942 52 15 ih ih X cana-5942 52 16 iq iq PROPN cana-5942 52 17 communications communication NOUN cana-5942 52 18 on on ADP cana-5942 52 19 applied apply VERB cana-5942 52 20 nonlinear nonlinear ADJ cana-5942 52 21 analysis analysis NOUN cana-5942 52 22 issn issn NOUN cana-5942 52 23 : : PUNCT cana-5942 52 24 1074 1074 NUM cana-5942 52 25 - - PUNCT cana-5942 52 26 133x 133x NUM cana-5942 52 27 vol vol NOUN cana-5942 52 28 31 31 NUM cana-5942 52 29 no no NOUN cana-5942 52 30 . . PUNCT cana-5942 53 1 8s 8s PROPN cana-5942 53 2 ( ( PUNCT cana-5942 53 3 2024 2024 NUM cana-5942 53 4 ) ) PUNCT cana-5942 53 5 1131 1131 NUM cana-5942 53 6 https://internationalpubls.com https://internationalpubls.com X cana-5942 53 7 insulated insulated ADJ cana-5942 53 8 and and CCONJ cana-5942 53 9 initially initially ADV cana-5942 53 10 at at ADP cana-5942 53 11 uniform uniform ADJ cana-5942 53 12 temperature temperature NOUN cana-5942 53 13 t0 t0 PROPN cana-5942 53 14 . . PUNCT cana-5942 54 1 rectangular rectangular ADJ cana-5942 54 2 cartesian cartesian ADJ cana-5942 54 3 coordinate coordinate NOUN cana-5942 54 4 system system NOUN cana-5942 54 5 has have AUX cana-5942 54 6 been be AUX cana-5942 54 7 chosen choose VERB cana-5942 54 8 with with ADP cana-5942 54 9 origin origin NOUN cana-5942 54 10 on on ADP cana-5942 54 11 the the DET cana-5942 54 12 surface surface NOUN cana-5942 54 13 𝑧 𝑧 NOUN cana-5942 54 14 = = NOUN cana-5942 54 15 0 0 NUM cana-5942 54 16 . . PUNCT cana-5942 55 1 the the DET cana-5942 55 2 z z NOUN cana-5942 55 3 - - PUNCT cana-5942 55 4 axis axis NOUN cana-5942 55 5 is be AUX cana-5942 55 6 pointing point VERB cana-5942 55 7 vertically vertically ADV cana-5942 55 8 downward downward ADV cana-5942 55 9 into into ADP cana-5942 55 10 the the DET cana-5942 55 11 medium medium NOUN cana-5942 55 12 . . PUNCT cana-5942 56 1 assuming assume VERB cana-5942 56 2 xz xz NOUN cana-5942 56 3 - - PUNCT cana-5942 56 4 plane plane NOUN cana-5942 56 5 as as ADP cana-5942 56 6 the the DET cana-5942 56 7 plane plane NOUN cana-5942 56 8 of of ADP cana-5942 56 9 incidence incidence NOUN cana-5942 56 10 , , PUNCT cana-5942 56 11 the the DET cana-5942 56 12 wave wave NOUN cana-5942 56 13 motion motion NOUN cana-5942 56 14 will will AUX cana-5942 56 15 be be AUX cana-5942 56 16 the the DET cana-5942 56 17 same same ADJ cana-5942 56 18 in in ADP cana-5942 56 19 every every DET cana-5942 56 20 plane plane NOUN cana-5942 56 21 parallel parallel NOUN cana-5942 56 22 to to ADP cana-5942 56 23 xz xz NOUN cana-5942 56 24 - - PUNCT cana-5942 56 25 plane plane NOUN cana-5942 56 26 which which PRON cana-5942 56 27 in in ADP cana-5942 56 28 turn turn NOUN cana-5942 56 29 implies imply VERB cana-5942 56 30 that that SCONJ cana-5942 56 31 all all DET cana-5942 56 32 physical physical ADJ cana-5942 56 33 quantities quantity NOUN cana-5942 56 34 will will AUX cana-5942 56 35 be be AUX cana-5942 56 36 independent independent ADJ cana-5942 56 37 of of ADP cana-5942 56 38 ycoordinate ycoordinate NOUN cana-5942 56 39 . . PUNCT cana-5942 57 1 thus thus ADV cana-5942 57 2 the the DET cana-5942 57 3 displacement displacement ADJ cana-5942 57 4 vector vector NOUN cana-5942 57 5 ( ( PUNCT cana-5942 57 6 οΏ½ οΏ½ NOUN cana-5942 57 7 βƒ—βƒ— βƒ—βƒ— PROPN cana-5942 57 8 οΏ½ οΏ½ PROPN cana-5942 57 9 = = SYM cana-5942 57 10 ( ( PUNCT cana-5942 57 11 𝑒 𝑒 X cana-5942 57 12 , , PUNCT cana-5942 57 13 0 0 NUM cana-5942 57 14 , , PUNCT cana-5942 57 15 𝑀 𝑀 NOUN cana-5942 57 16 ) ) PUNCT cana-5942 57 17 ) ) PUNCT cana-5942 57 18 components component NOUN cana-5942 57 19 , , PUNCT cana-5942 57 20 change change NOUN cana-5942 57 21 in in ADP cana-5942 57 22 void void ADJ cana-5942 57 23 volume volume NOUN cana-5942 57 24 fraction fraction NOUN cana-5942 57 25 πœ™ πœ™ NOUN cana-5942 57 26 and and CCONJ cana-5942 57 27 thermal thermal ADJ cana-5942 57 28 parameter parameter NOUN cana-5942 57 29 𝛩 𝛩 PROPN cana-5942 57 30 are be AUX cana-5942 57 31 𝑒 𝑒 X cana-5942 57 32 = = SYM cana-5942 57 33 𝑒(π‘₯ 𝑒(π‘₯ PROPN cana-5942 57 34 , , PUNCT cana-5942 57 35 𝑧 𝑧 PROPN cana-5942 57 36 , , PUNCT cana-5942 57 37 𝑑 𝑑 NOUN cana-5942 57 38 ) ) PUNCT cana-5942 57 39 , , PUNCT cana-5942 57 40 𝑣 𝑣 X cana-5942 57 41 = = SYM cana-5942 57 42 0 0 NUM cana-5942 57 43 , , PUNCT cana-5942 57 44 𝑀 𝑀 ADP cana-5942 57 45 = = SYM cana-5942 57 46 𝑀(π‘₯ 𝑀(π‘₯ NOUN cana-5942 57 47 , , PUNCT cana-5942 57 48 𝑧 𝑧 NOUN cana-5942 57 49 , , PUNCT cana-5942 57 50 𝑑 𝑑 NOUN cana-5942 57 51 ) ) PUNCT cana-5942 57 52 , , PUNCT cana-5942 57 53 πœ™ πœ™ X cana-5942 57 54 = = PUNCT cana-5942 57 55 πœ™(π‘₯ πœ™(π‘₯ PROPN cana-5942 57 56 , , PUNCT cana-5942 57 57 𝑧 𝑧 NOUN cana-5942 57 58 , , PUNCT cana-5942 57 59 𝑑 𝑑 NOUN cana-5942 57 60 ) ) PUNCT cana-5942 57 61 and and CCONJ cana-5942 57 62 𝛩 𝛩 NOUN cana-5942 57 63 = = SYM cana-5942 57 64 𝛩(π‘₯ 𝛩(π‘₯ PROPN cana-5942 57 65 , , PUNCT cana-5942 57 66 𝑧 𝑧 NOUN cana-5942 57 67 , , PUNCT cana-5942 57 68 𝑑 𝑑 NOUN cana-5942 57 69 ) ) PUNCT cana-5942 57 70 . . PUNCT cana-5942 58 1 hence hence ADV cana-5942 58 2 the the DET cana-5942 58 3 dynamical dynamical ADJ cana-5942 58 4 equations equation NOUN cana-5942 58 5 for for ADP cana-5942 58 6 xz xz PROPN cana-5942 58 7 - - PUNCT cana-5942 58 8 plane plane NOUN cana-5942 58 9 may may AUX cana-5942 58 10 be be AUX cana-5942 58 11 obtained obtain VERB cana-5942 58 12 as as SCONJ cana-5942 58 13 follows follow VERB cana-5942 58 14 𝜌 𝜌 ADP cana-5942 58 15 πœ•2𝑒 πœ•2𝑒 PROPN cana-5942 58 16 πœ•π‘‘2 πœ•π‘‘2 NOUN cana-5942 58 17 = = SYM cana-5942 58 18 ( ( PUNCT cana-5942 58 19 πœ† πœ† NOUN cana-5942 58 20 + + X cana-5942 58 21 πœ‡ πœ‡ NOUN cana-5942 58 22 ) ) PUNCT cana-5942 58 23 πœ•π‘’ πœ•π‘’ NOUN cana-5942 58 24 πœ•π‘₯ πœ•π‘₯ NOUN cana-5942 58 25 + + CCONJ cana-5942 58 26 πœ‡π›»2𝑒 πœ‡π›»2𝑒 VERB cana-5942 58 27 βˆ’ βˆ’ PROPN cana-5942 58 28 𝛽 𝛽 PROPN cana-5942 58 29 πœ•π›© πœ•π›© ADJ cana-5942 58 30 πœ•π‘₯ πœ•π‘₯ NOUN cana-5942 58 31 + + CCONJ cana-5942 58 32 𝑏 𝑏 ADP cana-5942 58 33 πœ•πœ™ πœ•πœ™ ADJ cana-5942 58 34 πœ•π‘₯ πœ•π‘₯ NOUN cana-5942 58 35 , , PUNCT cana-5942 58 36 ( ( PUNCT cana-5942 58 37 5 5 X cana-5942 58 38 ) ) PUNCT cana-5942 58 39 𝜌 𝜌 ADP cana-5942 58 40 πœ•2𝑀 πœ•2𝑀 PROPN cana-5942 58 41 πœ•π‘‘2 πœ•π‘‘2 NOUN cana-5942 58 42 = = PUNCT cana-5942 58 43 ( ( PUNCT cana-5942 58 44 πœ† πœ† NOUN cana-5942 58 45 + + X cana-5942 58 46 πœ‡ πœ‡ NOUN cana-5942 58 47 ) ) PUNCT cana-5942 58 48 πœ•π‘’ πœ•π‘’ NOUN cana-5942 58 49 πœ•π‘§ πœ•π‘§ NOUN cana-5942 58 50 + + CCONJ cana-5942 58 51 πœ‡π›»2𝑀 πœ‡π›»2𝑀 NUM cana-5942 58 52 βˆ’ βˆ’ PROPN cana-5942 58 53 𝛽 𝛽 NOUN cana-5942 58 54 πœ•π›© πœ•π›© NOUN cana-5942 58 55 πœ•π‘§ πœ•π‘§ NOUN cana-5942 58 56 + + CCONJ cana-5942 58 57 𝑏 𝑏 DET cana-5942 58 58 πœ•πœ™ πœ•πœ™ ADJ cana-5942 58 59 πœ•π‘§ πœ•π‘§ NOUN cana-5942 58 60 . . PUNCT cana-5942 59 1 ( ( PUNCT cana-5942 59 2 6 6 NUM cana-5942 59 3 ) ) PUNCT cana-5942 59 4 for for ADP cana-5942 59 5 convenience convenience NOUN cana-5942 59 6 , , PUNCT cana-5942 59 7 we we PRON cana-5942 59 8 will will AUX cana-5942 59 9 make make VERB cana-5942 59 10 use use NOUN cana-5942 59 11 of of ADP cana-5942 59 12 the the DET cana-5942 59 13 following follow VERB cana-5942 59 14 non non ADJ cana-5942 59 15 - - ADJ cana-5942 59 16 dimensional dimensional ADJ cana-5942 59 17 quantities quantity NOUN cana-5942 59 18 ( ( PUNCT cana-5942 59 19 π‘₯β€² π‘₯β€² PROPN cana-5942 59 20 , , PUNCT cana-5942 59 21 𝑧′ 𝑧′ NUM cana-5942 59 22 ) ) PUNCT cana-5942 60 1 = = PUNCT cana-5942 60 2 οΏ½ οΏ½ PROPN cana-5942 60 3 Μ„ Μ„ NOUN cana-5942 60 4 οΏ½ οΏ½ PROPN cana-5942 60 5 𝑐1 𝑐1 NOUN cana-5942 60 6 ( ( PUNCT cana-5942 60 7 π‘₯ π‘₯ PROPN cana-5942 60 8 , , PUNCT cana-5942 60 9 𝑧 𝑧 NOUN cana-5942 60 10 ) ) PUNCT cana-5942 60 11 , , PUNCT cana-5942 60 12 ( ( PUNCT cana-5942 60 13 𝑑′ 𝑑′ NUM cana-5942 60 14 , , PUNCT cana-5942 60 15 πœπ‘ž πœπ‘ž PRON cana-5942 60 16 β€² β€² NUM cana-5942 60 17 , , PUNCT cana-5942 60 18 𝜏𝜐 𝜏𝜐 INTJ cana-5942 60 19 β€² β€² NUM cana-5942 60 20 , , PUNCT cana-5942 60 21 πœπ‘‡ πœπ‘‡ NOUN cana-5942 60 22 β€² β€² NOUN cana-5942 60 23 , , PUNCT cana-5942 60 24 𝛽′ 𝛽′ PROPN cana-5942 60 25 ) ) PUNCT cana-5942 60 26 = = PUNCT cana-5942 60 27 οΏ½ οΏ½ PROPN cana-5942 60 28 Μ„ Μ„ PROPN cana-5942 60 29 οΏ½ οΏ½ PROPN cana-5942 60 30 (𝑑 (𝑑 PROPN cana-5942 60 31 , , PUNCT cana-5942 60 32 πœπ‘ž πœπ‘ž NOUN cana-5942 60 33 , , PUNCT cana-5942 60 34 𝜏𝜐 𝜏𝜐 ADJ cana-5942 60 35 , , PUNCT cana-5942 60 36 πœπ‘‡ πœπ‘‡ NOUN cana-5942 60 37 , , PUNCT cana-5942 60 38 𝛽 𝛽 NOUN cana-5942 60 39 ) ) PUNCT cana-5942 60 40 , , PUNCT cana-5942 60 41 ( ( PUNCT cana-5942 60 42 𝑒′ 𝑒′ NUM cana-5942 60 43 , , PUNCT cana-5942 60 44 𝑀′ 𝑀′ NUM cana-5942 60 45 ) ) PUNCT cana-5942 60 46 = = PUNCT cana-5942 60 47 𝜌 𝜌 ADP cana-5942 60 48 οΏ½ οΏ½ PROPN cana-5942 60 49 Μ„ Μ„ NOUN cana-5942 60 50 οΏ½ οΏ½ PROPN cana-5942 60 51 𝑐1 𝑐1 NOUN cana-5942 60 52 𝛽𝑇0 𝛽𝑇0 PROPN cana-5942 60 53 ( ( PUNCT cana-5942 60 54 𝑒 𝑒 X cana-5942 60 55 , , PUNCT cana-5942 60 56 𝑀 𝑀 ADP cana-5942 60 57 ) ) PUNCT cana-5942 60 58 , , PUNCT cana-5942 60 59     SPACE cana-5942 60 60 𝛩′ 𝛩′ NOUN cana-5942 60 61 = = PUNCT cana-5942 60 62 𝛩 𝛩 X cana-5942 60 63 𝑇0 𝑇0 NOUN cana-5942 60 64 , , PUNCT cana-5942 60 65     SPACE cana-5942 60 66 πœ™β€² πœ™β€² X cana-5942 60 67 = = SYM cana-5942 60 68 οΏ½ οΏ½ PROPN cana-5942 60 69 Μ„ Μ„ NOUN cana-5942 60 70 οΏ½ οΏ½ NOUN cana-5942 60 71 2πœ’ 2πœ’ NOUN cana-5942 60 72 𝑐1 𝑐1 NOUN cana-5942 60 73 2 2 NUM cana-5942 60 74 πœ™ πœ™ NOUN cana-5942 60 75 , , PUNCT cana-5942 60 76 ( ( PUNCT cana-5942 60 77 πœπ‘§π‘₯ πœπ‘§π‘₯ AUX cana-5942 60 78 β€² β€² NUM cana-5942 60 79 , , PUNCT cana-5942 60 80 πœπ‘§π‘§ πœπ‘§π‘§ NOUN cana-5942 60 81 β€² β€² NOUN cana-5942 60 82 ) ) PUNCT cana-5942 61 1 = = SYM cana-5942 61 2 1 1 NUM cana-5942 61 3 𝛽𝑇0 𝛽𝑇0 PROPN cana-5942 61 4 ( ( PUNCT cana-5942 61 5 πœπ‘§π‘₯ πœπ‘§π‘₯ INTJ cana-5942 61 6 , , PUNCT cana-5942 61 7 πœπ‘§π‘§ πœπ‘§π‘§ PROPN cana-5942 61 8 ) ) PUNCT cana-5942 61 9 . . PUNCT cana-5942 62 1 ( ( PUNCT cana-5942 62 2 7 7 X cana-5942 62 3 ) ) PUNCT cana-5942 62 4 where where SCONJ cana-5942 62 5 οΏ½ οΏ½ PROPN cana-5942 62 6 Μ„ Μ„ NOUN cana-5942 62 7 οΏ½ οΏ½ PROPN cana-5942 62 8 = = SYM cana-5942 62 9 πœŒπ‘π‘’π‘1 πœŒπ‘π‘’π‘1 X cana-5942 62 10 2 2 NUM cana-5942 62 11 𝐾 𝐾 NOUN cana-5942 62 12 , , PUNCT cana-5942 62 13 𝑐1 𝑐1 NOUN cana-5942 62 14 = = NOUN cana-5942 62 15 √ √ PART cana-5942 62 16 πœ†+2πœ‡ πœ†+2πœ‡ NOUN cana-5942 62 17 𝜌 𝜌 NOUN cana-5942 62 18 are be AUX cana-5942 62 19 the the DET cana-5942 62 20 characteristic characteristic ADJ cana-5942 62 21 frequency frequency NOUN cana-5942 62 22 and and CCONJ cana-5942 62 23 longitudinal longitudinal ADJ cana-5942 62 24 wave wave NOUN cana-5942 62 25 velocity velocity NOUN cana-5942 62 26 in in ADP cana-5942 62 27 the the DET cana-5942 62 28 medium medium NOUN cana-5942 62 29 respectively respectively ADV cana-5942 62 30 . . PUNCT cana-5942 63 1 for for ADP cana-5942 63 2 investigation investigation NOUN cana-5942 63 3 of of ADP cana-5942 63 4 plane plane NOUN cana-5942 63 5 waves wave NOUN cana-5942 63 6 , , PUNCT cana-5942 63 7 the the DET cana-5942 63 8 potentials potential NOUN cana-5942 63 9 πœ“1(π‘₯ πœ“1(π‘₯ NOUN cana-5942 63 10 , , PUNCT cana-5942 63 11 𝑧 𝑧 NOUN cana-5942 63 12 , , PUNCT cana-5942 63 13 𝑑),πœ“2(π‘₯ 𝑑),πœ“2(π‘₯ PROPN cana-5942 63 14 , , PUNCT cana-5942 63 15 𝑧 𝑧 PROPN cana-5942 63 16 , , PUNCT cana-5942 63 17 𝑑 𝑑 X cana-5942 63 18 ) ) PUNCT cana-5942 63 19 are be AUX cana-5942 63 20 introduced introduce VERB cana-5942 63 21 . . PUNCT cana-5942 64 1 they they PRON cana-5942 64 2 are be AUX cana-5942 64 3 related relate VERB cana-5942 64 4 to to ADP cana-5942 64 5 displacement displacement ADJ cana-5942 64 6 components component NOUN cana-5942 64 7 u u NOUN cana-5942 64 8 and and CCONJ cana-5942 64 9 w w NOUN cana-5942 64 10 by by ADP cana-5942 64 11 the the DET cana-5942 64 12 relation relation NOUN cana-5942 64 13 𝑒 𝑒 X cana-5942 64 14 = = SYM cana-5942 64 15 πœ•πœ“1 πœ•πœ“1 ADJ cana-5942 64 16 πœ•π‘₯ πœ•π‘₯ X cana-5942 64 17 βˆ’ βˆ’ PROPN cana-5942 64 18 πœ•πœ“2 πœ•πœ“2 PROPN cana-5942 64 19 πœ•π‘§ πœ•π‘§ NOUN cana-5942 64 20 , , PUNCT cana-5942 64 21 𝑀 𝑀 ADP cana-5942 64 22 = = SYM cana-5942 65 1 πœ•πœ“1 πœ•πœ“1 NUM cana-5942 65 2 πœ•π‘§ πœ•π‘§ PUNCT cana-5942 65 3 + + CCONJ cana-5942 65 4 πœ•πœ“2 πœ•πœ“2 PROPN cana-5942 65 5 πœ•π‘₯ πœ•π‘₯ X cana-5942 65 6 . . PUNCT cana-5942 66 1 ( ( PUNCT cana-5942 66 2 8) 8) NUM cana-5942 66 3 plugging plug VERB cana-5942 66 4 the the DET cana-5942 66 5 above above ADJ cana-5942 66 6 potentials potential NOUN cana-5942 66 7 from from ADP cana-5942 66 8 eq.(8 eq.(8 ADJ cana-5942 66 9 ) ) PUNCT cana-5942 66 10 into into ADP cana-5942 66 11 non non ADJ cana-5942 66 12 - - ADJ cana-5942 66 13 dimensional dimensional ADJ cana-5942 66 14 form form NOUN cana-5942 66 15 of of ADP cana-5942 66 16 the the DET cana-5942 66 17 equations equation NOUN cana-5942 66 18 ( ( PUNCT cana-5942 66 19 1)-(4 1)-(4 NUM cana-5942 66 20 ) ) PUNCT cana-5942 66 21 , , PUNCT cana-5942 66 22 we we PRON cana-5942 66 23 get get VERB cana-5942 66 24 ( ( PUNCT cana-5942 66 25 1 1 NUM cana-5942 66 26 + + NUM cana-5942 66 27 𝛼1 𝛼1 PROPN cana-5942 66 28 πœ• πœ• PROPN cana-5942 66 29 πœ•π‘‘ πœ•π‘‘ NOUN cana-5942 66 30 ) ) PUNCT cana-5942 66 31 𝛻2πœ“2 𝛻2πœ“2 NOUN cana-5942 67 1 = = NOUN cana-5942 67 2 π‘Ž1 π‘Ž1 NOUN cana-5942 67 3 πœ•2πœ“2 πœ•2πœ“2 PUNCT cana-5942 67 4 πœ•π‘‘2 πœ•π‘‘2 NOUN cana-5942 67 5 , , PUNCT cana-5942 67 6 ( ( PUNCT cana-5942 67 7 9 9 NUM cana-5942 67 8 ) ) PUNCT cana-5942 67 9 [ [ PUNCT cana-5942 67 10 πœ†π‘’ πœ†π‘’ NOUN cana-5942 67 11 πœ‡π‘’ πœ‡π‘’ ADP cana-5942 67 12 + + NOUN cana-5942 67 13 2 2 X cana-5942 67 14 ] ] PUNCT cana-5942 67 15 𝛻2πœ“1 𝛻2πœ“1 DET cana-5942 67 16 βˆ’ βˆ’ NOUN cana-5942 68 1 π‘Ž1 π‘Ž1 X cana-5942 68 2 πœ•2πœ“1 πœ•2πœ“1 ADV cana-5942 68 3 πœ•π‘‘2 πœ•π‘‘2 VERB cana-5942 68 4 βˆ’ βˆ’ PROPN cana-5942 68 5 ( ( PUNCT cana-5942 68 6 1 1 NUM cana-5942 68 7 + + X cana-5942 68 8 𝛽 𝛽 PROPN cana-5942 68 9 πœ• πœ• PROPN cana-5942 68 10 πœ•π‘‘ πœ•π‘‘ NOUN cana-5942 68 11 ) ) PUNCT cana-5942 68 12 𝛾2𝛩 𝛾2𝛩 NOUN cana-5942 68 13 βˆ’ βˆ’ PROPN cana-5942 68 14 π‘Ž2πœ™ π‘Ž2πœ™ CCONJ cana-5942 68 15 = = SYM cana-5942 68 16 0 0 NUM cana-5942 68 17 , , PUNCT cana-5942 68 18 ( ( PUNCT cana-5942 68 19 10 10 NUM cana-5942 68 20 ) ) PUNCT cana-5942 68 21 𝛻2πœ™ 𝛻2πœ™ PROPN cana-5942 68 22 βˆ’ βˆ’ PROPN cana-5942 68 23 π‘Ž3(𝛻2πœ“1 π‘Ž3(𝛻2πœ“1 NOUN cana-5942 68 24 ) ) PUNCT cana-5942 68 25 βˆ’ βˆ’ PROPN cana-5942 68 26 π‘Ž4πœ™ π‘Ž4πœ™ PROPN cana-5942 68 27 + + CCONJ cana-5942 68 28 π‘Ž5𝛩 π‘Ž5𝛩 VERB cana-5942 68 29 βˆ’ βˆ’ PROPN cana-5942 68 30 π‘Ž6 π‘Ž6 PROPN cana-5942 68 31 οΏ½ οΏ½ PROPN cana-5942 68 32 ̈ ̈ NOUN cana-5942 68 33 οΏ½ οΏ½ PROPN cana-5942 68 34 = = SYM cana-5942 68 35 0 0 PROPN cana-5942 68 36 , , PUNCT cana-5942 68 37 ( ( PUNCT cana-5942 68 38 11 11 NUM cana-5942 68 39 ) ) PUNCT cana-5942 69 1 [ [ X cana-5942 69 2 π‘Ž7 π‘Ž7 NOUN cana-5942 69 3 ( ( PUNCT cana-5942 69 4 1 1 NUM cana-5942 69 5 + + NUM cana-5942 69 6 𝜏𝜐 𝜏𝜐 INTJ cana-5942 69 7 πœ• πœ• PROPN cana-5942 69 8 πœ•π‘‘ πœ•π‘‘ PROPN cana-5942 69 9 ) ) PUNCT cana-5942 70 1 + + CCONJ cana-5942 70 2 πœ• πœ• PROPN cana-5942 70 3 πœ•π‘‘ πœ•π‘‘ NOUN cana-5942 70 4 ( ( PUNCT cana-5942 70 5 1 1 NUM cana-5942 70 6 + + NUM cana-5942 70 7 πœπ‘‡ πœπ‘‡ NOUN cana-5942 70 8 πœ• πœ• PROPN cana-5942 70 9 πœ•π‘‘ πœ•π‘‘ PROPN cana-5942 70 10 ) ) PUNCT cana-5942 70 11 ] ] PUNCT cana-5942 71 1 𝛻2𝛩 𝛻2𝛩 PROPN cana-5942 71 2 = = SYM cana-5942 71 3 ( ( PUNCT cana-5942 71 4 1 1 NUM cana-5942 71 5 + + NUM cana-5942 71 6 πœπ‘ž πœπ‘ž NOUN cana-5942 71 7 + + NUM cana-5942 71 8 πœπ‘ž πœπ‘ž NOUN cana-5942 71 9 2 2 NUM cana-5942 71 10 2 2 NUM cana-5942 71 11 πœ•2 πœ•2 NOUN cana-5942 71 12 πœ•π‘‘2 πœ•π‘‘2 NOUN cana-5942 71 13 ) ) PUNCT cana-5942 71 14 ( ( PUNCT cana-5942 71 15 οΏ½ οΏ½ PROPN cana-5942 71 16 ̈ ̈ X cana-5942 71 17 οΏ½ οΏ½ PROPN cana-5942 71 18 + + CCONJ cana-5942 71 19 π‘Ž8(1 π‘Ž8(1 NOUN cana-5942 71 20 + + CCONJ cana-5942 71 21 𝛽 𝛽 NOUN cana-5942 71 22 πœ• πœ• PROPN cana-5942 71 23 πœ•π‘‘ πœ•π‘‘ PROPN cana-5942 71 24 ) ) PUNCT cana-5942 71 25 οΏ½ οΏ½ PROPN cana-5942 71 26 ̈ ̈ X cana-5942 71 27 οΏ½ οΏ½ NOUN cana-5942 71 28 1 1 NUM cana-5942 71 29 + + CCONJ cana-5942 71 30 π‘Ž9 π‘Ž9 PROPN cana-5942 71 31 οΏ½ οΏ½ PROPN cana-5942 71 32 ̈ ̈ X cana-5942 71 33 οΏ½ οΏ½ PROPN cana-5942 71 34 ) ) PUNCT cana-5942 71 35 . . PUNCT cana-5942 72 1 ( ( PUNCT cana-5942 72 2 12 12 NUM cana-5942 72 3 ) ) PUNCT cana-5942 72 4 where where SCONJ cana-5942 72 5 𝛾2 𝛾2 NOUN cana-5942 72 6 = = SYM cana-5942 72 7 πœŒπ‘1 πœŒπ‘1 NOUN cana-5942 72 8 2 2 NUM cana-5942 72 9 πœ‡π‘’ πœ‡π‘’ NOUN cana-5942 72 10 , , PUNCT cana-5942 72 11 π‘Ž1 π‘Ž1 NOUN cana-5942 72 12 = = NOUN cana-5942 72 13 𝛾2 𝛾2 VERB cana-5942 72 14 , , PUNCT cana-5942 72 15 π‘Ž2 π‘Ž2 NOUN cana-5942 72 16 = = SYM cana-5942 72 17 πœŒπ‘π‘1 πœŒπ‘π‘1 PROPN cana-5942 72 18 4 4 NUM cana-5942 73 1 𝛽𝑇0πœ‡ 𝛽𝑇0πœ‡ NOUN cana-5942 73 2 οΏ½ οΏ½ NOUN cana-5942 73 3 Μ„ Μ„ NOUN cana-5942 73 4 οΏ½ οΏ½ NOUN cana-5942 73 5 2πœ’ 2πœ’ NOUN cana-5942 73 6 , , PUNCT cana-5942 73 7 π‘Ž3 π‘Ž3 PROPN cana-5942 73 8 = = SYM cana-5942 73 9 π‘πœ’π›½π‘‡0 π‘πœ’π›½π‘‡0 PROPN cana-5942 73 10 π›ΌπœŒπ‘1 π›ΌπœŒπ‘1 NOUN cana-5942 73 11 2 2 NUM cana-5942 73 12 , , PUNCT cana-5942 73 13 π‘Ž4 π‘Ž4 PROPN cana-5942 73 14 = = SYM cana-5942 73 15 πœ‰1𝑐1 πœ‰1𝑐1 PROPN cana-5942 73 16 2 2 NUM cana-5942 73 17 οΏ½ οΏ½ PROPN cana-5942 73 18 Μ„ Μ„ PROPN cana-5942 73 19 οΏ½ οΏ½ PROPN cana-5942 73 20 2𝛼 2𝛼 PROPN cana-5942 73 21 , , PUNCT cana-5942 73 22     SPACE cana-5942 73 23 π‘Ž5 π‘Ž5 PROPN cana-5942 73 24 = = PUNCT cana-5942 73 25 π‘šπ‘‡0πœ’ π‘šπ‘‡0πœ’ VERB cana-5942 73 26 𝛼 𝛼 X cana-5942 73 27 , , PUNCT cana-5942 73 28 communications communication NOUN cana-5942 73 29 on on ADP cana-5942 73 30 applied apply VERB cana-5942 73 31 nonlinear nonlinear ADJ cana-5942 73 32 analysis analysis NOUN cana-5942 73 33 issn issn NOUN cana-5942 73 34 : : PUNCT cana-5942 73 35 1074 1074 NUM cana-5942 73 36 - - PUNCT cana-5942 73 37 133x 133x NUM cana-5942 73 38 vol vol NOUN cana-5942 73 39 31 31 NUM cana-5942 73 40 no no NOUN cana-5942 73 41 . . PUNCT cana-5942 74 1 8s 8s PROPN cana-5942 74 2 ( ( PUNCT cana-5942 74 3 2024 2024 NUM cana-5942 74 4 ) ) PUNCT cana-5942 74 5 1132 1132 NUM cana-5942 74 6 https://internationalpubls.com https://internationalpubls.com X cana-5942 74 7 π‘Ž6 π‘Ž6 NOUN cana-5942 74 8 = = PUNCT cana-5942 74 9 πœŒπ‘1 πœŒπ‘1 NOUN cana-5942 74 10 2πœ’ 2πœ’ X cana-5942 74 11 𝛼 𝛼 X cana-5942 74 12 , , PUNCT cana-5942 74 13 π‘Ž7 π‘Ž7 NOUN cana-5942 74 14 = = PUNCT cana-5942 74 15 πΎβˆ— πΎβˆ— NUM cana-5942 74 16 𝐾 𝐾 PROPN cana-5942 74 17 οΏ½ οΏ½ PROPN cana-5942 74 18 Μ„ Μ„ NOUN cana-5942 74 19 οΏ½ οΏ½ PROPN cana-5942 74 20 , , PUNCT cana-5942 74 21     SPACE cana-5942 74 22 π‘Ž8 π‘Ž8 NOUN cana-5942 74 23 = = PUNCT cana-5942 74 24 𝛽2𝑇0 𝛽2𝑇0 PUNCT cana-5942 74 25 𝐾𝜌 𝐾𝜌 NOUN cana-5942 74 26 οΏ½ οΏ½ PROPN cana-5942 74 27 Μ„ Μ„ NOUN cana-5942 74 28 οΏ½ οΏ½ PROPN cana-5942 74 29 , , PUNCT cana-5942 74 30     SPACE cana-5942 74 31 π‘Ž9 π‘Ž9 ADJ cana-5942 74 32 = = PUNCT cana-5942 74 33 π‘šπ‘1 π‘šπ‘1 NOUN cana-5942 74 34 4 4 NUM cana-5942 74 35 πΎπœ’ πΎπœ’ PROPN cana-5942 74 36 οΏ½ οΏ½ PROPN cana-5942 74 37 Μ„ Μ„ NOUN cana-5942 74 38 οΏ½ οΏ½ PROPN cana-5942 74 39 3 3 NUM cana-5942 74 40 . . PUNCT cana-5942 75 1 eq eq ADP cana-5942 75 2 . . PUNCT cana-5942 76 1 ( ( PUNCT cana-5942 76 2 9 9 NUM cana-5942 76 3 ) ) PUNCT cana-5942 76 4 is be AUX cana-5942 76 5 uncoupled uncoupled ADJ cana-5942 76 6 while while SCONJ cana-5942 76 7 equations equation NOUN cana-5942 76 8 ( ( PUNCT cana-5942 76 9 10)-(12 10)-(12 NUM cana-5942 76 10 ) ) PUNCT cana-5942 76 11 are be AUX cana-5942 76 12 coupled couple VERB cana-5942 76 13 in in ADP cana-5942 76 14 πœ“1 πœ“1 NOUN cana-5942 76 15 , , PUNCT cana-5942 76 16 𝛩 𝛩 NOUN cana-5942 76 17 and and CCONJ cana-5942 76 18 πœ™. πœ™. NOUN cana-5942 76 19 to to PART cana-5942 76 20 suit suit VERB cana-5942 76 21 the the DET cana-5942 76 22 actual actual ADJ cana-5942 76 23 situation situation NOUN cana-5942 76 24 of of ADP cana-5942 76 25 the the DET cana-5942 76 26 problem problem NOUN cana-5942 76 27 , , PUNCT cana-5942 76 28 we we PRON cana-5942 76 29 seek seek VERB cana-5942 76 30 solutions solution NOUN cana-5942 76 31 of of ADP cana-5942 76 32 differential differential ADJ cana-5942 76 33 equations equation NOUN cana-5942 76 34 ( ( PUNCT cana-5942 76 35 9)-(12 9)-(12 NUM cana-5942 76 36 ) ) PUNCT cana-5942 76 37 in in ADP cana-5942 76 38 the the DET cana-5942 76 39 following follow VERB cana-5942 76 40 forms form NOUN cana-5942 76 41 : : PUNCT cana-5942 76 42 [ [ X cana-5942 76 43 πœ“1 πœ“1 NOUN cana-5942 76 44 , , PUNCT cana-5942 76 45 πœ“2 πœ“2 NOUN cana-5942 76 46 , , PUNCT cana-5942 76 47 𝛩 𝛩 PROPN cana-5942 76 48 , , PUNCT cana-5942 76 49 πœ™](π‘₯ πœ™](π‘₯ NOUN cana-5942 76 50 , , PUNCT cana-5942 76 51 𝑧 𝑧 PROPN cana-5942 76 52 , , PUNCT cana-5942 76 53 𝑑 𝑑 NOUN cana-5942 76 54 ) ) PUNCT cana-5942 76 55 = = PUNCT cana-5942 77 1 [ [ X cana-5942 77 2 οΏ½ οΏ½ PROPN cana-5942 77 3 Μ„ Μ„ NOUN cana-5942 77 4 οΏ½ οΏ½ PROPN cana-5942 77 5 1 1 NUM cana-5942 77 6 , , PUNCT cana-5942 77 7 οΏ½ οΏ½ PROPN cana-5942 77 8 Μ„ Μ„ NOUN cana-5942 77 9 οΏ½ οΏ½ PROPN cana-5942 77 10 2 2 NUM cana-5942 77 11 , , PUNCT cana-5942 77 12 οΏ½ οΏ½ PROPN cana-5942 77 13 Μ„ Μ„ PROPN cana-5942 77 14 οΏ½ οΏ½ PROPN cana-5942 77 15 , , PUNCT cana-5942 77 16 οΏ½ οΏ½ PROPN cana-5942 77 17 Μ„ Μ„ PROPN cana-5942 77 18 οΏ½ οΏ½ PROPN cana-5942 77 19 ] ] PUNCT cana-5942 77 20 𝑒π‘₯𝑝 𝑒π‘₯𝑝 NOUN cana-5942 77 21 { { PUNCT cana-5942 77 22 πœ„π‘˜(π‘₯ πœ„π‘˜(π‘₯ NUM cana-5942 77 23 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5942 77 24 πœƒ πœƒ X cana-5942 77 25 βˆ’ βˆ’ NOUN cana-5942 77 26 𝑧 𝑧 PRON cana-5942 77 27 π‘π‘œπ‘  π‘π‘œπ‘  NOUN cana-5942 77 28 πœƒ πœƒ NOUN cana-5942 77 29 ) ) PUNCT cana-5942 77 30 βˆ’ βˆ’ PROPN cana-5942 77 31 πœ„πœ”π‘‘ πœ„πœ”π‘‘ PROPN cana-5942 77 32 } } PUNCT cana-5942 77 33 , , PUNCT cana-5942 77 34 ( ( PUNCT cana-5942 77 35 13 13 NUM cana-5942 77 36 ) ) PUNCT cana-5942 77 37 where where SCONJ cana-5942 77 38 k k PROPN cana-5942 77 39 is be AUX cana-5942 77 40 the the DET cana-5942 77 41 wave wave NOUN cana-5942 77 42 number number NOUN cana-5942 77 43 and and CCONJ cana-5942 77 44 πœ” πœ” NOUN cana-5942 77 45 is be AUX cana-5942 77 46 angular angular ADJ cana-5942 77 47 frequency frequency NOUN cana-5942 77 48 connected connect VERB cana-5942 77 49 by by ADP cana-5942 77 50 the the DET cana-5942 77 51 relation relation NOUN cana-5942 77 52 πœ” πœ” ADP cana-5942 77 53 = = SYM cana-5942 77 54 π‘˜π‘‰ π‘˜π‘‰ NOUN cana-5942 77 55 , , PUNCT cana-5942 77 56 v v ADP cana-5942 77 57 being be AUX cana-5942 77 58 the the DET cana-5942 77 59 phase phase NOUN cana-5942 77 60 velocity velocity NOUN cana-5942 77 61 and and CCONJ cana-5942 77 62 ( ( PUNCT cana-5942 77 63 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5942 77 64 πœƒ πœƒ X cana-5942 77 65 , , PUNCT cana-5942 77 66 βˆ’ βˆ’ PROPN cana-5942 77 67 π‘π‘œπ‘  π‘π‘œπ‘  ADJ cana-5942 77 68 πœƒ πœƒ NOUN cana-5942 77 69 ) ) PUNCT cana-5942 77 70 denotes denote VERB cana-5942 77 71 the the DET cana-5942 77 72 projection projection NOUN cana-5942 77 73 of of ADP cana-5942 77 74 wave wave NOUN cana-5942 77 75 normal normal ADJ cana-5942 77 76 of of ADP cana-5942 77 77 incident incident NOUN cana-5942 77 78 wave wave NOUN cana-5942 77 79 onto onto ADP cana-5942 77 80 the the DET cana-5942 77 81 xz xz NOUN cana-5942 77 82 - - PUNCT cana-5942 77 83 plane plane NOUN cana-5942 77 84 . . PUNCT cana-5942 78 1 barred bar VERB cana-5942 78 2 quantities quantity NOUN cana-5942 78 3 are be AUX cana-5942 78 4 the the DET cana-5942 78 5 amplitudes amplitude NOUN cana-5942 78 6 of of ADP cana-5942 78 7 the the DET cana-5942 78 8 field field NOUN cana-5942 78 9 quantities quantity NOUN cana-5942 78 10 . . PUNCT cana-5942 79 1 injecting inject VERB cana-5942 79 2 eq.(13 eq.(13 NOUN cana-5942 79 3 ) ) PUNCT cana-5942 79 4 into into ADP cana-5942 79 5 eqs.(10)-(12 eqs.(10)-(12 NUM cana-5942 79 6 ) ) PUNCT cana-5942 79 7 , , PUNCT cana-5942 79 8 we we PRON cana-5942 79 9 get get VERB cana-5942 79 10 respectively respectively ADV cana-5942 79 11 ( ( PUNCT cana-5942 79 12 π‘Ž1πœ”2𝑉2 π‘Ž1πœ”2𝑉2 NOUN cana-5942 79 13 + + CCONJ cana-5942 79 14 πœ„πœ”3π‘Ž10) πœ„πœ”3π‘Ž10) NUM cana-5942 79 15 οΏ½ οΏ½ SYM cana-5942 79 16 Μ„ Μ„ NOUN cana-5942 79 17 οΏ½ οΏ½ NOUN cana-5942 79 18 1 1 NUM cana-5942 79 19 + + CCONJ cana-5942 79 20 ( ( PUNCT cana-5942 79 21 πœ„πœ”π‘‰2𝛾2𝛽00) πœ„πœ”π‘‰2𝛾2𝛽00) PROPN cana-5942 79 22 οΏ½ οΏ½ PROPN cana-5942 79 23 Μ„ Μ„ NOUN cana-5942 79 24 οΏ½ οΏ½ PROPN cana-5942 79 25 + + CCONJ cana-5942 79 26 π‘Ž2𝑉2 π‘Ž2𝑉2 NOUN cana-5942 79 27 οΏ½ οΏ½ PROPN cana-5942 79 28 Μ„ Μ„ NOUN cana-5942 79 29 οΏ½ οΏ½ PROPN cana-5942 79 30 = = SYM cana-5942 79 31 0 0 PROPN cana-5942 79 32 , , PUNCT cana-5942 79 33 ( ( PUNCT cana-5942 79 34 14 14 NUM cana-5942 79 35 ) ) PUNCT cana-5942 79 36 πœ”2π‘Ž3 πœ”2π‘Ž3 VERB cana-5942 79 37 οΏ½ οΏ½ PROPN cana-5942 79 38 Μ„ Μ„ NOUN cana-5942 79 39 οΏ½ οΏ½ NOUN cana-5942 79 40 1 1 NUM cana-5942 79 41 + + NUM cana-5942 79 42 π‘Ž5𝑉2 π‘Ž5𝑉2 PROPN cana-5942 79 43 οΏ½ οΏ½ PROPN cana-5942 79 44 Μ„ Μ„ NOUN cana-5942 79 45 οΏ½ οΏ½ PROPN cana-5942 79 46 + + CCONJ cana-5942 79 47 ( ( PUNCT cana-5942 79 48 π‘Ž6πœ”2𝑉2 π‘Ž6πœ”2𝑉2 NOUN cana-5942 79 49 βˆ’ βˆ’ NOUN cana-5942 79 50 πœ”2 πœ”2 ADJ cana-5942 79 51 βˆ’ βˆ’ PROPN cana-5942 79 52 π‘Ž4𝑉2) π‘Ž4𝑉2) ADV cana-5942 79 53 οΏ½ οΏ½ PROPN cana-5942 79 54 Μ„ Μ„ NOUN cana-5942 79 55 οΏ½ οΏ½ PROPN cana-5942 79 56 = = SYM cana-5942 79 57 0 0 PROPN cana-5942 79 58 , , PUNCT cana-5942 79 59 ( ( PUNCT cana-5942 79 60 15 15 NUM cana-5942 79 61 ) ) PUNCT cana-5942 79 62 and and CCONJ cana-5942 79 63 πœ„πœ”3𝛽00π‘Ž8π‘Ž11 πœ„πœ”3𝛽00π‘Ž8π‘Ž11 PROPN cana-5942 79 64 οΏ½ οΏ½ PROPN cana-5942 79 65 Μ„ Μ„ NOUN cana-5942 79 66 οΏ½ οΏ½ PROPN cana-5942 79 67 1 1 NUM cana-5942 79 68 + + CCONJ cana-5942 79 69 ( ( PUNCT cana-5942 79 70 πœ”2πœπ‘‡0 πœ”2πœπ‘‡0 X cana-5942 80 1 + + CCONJ cana-5942 80 2 πœ„πœ”π‘Ž7𝜏𝜐0 πœ„πœ”π‘Ž7𝜏𝜐0 PROPN cana-5942 80 3 + + NUM cana-5942 80 4 𝑉2π‘Ž11) 𝑉2π‘Ž11) NOUN cana-5942 80 5 οΏ½ οΏ½ PROPN cana-5942 80 6 Μ„ Μ„ PROPN cana-5942 80 7 οΏ½ οΏ½ PROPN cana-5942 80 8 + + CCONJ cana-5942 80 9 π‘Ž9π‘Ž11𝑉2 π‘Ž9π‘Ž11𝑉2 PROPN cana-5942 80 10 οΏ½ οΏ½ PROPN cana-5942 80 11 Μ„ Μ„ NOUN cana-5942 80 12 οΏ½ οΏ½ PROPN cana-5942 80 13 = = SYM cana-5942 80 14 0 0 PROPN cana-5942 80 15 , , PUNCT cana-5942 80 16 ( ( PUNCT cana-5942 80 17 16 16 NUM cana-5942 80 18 ) ) PUNCT cana-5942 80 19 where where SCONJ cana-5942 80 20 𝛽00 𝛽00 PROPN cana-5942 80 21 = = SYM cana-5942 80 22 𝛽 𝛽 PROPN cana-5942 80 23 + + X cana-5942 80 24 πœ„ πœ„ PROPN cana-5942 80 25 πœ” πœ” NOUN cana-5942 80 26 , , PUNCT cana-5942 80 27 𝜏𝜐0 𝜏𝜐0 PROPN cana-5942 80 28 = = SYM cana-5942 80 29 𝜏𝜐 𝜏𝜐 PROPN cana-5942 80 30 + + CCONJ cana-5942 80 31 πœ„ πœ„ PROPN cana-5942 80 32 πœ” πœ” X cana-5942 80 33 , , PUNCT cana-5942 80 34 πœπ‘‡0 πœπ‘‡0 NOUN cana-5942 80 35 = = PUNCT cana-5942 81 1 πœπ‘‡ πœπ‘‡ NOUN cana-5942 81 2 + + CCONJ cana-5942 81 3 πœ„ πœ„ PROPN cana-5942 81 4 πœ” πœ” NOUN cana-5942 81 5 , , PUNCT cana-5942 81 6 π‘Ž10 π‘Ž10 NOUN cana-5942 81 7 = = PUNCT cana-5942 81 8 ( ( PUNCT cana-5942 81 9 πœ†+2πœ‡ πœ†+2πœ‡ PROPN cana-5942 81 10 πœ‡ πœ‡ PROPN cana-5942 81 11 ) ) PUNCT cana-5942 81 12 , , PUNCT cana-5942 81 13 π‘Ž11 π‘Ž11 VERB cana-5942 81 14 = = SYM cana-5942 81 15 1 1 NUM cana-5942 81 16 βˆ’ βˆ’ NOUN cana-5942 81 17 πœ„πœ”πœπ‘ž πœ„πœ”πœπ‘ž NOUN cana-5942 81 18 βˆ’ βˆ’ NOUN cana-5942 81 19 πœπ‘ž πœπ‘ž NOUN cana-5942 81 20 2πœ”2 2πœ”2 NUM cana-5942 81 21 2 2 NUM cana-5942 81 22 . . PUNCT cana-5942 82 1 the the DET cana-5942 82 2 condition condition NOUN cana-5942 82 3 for for ADP cana-5942 82 4 the the DET cana-5942 82 5 existence existence NOUN cana-5942 82 6 of of ADP cana-5942 82 7 non non ADJ cana-5942 82 8 - - ADJ cana-5942 82 9 trivial trivial ADJ cana-5942 82 10 solution solution NOUN cana-5942 82 11 of of ADP cana-5942 82 12 the the DET cana-5942 82 13 system system NOUN cana-5942 82 14 of of ADP cana-5942 82 15 equations equation NOUN cana-5942 82 16 ( ( PUNCT cana-5942 82 17 14)-(16 14)-(16 NOUN cana-5942 82 18 ) ) PUNCT cana-5942 82 19 provides provide VERB cana-5942 82 20 us we PRON cana-5942 82 21 𝑉6 𝑉6 NOUN cana-5942 82 22 + + CCONJ cana-5942 82 23 𝐴𝑉4 𝐴𝑉4 PROPN cana-5942 82 24 + + CCONJ cana-5942 82 25 𝐡𝑉2 𝐡𝑉2 PROPN cana-5942 82 26 + + CCONJ cana-5942 82 27 𝐢 𝐢 PROPN cana-5942 82 28 = = SYM cana-5942 82 29 0 0 NUM cana-5942 82 30 ( ( PUNCT cana-5942 82 31 17 17 NUM cana-5942 82 32 ) ) PUNCT cana-5942 82 33 where where SCONJ cana-5942 82 34 𝐴 𝐴 PROPN cana-5942 82 35 = = SYM cana-5942 82 36 𝐴′ 𝐴′ PROPN cana-5942 82 37 𝐹 𝐹 PROPN cana-5942 82 38 , , PUNCT cana-5942 82 39 𝐡 𝐡 NOUN cana-5942 82 40 = = PUNCT cana-5942 82 41 𝐡′ 𝐡′ NUM cana-5942 82 42 𝐹 𝐹 PROPN cana-5942 82 43 , , PUNCT cana-5942 82 44 𝐢 𝐢 PROPN cana-5942 83 1 = = PRON cana-5942 83 2 𝐢′ 𝐢′ PROPN cana-5942 83 3 𝐹 𝐹 PROPN cana-5942 83 4 , , PUNCT cana-5942 83 5 𝐹 𝐹 PROPN cana-5942 83 6 = = SYM cana-5942 83 7 π‘Ž1π‘Ž6π‘Ž11πœ”4 π‘Ž1π‘Ž6π‘Ž11πœ”4 NUM cana-5942 83 8 βˆ’ βˆ’ NOUN cana-5942 83 9 π‘Ž1π‘Ž4π‘Ž11πœ”2 π‘Ž1π‘Ž4π‘Ž11πœ”2 NUM cana-5942 83 10 βˆ’ βˆ’ PROPN cana-5942 83 11 π‘Ž1π‘Ž5π‘Ž9π‘Ž11πœ”2 π‘Ž1π‘Ž5π‘Ž9π‘Ž11πœ”2 PROPN cana-5942 83 12 , , PUNCT cana-5942 83 13 𝐴′ 𝐴′ PROPN cana-5942 83 14 = = SYM cana-5942 83 15 π‘Ž1π‘Ž6πœπ‘‡0πœ”6 π‘Ž1π‘Ž6πœπ‘‡0πœ”6 PROPN cana-5942 83 16 + + CCONJ cana-5942 83 17 ( ( PUNCT cana-5942 83 18 πœ„π‘Ž1π‘Ž6π‘Ž7𝜏𝜐0 πœ„π‘Ž1π‘Ž6π‘Ž7𝜏𝜐0 NOUN cana-5942 83 19 + + X cana-5942 83 20 πœ„π‘Ž6π‘Ž10π‘Ž11)πœ”5 πœ„π‘Ž6π‘Ž10π‘Ž11)πœ”5 ADJ cana-5942 83 21 βˆ’ βˆ’ ADP cana-5942 83 22 ( ( PUNCT cana-5942 83 23 π‘Ž1π‘Ž4πœπ‘‡0 π‘Ž1π‘Ž4πœπ‘‡0 NOUN cana-5942 83 24 + + NOUN cana-5942 83 25 π‘Ž1π‘Ž11)πœ”4 π‘Ž1π‘Ž11)πœ”4 NOUN cana-5942 83 26 βˆ’(πœ„π‘Ž1π‘Ž4π‘Ž7𝜏𝜐0 βˆ’(πœ„π‘Ž1π‘Ž4π‘Ž7𝜏𝜐0 PROPN cana-5942 83 27 + + NOUN cana-5942 83 28 πœ„π‘Ž4π‘Ž10π‘Ž11 πœ„π‘Ž4π‘Ž10π‘Ž11 NUM cana-5942 83 29 βˆ’ βˆ’ PROPN cana-5942 83 30 πœ„π‘Ž3π‘Ž9π‘Ž11𝛽00𝛾2 πœ„π‘Ž3π‘Ž9π‘Ž11𝛽00𝛾2 DET cana-5942 83 31 + + NUM cana-5942 83 32 πœ„π‘Ž5π‘Ž9π‘Ž10π‘Ž11)πœ”3 πœ„π‘Ž5π‘Ž9π‘Ž10π‘Ž11)πœ”3 NOUN cana-5942 83 33 βˆ’ βˆ’ PROPN cana-5942 83 34 π‘Ž2π‘Ž3π‘Ž11πœ”2 π‘Ž2π‘Ž3π‘Ž11πœ”2 PROPN cana-5942 83 35 , , PUNCT cana-5942 83 36 𝐡′ 𝐡′ PROPN cana-5942 83 37 = = PUNCT cana-5942 83 38 πœ„π‘Ž6π‘Ž10πœπ‘‡0πœ”7 πœ„π‘Ž6π‘Ž10πœπ‘‡0πœ”7 NUM cana-5942 83 39 βˆ’ βˆ’ PROPN cana-5942 83 40 ( ( PUNCT cana-5942 83 41 π‘Ž1πœπ‘‡0 π‘Ž1πœπ‘‡0 X cana-5942 83 42 + + CCONJ cana-5942 83 43 π‘Ž6π‘Ž7π‘Ž10𝜏𝜐0)πœ”6 π‘Ž6π‘Ž7π‘Ž10𝜏𝜐0)πœ”6 X cana-5942 83 44 βˆ’ βˆ’ NOUN cana-5942 83 45 ( ( PUNCT cana-5942 83 46 πœ„π‘Ž4π‘Ž10πœπ‘‡0 πœ„π‘Ž4π‘Ž10πœπ‘‡0 NOUN cana-5942 83 47 + + CCONJ cana-5942 83 48 πœ„π‘Ž1π‘Ž7𝜏𝜐0 πœ„π‘Ž1π‘Ž7𝜏𝜐0 NOUN cana-5942 83 49 + + CCONJ cana-5942 83 50 πœ„π‘Ž10π‘Ž11)πœ”5 πœ„π‘Ž10π‘Ž11)πœ”5 ADJ cana-5942 83 51 βˆ’(π‘Ž2π‘Ž3πœπ‘‡0 βˆ’(π‘Ž2π‘Ž3πœπ‘‡0 NOUN cana-5942 83 52 βˆ’ βˆ’ PROPN cana-5942 83 53 π‘Ž4π‘Ž7π‘Ž10𝜏𝜐0)πœ”4 π‘Ž4π‘Ž7π‘Ž10𝜏𝜐0)πœ”4 ADP cana-5942 83 54 βˆ’ βˆ’ PROPN cana-5942 83 55 πœ„π‘Ž2π‘Ž3π‘Ž7𝜏𝜐0πœ”3 πœ„π‘Ž2π‘Ž3π‘Ž7𝜏𝜐0πœ”3 NOUN cana-5942 83 56 , , PUNCT cana-5942 83 57 𝐢′ 𝐢′ NOUN cana-5942 83 58 = = PUNCT cana-5942 83 59 π‘Ž7π‘Ž10𝜏𝜐0πœ”6 π‘Ž7π‘Ž10𝜏𝜐0πœ”6 PROPN cana-5942 83 60 + + CCONJ cana-5942 83 61 ( ( PUNCT cana-5942 83 62 πœ„π‘Ž8π‘Ž11𝛽00 πœ„π‘Ž8π‘Ž11𝛽00 NOUN cana-5942 83 63 βˆ’ βˆ’ PROPN cana-5942 83 64 πœ„π‘Ž10πœπ‘‡0)πœ”3 πœ„π‘Ž10πœπ‘‡0)πœ”3 NOUN cana-5942 83 65 . . PUNCT cana-5942 84 1 using use VERB cana-5942 84 2 the the DET cana-5942 84 3 transformation transformation NOUN cana-5942 84 4 𝑉2 𝑉2 NOUN cana-5942 84 5 = = PROPN cana-5942 84 6 π‘Œ π‘Œ PROPN cana-5942 84 7 in in ADP cana-5942 84 8 eq eq ADP cana-5942 84 9 . . PUNCT cana-5942 85 1 ( ( PUNCT cana-5942 85 2 17 17 NUM cana-5942 85 3 ) ) PUNCT cana-5942 85 4 , , PUNCT cana-5942 85 5 we we PRON cana-5942 85 6 obtain obtain VERB cana-5942 85 7 π‘Œ3 π‘Œ3 NOUN cana-5942 86 1 + + CCONJ cana-5942 86 2 π΄π‘Œ2 π΄π‘Œ2 PROPN cana-5942 86 3 + + CCONJ cana-5942 86 4 π΅π‘Œ π΅π‘Œ PROPN cana-5942 86 5 + + CCONJ cana-5942 86 6 𝐢 𝐢 PROPN cana-5942 86 7 = = SYM cana-5942 86 8 0 0 PROPN cana-5942 86 9 . . PUNCT cana-5942 87 1 ( ( PUNCT cana-5942 87 2 18 18 NUM cana-5942 87 3 ) ) PUNCT cana-5942 87 4 eq eq NOUN cana-5942 87 5 . . PUNCT cana-5942 88 1 ( ( PUNCT cana-5942 88 2 18 18 NUM cana-5942 88 3 ) ) PUNCT cana-5942 88 4 is be AUX cana-5942 88 5 a a DET cana-5942 88 6 cubic cubic ADJ cana-5942 88 7 in in ADP cana-5942 88 8 𝑉2 𝑉2 NOUN cana-5942 88 9 , , PUNCT cana-5942 88 10 which which PRON cana-5942 88 11 implies imply VERB cana-5942 88 12 that that SCONJ cana-5942 88 13 there there PRON cana-5942 88 14 shall shall AUX cana-5942 88 15 be be AUX cana-5942 88 16 three three NUM cana-5942 88 17 dilatational dilatational ADJ cana-5942 88 18 waves wave NOUN cana-5942 88 19 travelling travel VERB cana-5942 88 20 with with ADP cana-5942 88 21 three three NUM cana-5942 88 22 different different ADJ cana-5942 88 23 velocities velocity NOUN cana-5942 88 24 . . PUNCT cana-5942 89 1 communications communication NOUN cana-5942 89 2 on on ADP cana-5942 89 3 applied apply VERB cana-5942 89 4 nonlinear nonlinear ADJ cana-5942 89 5 analysis analysis NOUN cana-5942 89 6 issn issn NOUN cana-5942 89 7 : : PUNCT cana-5942 89 8 1074 1074 NUM cana-5942 89 9 - - PUNCT cana-5942 89 10 133x 133x NUM cana-5942 89 11 vol vol NOUN cana-5942 89 12 31 31 NUM cana-5942 89 13 no no NOUN cana-5942 89 14 . . PUNCT cana-5942 90 1 8s 8s PROPN cana-5942 90 2 ( ( PUNCT cana-5942 90 3 2024 2024 NUM cana-5942 90 4 ) ) PUNCT cana-5942 90 5 1133 1133 NUM cana-5942 90 6 https://internationalpubls.com https://internationalpubls.com X cana-5942 90 7 using use VERB cana-5942 90 8 cardan cardan PROPN cana-5942 90 9 ’s ’s PART cana-5942 90 10 method method NOUN cana-5942 90 11 in in ADP cana-5942 90 12 eq.(18 eq.(18 NOUN cana-5942 90 13 ) ) PUNCT cana-5942 90 14 , , PUNCT cana-5942 90 15 we we PRON cana-5942 90 16 get get VERB cana-5942 90 17 𝑍3 𝑍3 NOUN cana-5942 90 18 + + CCONJ cana-5942 91 1 3𝐻𝑍 3𝐻𝑍 NUM cana-5942 91 2 + + CCONJ cana-5942 91 3 𝐺 𝐺 NOUN cana-5942 91 4 = = SYM cana-5942 91 5 0 0 PROPN cana-5942 91 6 , , PUNCT cana-5942 91 7 ( ( PUNCT cana-5942 91 8 19 19 NUM cana-5942 91 9 ) ) PUNCT cana-5942 92 1 where where SCONJ cana-5942 92 2 𝑍 𝑍 NOUN cana-5942 92 3 = = SYM cana-5942 92 4 π‘Œ π‘Œ PROPN cana-5942 92 5 + + CCONJ cana-5942 92 6 𝐴 𝐴 PROPN cana-5942 92 7 3 3 NUM cana-5942 92 8 , , PUNCT cana-5942 92 9     SPACE cana-5942 92 10 𝐻 𝐻 NOUN cana-5942 92 11 = = SYM cana-5942 92 12 𝐡 𝐡 NOUN cana-5942 92 13 3 3 NUM cana-5942 92 14 βˆ’ βˆ’ NOUN cana-5942 92 15 𝐴2 𝐴2 PROPN cana-5942 92 16 9 9 NUM cana-5942 92 17 and and CCONJ cana-5942 92 18 𝐺 𝐺 PROPN cana-5942 92 19 = = SYM cana-5942 92 20 2𝐴3 2𝐴3 NUM cana-5942 92 21 27 27 NUM cana-5942 92 22 βˆ’ βˆ’ NOUN cana-5942 92 23 𝐴𝐡 𝐴𝐡 NOUN cana-5942 92 24 3 3 NUM cana-5942 92 25 + + CCONJ cana-5942 92 26 𝐢. 𝐢. PROPN cana-5942 92 27 since since SCONJ cana-5942 92 28 the the DET cana-5942 92 29 quantities quantity NOUN cana-5942 92 30 a a PRON cana-5942 92 31 , , PUNCT cana-5942 92 32 b b NOUN cana-5942 92 33 and and CCONJ cana-5942 92 34 c c PROPN cana-5942 92 35 are be AUX cana-5942 92 36 complex complex ADJ cana-5942 92 37 , , PUNCT cana-5942 92 38 therefore therefore ADV cana-5942 92 39 the the DET cana-5942 92 40 coefficients coefficient NOUN cana-5942 92 41 h h NOUN cana-5942 92 42 and and CCONJ cana-5942 92 43 g g PROPN cana-5942 92 44 are be AUX cana-5942 92 45 complex complex ADJ cana-5942 92 46 . . PUNCT cana-5942 93 1 thus thus ADV cana-5942 93 2 the the DET cana-5942 93 3 roots root NOUN cana-5942 93 4 of of ADP cana-5942 93 5 eq.(19 eq.(19 NOUN cana-5942 93 6 ) ) PUNCT cana-5942 93 7 and and CCONJ cana-5942 93 8 hence hence ADV cana-5942 93 9 of of ADP cana-5942 93 10 eq.(18 eq.(18 NOUN cana-5942 93 11 ) ) PUNCT cana-5942 93 12 are be AUX cana-5942 93 13 given give VERB cana-5942 93 14 by by ADP cana-5942 93 15 𝑉1 𝑉1 PROPN cana-5942 93 16 2 2 NUM cana-5942 93 17 = = SYM cana-5942 93 18 𝑆 𝑆 PROPN cana-5942 93 19 βˆ’ βˆ’ PROPN cana-5942 93 20 𝐴 𝐴 PROPN cana-5942 93 21 3 3 NUM cana-5942 93 22 , , PUNCT cana-5942 93 23 𝑉2 𝑉2 NOUN cana-5942 93 24 2 2 NUM cana-5942 93 25 = = SYM cana-5942 93 26 βˆ’ βˆ’ PROPN cana-5942 93 27 1 1 NUM cana-5942 93 28 2 2 NUM cana-5942 93 29 𝑆 𝑆 PROPN cana-5942 93 30 + + CCONJ cana-5942 93 31 πœ„ πœ„ PROPN cana-5942 93 32 √3 √3 PROPN cana-5942 93 33 2 2 NUM cana-5942 93 34 𝑇 𝑇 PROPN cana-5942 93 35 βˆ’ βˆ’ PROPN cana-5942 93 36 𝐴 𝐴 PROPN cana-5942 93 37 3 3 NUM cana-5942 93 38 , , PUNCT cana-5942 93 39 𝑉3 𝑉3 NOUN cana-5942 93 40 2 2 NUM cana-5942 93 41 = = SYM cana-5942 93 42 βˆ’ βˆ’ PROPN cana-5942 93 43 1 1 NUM cana-5942 93 44 2 2 NUM cana-5942 93 45 𝑆 𝑆 PROPN cana-5942 93 46 βˆ’ βˆ’ NOUN cana-5942 93 47 πœ„ πœ„ PROPN cana-5942 93 48 √3 √3 PROPN cana-5942 93 49 2 2 NUM cana-5942 93 50 𝑇 𝑇 PROPN cana-5942 93 51 βˆ’ βˆ’ PROPN cana-5942 93 52 𝐴 𝐴 PROPN cana-5942 93 53 3 3 NUM cana-5942 93 54 ( ( PUNCT cana-5942 93 55 20 20 NUM cana-5942 93 56 ) ) PUNCT cana-5942 93 57 where where SCONJ cana-5942 93 58 𝑆 𝑆 PROPN cana-5942 93 59 = = SYM cana-5942 93 60 π‘ˆ π‘ˆ PROPN cana-5942 93 61 + + SYM cana-5942 93 62 𝑉 𝑉 PROPN cana-5942 93 63 , , PUNCT cana-5942 93 64 𝑇 𝑇 PROPN cana-5942 93 65 = = SYM cana-5942 93 66 π‘ˆ π‘ˆ PROPN cana-5942 93 67 βˆ’ βˆ’ PROPN cana-5942 93 68 𝑉 𝑉 PROPN cana-5942 93 69 , , PUNCT cana-5942 93 70 π‘ˆ3 π‘ˆ3 NOUN cana-5942 93 71 = = SYM cana-5942 93 72 1 1 NUM cana-5942 93 73 2 2 NUM cana-5942 94 1 [ [ X cana-5942 94 2 βˆ’πΊ βˆ’πΊ NOUN cana-5942 94 3 + + NUM cana-5942 94 4 √𝐺2 √𝐺2 NOUN cana-5942 94 5 + + SYM cana-5942 94 6 4𝐻3 4𝐻3 NUM cana-5942 94 7 ] ] PUNCT cana-5942 94 8 and and CCONJ cana-5942 94 9 𝑉 𝑉 PROPN cana-5942 94 10 = = PROPN cana-5942 94 11 βˆ’π» βˆ’π» PROPN cana-5942 94 12 π‘ˆ π‘ˆ PROPN cana-5942 94 13 . . PUNCT cana-5942 95 1 𝑉1,2,3 𝑉1,2,3 NUM cana-5942 95 2 are be AUX cana-5942 95 3 the the DET cana-5942 95 4 speeds speed NOUN cana-5942 95 5 of of ADP cana-5942 95 6 propagation propagation NOUN cana-5942 95 7 of of ADP cana-5942 95 8 three three NUM cana-5942 95 9 coupled couple VERB cana-5942 95 10 dilatational dilatational ADJ cana-5942 95 11 waves wave NOUN cana-5942 95 12 namely namely ADV cana-5942 95 13 longitudinal longitudinal ADJ cana-5942 95 14 displacement displacement ADJ cana-5942 95 15 wave wave NOUN cana-5942 95 16 ( ( PUNCT cana-5942 95 17 𝑃1 𝑃1 NOUN cana-5942 95 18 ) ) PUNCT cana-5942 95 19 , , PUNCT cana-5942 95 20 thermal thermal ADJ cana-5942 95 21 wave wave NOUN cana-5942 95 22 ( ( PUNCT cana-5942 95 23 𝑃2 𝑃2 PROPN cana-5942 95 24 ) ) PUNCT cana-5942 95 25 and and CCONJ cana-5942 95 26 longitudinal longitudinal ADJ cana-5942 95 27 void void ADJ cana-5942 95 28 volume volume NOUN cana-5942 95 29 fraction fraction NOUN cana-5942 95 30 wave wave NOUN cana-5942 95 31 ( ( PUNCT cana-5942 95 32 𝑃3 𝑃3 NOUN cana-5942 95 33 ) ) PUNCT cana-5942 95 34 . . PUNCT cana-5942 96 1 it it PRON cana-5942 96 2 can can AUX cana-5942 96 3 be be AUX cana-5942 96 4 easily easily ADV cana-5942 96 5 observed observe VERB cana-5942 96 6 that that SCONJ cana-5942 96 7 speeds speed NOUN cana-5942 96 8 of of ADP cana-5942 96 9 all all DET cana-5942 96 10 the the DET cana-5942 96 11 coupled couple VERB cana-5942 96 12 longitudinal longitudinal ADJ cana-5942 96 13 waves wave NOUN cana-5942 96 14 are be AUX cana-5942 96 15 influenced influence VERB cana-5942 96 16 by by ADP cana-5942 96 17 three three NUM cana-5942 96 18 phase phase NOUN cana-5942 96 19 lags lag VERB cana-5942 96 20 ( ( PUNCT cana-5942 96 21 πœπ‘ž,𝜏𝜐andπœπ‘‡ πœπ‘ž,𝜏𝜐andπœπ‘‡ PROPN cana-5942 96 22 ) ) PUNCT cana-5942 96 23 and and CCONJ cana-5942 96 24 void void ADJ cana-5942 96 25 parameters parameter NOUN cana-5942 96 26 . . PUNCT cana-5942 97 1 eq eq ADJ cana-5942 97 2 . . PUNCT cana-5942 98 1 ( ( PUNCT cana-5942 98 2 9 9 X cana-5942 98 3 ) ) PUNCT cana-5942 98 4 corresponds correspond NOUN cana-5942 98 5 to to ADP cana-5942 98 6 the the DET cana-5942 98 7 uncoupled uncoupled ADJ cana-5942 98 8 transverse transverse NOUN cana-5942 98 9 displacement displacement ADJ cana-5942 98 10 wave wave NOUN cana-5942 98 11 ( ( PUNCT cana-5942 98 12 sv sv NOUN cana-5942 98 13 ) ) PUNCT cana-5942 98 14 whose whose DET cana-5942 98 15 velocity velocity NOUN cana-5942 98 16 is be AUX cana-5942 98 17 given give VERB cana-5942 98 18 by by ADP cana-5942 98 19 𝑉4 𝑉4 NOUN cana-5942 98 20 = = SYM cana-5942 98 21 √ √ ADP cana-5942 98 22 1 1 NUM cana-5942 98 23 π‘Ž1 π‘Ž1 NOUN cana-5942 98 24 . . PUNCT cana-5942 99 1 ( ( PUNCT cana-5942 99 2 21 21 NUM cana-5942 99 3 ) ) PUNCT cana-5942 99 4 reflection reflection NOUN cana-5942 99 5 at at ADP cana-5942 99 6 the the DET cana-5942 99 7 free free ADJ cana-5942 99 8 surface surface NOUN cana-5942 99 9 here here ADV cana-5942 99 10 , , PUNCT cana-5942 99 11 we we PRON cana-5942 99 12 shall shall AUX cana-5942 99 13 consider consider VERB cana-5942 99 14 the the DET cana-5942 99 15 problem problem NOUN cana-5942 99 16 of of ADP cana-5942 99 17 incidence incidence NOUN cana-5942 99 18 of of ADP cana-5942 99 19 a a DET cana-5942 99 20 coupled couple VERB cana-5942 99 21 longitudinal longitudinal ADJ cana-5942 99 22 wave wave NOUN cana-5942 99 23 on on ADP cana-5942 99 24 the the DET cana-5942 99 25 free free ADJ cana-5942 99 26 and and CCONJ cana-5942 99 27 thermally thermally ADV cana-5942 99 28 insulated insulate VERB cana-5942 99 29 boundary boundary NOUN cana-5942 99 30 of of ADP cana-5942 99 31 a a DET cana-5942 99 32 thermoelastic thermoelastic ADJ cana-5942 99 33 half half ADJ cana-5942 99 34 - - PUNCT cana-5942 99 35 space space NOUN cana-5942 99 36 with with ADP cana-5942 99 37 voids voids NOUN cana-5942 99 38 . . PUNCT cana-5942 100 1 fig fig NOUN cana-5942 100 2 . . PUNCT cana-5942 101 1 1 1 NUM cana-5942 101 2 schematic schematic ADJ cana-5942 101 3 diagram diagram NOUN cana-5942 101 4 for for ADP cana-5942 101 5 the the DET cana-5942 101 6 problem problem NOUN cana-5942 101 7 we we PRON cana-5942 101 8 assume assume VERB cana-5942 101 9 that that SCONJ cana-5942 101 10 a a DET cana-5942 101 11 set set NOUN cana-5942 101 12 of of ADP cana-5942 101 13 coupled couple VERB cana-5942 101 14 longitudinal longitudinal ADJ cana-5942 101 15 waves wave NOUN cana-5942 101 16 of of ADP cana-5942 101 17 amplitude amplitude NOUN cana-5942 101 18 a0 a0 NOUN cana-5942 101 19 propagating propagate VERB cana-5942 101 20 with with ADP cana-5942 101 21 the the DET cana-5942 101 22 phase phase NOUN cana-5942 101 23 velocity velocity NOUN cana-5942 101 24 v1 v1 NOUN cana-5942 101 25 becomes become VERB cana-5942 101 26 incident incident NOUN cana-5942 101 27 obliquely obliquely ADV cana-5942 101 28 at at ADP cana-5942 101 29 the the DET cana-5942 101 30 free free ADJ cana-5942 101 31 plane plane NOUN cana-5942 101 32 surface surface NOUN cana-5942 101 33 , , PUNCT cana-5942 101 34 making make VERB cana-5942 101 35 an an DET cana-5942 101 36 angle angle NOUN cana-5942 101 37 ΞΈ0 ΞΈ0 NOUN cana-5942 101 38 with with ADP cana-5942 101 39 the the DET cana-5942 101 40 normal normal ADJ cana-5942 101 41 . . PUNCT cana-5942 102 1 in in ADP cana-5942 102 2 order order NOUN cana-5942 102 3 to to PART cana-5942 102 4 satisfy satisfy VERB cana-5942 102 5 the the DET cana-5942 102 6 boundary boundary ADJ cana-5942 102 7 conditions condition NOUN cana-5942 102 8 , , PUNCT cana-5942 102 9 this this DET cana-5942 102 10 incident incident NOUN cana-5942 102 11 coupled couple VERB cana-5942 102 12 longitudinal longitudinal ADJ cana-5942 102 13 wave wave NOUN cana-5942 102 14 gives give VERB cana-5942 102 15 rise rise NOUN cana-5942 102 16 to to ADP cana-5942 102 17 the the DET cana-5942 102 18 following follow VERB cana-5942 102 19 reflected reflect VERB cana-5942 102 20 waves wave NOUN cana-5942 102 21 : : PUNCT cana-5942 102 22 ( ( PUNCT cana-5942 102 23 i i NOUN cana-5942 102 24 ) ) PUNCT cana-5942 102 25 three three NUM cana-5942 102 26 coupled couple VERB cana-5942 102 27 longitudinal longitudinal ADJ cana-5942 102 28 waves wave NOUN cana-5942 102 29 with with ADP cana-5942 102 30 amplitudes amplitude NOUN cana-5942 102 31 a1 a1 NOUN cana-5942 102 32 , , PUNCT cana-5942 102 33 a2 a2 PROPN cana-5942 102 34 and and CCONJ cana-5942 102 35 a3 a3 NOUN cana-5942 102 36 propagating propagate VERB cana-5942 102 37 with with ADP cana-5942 102 38 speeds speed NOUN cana-5942 102 39 v1,2,3 v1,2,3 PROPN cana-5942 102 40 and and CCONJ cana-5942 102 41 making make VERB cana-5942 102 42 angles angle NOUN cana-5942 102 43 ΞΈ1,2,3 ΞΈ1,2,3 PROPN cana-5942 102 44 respectively respectively ADV cana-5942 102 45 with with ADP cana-5942 102 46 the the DET cana-5942 102 47 normal normal ADJ cana-5942 102 48 . . PUNCT cana-5942 103 1 communications communication NOUN cana-5942 103 2 on on ADP cana-5942 103 3 applied apply VERB cana-5942 103 4 nonlinear nonlinear ADJ cana-5942 103 5 analysis analysis NOUN cana-5942 103 6 issn issn NOUN cana-5942 103 7 : : PUNCT cana-5942 103 8 1074 1074 NUM cana-5942 103 9 - - PUNCT cana-5942 103 10 133x 133x NUM cana-5942 103 11 vol vol NOUN cana-5942 103 12 31 31 NUM cana-5942 103 13 no no NOUN cana-5942 103 14 . . PUNCT cana-5942 104 1 8s 8s PROPN cana-5942 104 2 ( ( PUNCT cana-5942 104 3 2024 2024 NUM cana-5942 104 4 ) ) PUNCT cana-5942 104 5 1134 1134 NUM cana-5942 104 6 https://internationalpubls.com https://internationalpubls.com X cana-5942 104 7 ( ( PUNCT cana-5942 104 8 ii ii NOUN cana-5942 104 9 ) ) PUNCT cana-5942 104 10 a a DET cana-5942 104 11 transverse transverse NOUN cana-5942 104 12 wave wave NOUN cana-5942 104 13 of of ADP cana-5942 104 14 amplitude amplitude NOUN cana-5942 104 15 b1 b1 NOUN cana-5942 104 16 propagating propagate VERB cana-5942 104 17 with with ADP cana-5942 104 18 speed speed NOUN cana-5942 104 19 v4 v4 NOUN cana-5942 104 20 making make VERB cana-5942 104 21 an an DET cana-5942 104 22 angle angle NOUN cana-5942 104 23 ΞΈ4 ΞΈ4 NOUN cana-5942 104 24 with with ADP cana-5942 104 25 the the DET cana-5942 104 26 normal normal ADJ cana-5942 104 27 . . PUNCT cana-5942 105 1 full full ADJ cana-5942 105 2 structure structure NOUN cana-5942 105 3 of of ADP cana-5942 105 4 the the DET cana-5942 105 5 wave wave NOUN cana-5942 105 6 field field NOUN cana-5942 105 7 consisting consist VERB cana-5942 105 8 of of ADP cana-5942 105 9 the the DET cana-5942 105 10 incident incident NOUN cana-5942 105 11 and and CCONJ cana-5942 105 12 reflected reflect VERB cana-5942 105 13 waves wave NOUN cana-5942 105 14 can can AUX cana-5942 105 15 be be AUX cana-5942 105 16 written write VERB cana-5942 105 17 as as ADP cana-5942 105 18 : : PUNCT cana-5942 105 19 πœ“1 πœ“1 NOUN cana-5942 105 20 = = PUNCT cana-5942 105 21 𝐴0 𝐴0 PROPN cana-5942 105 22 𝑒π‘₯𝑝{πœ„π‘˜1(π‘₯ 𝑒π‘₯𝑝{πœ„π‘˜1(π‘₯ PROPN cana-5942 105 23 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5942 105 24 πœƒ0 πœƒ0 VERB cana-5942 105 25 βˆ’ βˆ’ PROPN cana-5942 105 26 𝑧 𝑧 DET cana-5942 105 27 π‘π‘œπ‘  π‘π‘œπ‘  NOUN cana-5942 105 28 πœƒ0 πœƒ0 NOUN cana-5942 105 29 ) ) PUNCT cana-5942 105 30 βˆ’ βˆ’ PROPN cana-5942 105 31 πœ„πœ”π‘‘ πœ„πœ”π‘‘ PROPN cana-5942 105 32 } } PUNCT cana-5942 105 33 + + CCONJ cana-5942 105 34 βˆ‘ βˆ‘ PROPN cana-5942 105 35 𝐴𝑖 𝐴𝑖 PROPN cana-5942 105 36 𝑒π‘₯𝑝{πœ„π‘˜π‘–(π‘₯ 𝑒π‘₯𝑝{πœ„π‘˜π‘–(π‘₯ PROPN cana-5942 105 37 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5942 105 38 πœƒπ‘– πœƒπ‘– ADP cana-5942 105 39 + + NOUN cana-5942 105 40 𝑧 𝑧 PROPN cana-5942 105 41 π‘π‘œπ‘  π‘π‘œπ‘  NOUN cana-5942 105 42 πœƒπ‘– πœƒπ‘– NOUN cana-5942 105 43 ) ) PUNCT cana-5942 105 44 βˆ’ βˆ’ PROPN cana-5942 105 45 πœ„πœ”π‘‘}3 πœ„πœ”π‘‘}3 INTJ cana-5942 105 46 𝑖=1 𝑖=1 PUNCT cana-5942 105 47 , , PUNCT cana-5942 105 48 ( ( PUNCT cana-5942 105 49 22 22 X cana-5942 105 50 ) ) PUNCT cana-5942 105 51 𝛩 𝛩 NOUN cana-5942 105 52 = = NOUN cana-5942 106 1 π‘Ž1 π‘Ž1 INTJ cana-5942 106 2 βˆ—π΄0 βˆ—π΄0 PROPN cana-5942 106 3 𝑒π‘₯𝑝{πœ„π‘˜1(π‘₯ 𝑒π‘₯𝑝{πœ„π‘˜1(π‘₯ PROPN cana-5942 106 4 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5942 106 5 πœƒ0 πœƒ0 VERB cana-5942 106 6 βˆ’ βˆ’ PROPN cana-5942 106 7 𝑧 𝑧 PRON cana-5942 106 8 π‘π‘œπ‘  π‘π‘œπ‘  NOUN cana-5942 106 9 πœƒ0 πœƒ0 NOUN cana-5942 106 10 ) ) PUNCT cana-5942 106 11 βˆ’ βˆ’ PROPN cana-5942 106 12 πœ„πœ”π‘‘ πœ„πœ”π‘‘ PROPN cana-5942 106 13 } } PUNCT cana-5942 106 14 + + CCONJ cana-5942 106 15 βˆ‘ βˆ‘ PROPN cana-5942 106 16 π‘Žπ‘– π‘Žπ‘– DET cana-5942 106 17 βˆ—π΄π‘– βˆ—π΄π‘– ADJ cana-5942 106 18 𝑒π‘₯𝑝{πœ„π‘˜π‘–(π‘₯ 𝑒π‘₯𝑝{πœ„π‘˜π‘–(π‘₯ PROPN cana-5942 106 19 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5942 106 20 πœƒπ‘– πœƒπ‘– ADP cana-5942 106 21 + + NOUN cana-5942 106 22 𝑧 𝑧 PROPN cana-5942 106 23 π‘π‘œπ‘  π‘π‘œπ‘  NOUN cana-5942 106 24 πœƒπ‘– πœƒπ‘– NOUN cana-5942 106 25 ) ) PUNCT cana-5942 106 26 βˆ’ βˆ’ PROPN cana-5942 106 27 πœ„πœ”π‘‘}3 πœ„πœ”π‘‘}3 INTJ cana-5942 106 28 𝑖=1 𝑖=1 PUNCT cana-5942 106 29 , , PUNCT cana-5942 106 30 ( ( PUNCT cana-5942 106 31 23 23 NUM cana-5942 106 32 ) ) PUNCT cana-5942 106 33 πœ™ πœ™ NOUN cana-5942 106 34 = = SYM cana-5942 106 35 𝑏1 𝑏1 ADJ cana-5942 106 36 βˆ—π΄0 βˆ—π΄0 NOUN cana-5942 106 37 𝑒π‘₯𝑝{πœ„π‘˜1(π‘₯ 𝑒π‘₯𝑝{πœ„π‘˜1(π‘₯ PROPN cana-5942 106 38 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5942 106 39 πœƒ0 πœƒ0 VERB cana-5942 106 40 βˆ’ βˆ’ PROPN cana-5942 106 41 𝑧 𝑧 PRON cana-5942 106 42 π‘π‘œπ‘  π‘π‘œπ‘  NOUN cana-5942 106 43 πœƒ0 πœƒ0 NOUN cana-5942 106 44 ) ) PUNCT cana-5942 106 45 βˆ’ βˆ’ PROPN cana-5942 106 46 πœ„πœ”π‘‘ πœ„πœ”π‘‘ PROPN cana-5942 106 47 } } PUNCT cana-5942 106 48 + + CCONJ cana-5942 106 49 βˆ‘ βˆ‘ PROPN cana-5942 106 50 𝑏𝑖 𝑏𝑖 ADP cana-5942 106 51 βˆ—π΄π‘– βˆ—π΄π‘– ADJ cana-5942 106 52 𝑒π‘₯𝑝{πœ„π‘˜π‘–(π‘₯ 𝑒π‘₯𝑝{πœ„π‘˜π‘–(π‘₯ PROPN cana-5942 106 53 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5942 106 54 πœƒπ‘– πœƒπ‘– ADP cana-5942 106 55 + + NOUN cana-5942 106 56 𝑧 𝑧 PROPN cana-5942 106 57 π‘π‘œπ‘  π‘π‘œπ‘  NOUN cana-5942 106 58 πœƒπ‘– πœƒπ‘– NOUN cana-5942 106 59 ) ) PUNCT cana-5942 106 60 βˆ’ βˆ’ PROPN cana-5942 106 61 πœ„πœ”π‘‘}3 πœ„πœ”π‘‘}3 INTJ cana-5942 106 62 𝑖=1 𝑖=1 PUNCT cana-5942 106 63 , , PUNCT cana-5942 106 64 ( ( PUNCT cana-5942 106 65 24 24 NUM cana-5942 106 66 ) ) PUNCT cana-5942 106 67 πœ“2 πœ“2 NOUN cana-5942 106 68 = = PUNCT cana-5942 106 69 𝐡1 𝐡1 NOUN cana-5942 106 70 𝑒π‘₯𝑝{πœ„π‘˜4(π‘₯ 𝑒π‘₯𝑝{πœ„π‘˜4(π‘₯ NOUN cana-5942 106 71 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5942 106 72 πœƒ4 πœƒ4 PROPN cana-5942 106 73 + + CCONJ cana-5942 106 74 𝑧 𝑧 PRON cana-5942 106 75 π‘π‘œπ‘  π‘π‘œπ‘  NOUN cana-5942 106 76 πœƒ4 πœƒ4 PROPN cana-5942 106 77 ) ) PUNCT cana-5942 106 78 βˆ’ βˆ’ PROPN cana-5942 106 79 πœ„πœ”π‘‘ πœ„πœ”π‘‘ PROPN cana-5942 106 80 } } PUNCT cana-5942 106 81 , , PUNCT cana-5942 106 82 ( ( PUNCT cana-5942 106 83 25 25 NUM cana-5942 106 84 ) ) PUNCT cana-5942 106 85 where where SCONJ cana-5942 106 86 π‘Žπ‘– π‘Žπ‘– DET cana-5942 106 87 βˆ— βˆ— NOUN cana-5942 106 88 and and CCONJ cana-5942 106 89 𝑏𝑖 𝑏𝑖 ADP cana-5942 106 90 βˆ— βˆ— NOUN cana-5942 106 91 ( ( PUNCT cana-5942 106 92 𝑖 𝑖 SYM cana-5942 106 93 = = NOUN cana-5942 106 94 1,2,3 1,2,3 NUM cana-5942 106 95 ) ) PUNCT cana-5942 106 96 are be AUX cana-5942 106 97 the the DET cana-5942 106 98 coupling coupling NOUN cana-5942 106 99 parameters parameter NOUN cana-5942 106 100 . . PUNCT cana-5942 107 1 their their PRON cana-5942 107 2 expressions expression NOUN cana-5942 107 3 are be AUX cana-5942 107 4 given give VERB cana-5942 107 5 by by ADP cana-5942 107 6 π‘Žπ‘– π‘Žπ‘– ADP cana-5942 107 7 βˆ— βˆ— NOUN cana-5942 107 8 = = PUNCT cana-5942 107 9 ( ( PUNCT cana-5942 107 10 π‘Ž1π‘Ž6πœ”4βˆ’π‘Ž1π‘Ž4πœ”2)𝑉𝑖 π‘Ž1π‘Ž6πœ”4βˆ’π‘Ž1π‘Ž4πœ”2)𝑉𝑖 PROPN cana-5942 107 11 4+(πœ„π‘Ž6π‘Ž10πœ”5βˆ’π‘Ž1πœ”4βˆ’πœ„π‘Ž4π‘Ž10πœ”3βˆ’π‘Ž2π‘Ž3πœ”2)𝑉𝑖 4+(πœ„π‘Ž6π‘Ž10πœ”5βˆ’π‘Ž1πœ”4βˆ’πœ„π‘Ž4π‘Ž10πœ”3βˆ’π‘Ž2π‘Ž3πœ”2)𝑉𝑖 NUM cana-5942 107 12 2βˆ’πœ„π‘Ž10πœ”5 2βˆ’πœ„π‘Ž10πœ”5 NUM cana-5942 107 13 ( ( PUNCT cana-5942 107 14 𝑔1𝑉𝑖 𝑔1𝑉𝑖 X cana-5942 107 15 4+𝑔2𝑉𝑖 4+𝑔2𝑉𝑖 NUM cana-5942 107 16 2 2 NUM cana-5942 107 17 ) ) PUNCT cana-5942 107 18 , , PUNCT cana-5942 107 19 𝑏𝑖 𝑏𝑖 ADP cana-5942 107 20 βˆ— βˆ— NOUN cana-5942 107 21 = = PUNCT cana-5942 107 22 ( ( PUNCT cana-5942 107 23 βˆ’π‘Ž1π‘Ž5πœ”2)𝑉𝑖 βˆ’π‘Ž1π‘Ž5πœ”2)𝑉𝑖 PROPN cana-5942 107 24 4+πœ„πœ”3(π‘Ž3𝛽00𝛾2βˆ’π‘Ž5π‘Ž10)𝑉𝑖 4+πœ„πœ”3(π‘Ž3𝛽00𝛾2βˆ’π‘Ž5π‘Ž10)𝑉𝑖 NUM cana-5942 107 25 2 2 NUM cana-5942 107 26 ( ( PUNCT cana-5942 107 27 𝑔1𝑉𝑖 𝑔1𝑉𝑖 X cana-5942 107 28 4+𝑔2𝑉𝑖 4+𝑔2𝑉𝑖 NUM cana-5942 107 29 2 2 NUM cana-5942 107 30 ) ) PUNCT cana-5942 107 31 , , PUNCT cana-5942 107 32 where where SCONJ cana-5942 107 33 𝑔1 𝑔1 PROPN cana-5942 107 34 = = PROPN cana-5942 107 35 πœ„πœ”π‘Ž4𝛽00𝛾2 πœ„πœ”π‘Ž4𝛽00𝛾2 PROPN cana-5942 107 36 βˆ’ βˆ’ NOUN cana-5942 107 37 πœ„πœ”3π‘Ž6𝛽00𝛾2 πœ„πœ”3π‘Ž6𝛽00𝛾2 PRON cana-5942 107 38 + + CCONJ cana-5942 107 39 π‘Ž2π‘Ž5 π‘Ž2π‘Ž5 PROPN cana-5942 107 40 , , PUNCT cana-5942 107 41 𝑔2 𝑔2 VERB cana-5942 107 42 = = SYM cana-5942 107 43 πœ„πœ”3𝛽00𝛾2 πœ„πœ”3𝛽00𝛾2 PROPN cana-5942 107 44 . . PUNCT cana-5942 108 1 the the DET cana-5942 108 2 amplitudes amplitude NOUN cana-5942 108 3 𝐴1 𝐴1 PROPN cana-5942 108 4 , , PUNCT cana-5942 108 5 𝐴2 𝐴2 PROPN cana-5942 108 6 , , PUNCT cana-5942 108 7 𝐴3 𝐴3 PROPN cana-5942 108 8 and and CCONJ cana-5942 108 9 𝐡1 𝐡1 NOUN cana-5942 108 10 can can AUX cana-5942 108 11 be be AUX cana-5942 108 12 determined determine VERB cana-5942 108 13 from from ADP cana-5942 108 14 the the DET cana-5942 108 15 boundary boundary ADJ cana-5942 108 16 conditions condition NOUN cana-5942 108 17 at at ADP cana-5942 108 18 𝑧 𝑧 PRON cana-5942 108 19 = = ADJ cana-5942 108 20 0 0 PROPN cana-5942 108 21 . . PUNCT cana-5942 109 1 the the DET cana-5942 109 2 surface surface NOUN cana-5942 109 3 𝑧 𝑧 PROPN cana-5942 109 4 = = SYM cana-5942 109 5 0 0 NUM cana-5942 109 6 is be AUX cana-5942 109 7 assumed assume VERB cana-5942 109 8 to to PART cana-5942 109 9 be be AUX cana-5942 109 10 traction traction NOUN cana-5942 109 11 free free ADJ cana-5942 109 12 and and CCONJ cana-5942 109 13 thermally thermally ADV cana-5942 109 14 insulated insulate VERB cana-5942 109 15 so so SCONJ cana-5942 109 16 that that SCONJ cana-5942 109 17 there there PRON cana-5942 109 18 is be VERB cana-5942 109 19 no no DET cana-5942 109 20 variation variation NOUN cana-5942 109 21 of of ADP cana-5942 109 22 temperature temperature NOUN cana-5942 109 23 and and CCONJ cana-5942 109 24 volume volume NOUN cana-5942 109 25 fraction fraction NOUN cana-5942 109 26 field field NOUN cana-5942 109 27 on on ADP cana-5942 109 28 it it PRON cana-5942 109 29 . . PUNCT cana-5942 110 1 therefore therefore ADV cana-5942 110 2 , , PUNCT cana-5942 110 3 the the DET cana-5942 110 4 boundary boundary ADJ cana-5942 110 5 conditions condition NOUN cana-5942 110 6 are be AUX cana-5942 110 7 written write VERB cana-5942 110 8 as as ADP cana-5942 110 9 πœπ‘§π‘§ πœπ‘§π‘§ NOUN cana-5942 110 10 = = SYM cana-5942 110 11 0 0 NUM cana-5942 110 12 , , PUNCT cana-5942 110 13 πœπ‘§π‘₯ πœπ‘§π‘₯ VERB cana-5942 110 14 = = SYM cana-5942 110 15 0 0 NUM cana-5942 110 16 , , PUNCT cana-5942 110 17 πœ•π›© πœ•π›© NOUN cana-5942 110 18 πœ•π‘§ πœ•π‘§ NOUN cana-5942 110 19 = = SYM cana-5942 110 20 0 0 NUM cana-5942 110 21 , , PUNCT cana-5942 110 22 πœ•πœ™ πœ•πœ™ ADJ cana-5942 110 23 πœ•π‘§ πœ•π‘§ NOUN cana-5942 110 24 = = SYM cana-5942 110 25 0 0 NUM cana-5942 110 26 at at ADP cana-5942 110 27 𝑧 𝑧 PROPN cana-5942 110 28 = = SYM cana-5942 110 29 0 0 NUM cana-5942 110 30 ( ( PUNCT cana-5942 110 31 26 26 NUM cana-5942 110 32 ) ) PUNCT cana-5942 110 33 the the DET cana-5942 110 34 above above ADJ cana-5942 110 35 boundary boundary ADJ cana-5942 110 36 conditions condition NOUN cana-5942 110 37 are be AUX cana-5942 110 38 identically identically ADV cana-5942 110 39 satisfied satisfied ADJ cana-5942 110 40 if if SCONJ cana-5942 110 41 and and CCONJ cana-5942 110 42 only only ADV cana-5942 110 43 if if SCONJ cana-5942 110 44 π‘˜1 π‘˜1 ADJ cana-5942 110 45 𝑠𝑖𝑛 𝑠𝑖𝑛 ADJ cana-5942 110 46 πœƒ1 πœƒ1 NOUN cana-5942 110 47 = = SYM cana-5942 110 48 π‘˜2 π‘˜2 PROPN cana-5942 110 49 𝑠𝑖𝑛 𝑠𝑖𝑛 ADJ cana-5942 111 1 πœƒ2 πœƒ2 PROPN cana-5942 111 2 = = SYM cana-5942 111 3 π‘˜3 π‘˜3 PROPN cana-5942 111 4 𝑠𝑖𝑛 𝑠𝑖𝑛 VERB cana-5942 111 5 πœƒ3 πœƒ3 ADJ cana-5942 111 6 = = SYM cana-5942 111 7 π‘˜4 π‘˜4 PROPN cana-5942 111 8 𝑠𝑖𝑛 𝑠𝑖𝑛 NOUN cana-5942 111 9 πœƒ4 πœƒ4 PROPN cana-5942 111 10 and and CCONJ cana-5942 111 11 π‘˜1𝑉1 π‘˜1𝑉1 SYM cana-5942 111 12 = = SYM cana-5942 111 13 π‘˜2𝑉2 π‘˜2𝑉2 PROPN cana-5942 111 14 = = SYM cana-5942 111 15 π‘˜3𝑉3 π‘˜3𝑉3 PROPN cana-5942 111 16 = = SYM cana-5942 111 17 π‘˜4𝑉4 π‘˜4𝑉4 PROPN cana-5942 111 18 . . PUNCT cana-5942 112 1 ( ( PUNCT cana-5942 112 2 27 27 NUM cana-5942 112 3 ) ) PUNCT cana-5942 112 4 now now ADV cana-5942 112 5 , , PUNCT cana-5942 112 6 substituting substitute VERB cana-5942 112 7 the the DET cana-5942 112 8 values value NOUN cana-5942 112 9 of of ADP cana-5942 112 10 potentials potential NOUN cana-5942 112 11 πœ“1 πœ“1 NOUN cana-5942 112 12 , , PUNCT cana-5942 112 13 𝛩 𝛩 PROPN cana-5942 112 14 , , PUNCT cana-5942 112 15 πœ™ πœ™ NOUN cana-5942 112 16 and and CCONJ cana-5942 112 17 πœ“2 πœ“2 NOUN cana-5942 112 18 from from ADP cana-5942 112 19 eqs.(22)-(25 eqs.(22)-(25 NUM cana-5942 112 20 ) ) PUNCT cana-5942 112 21 into into ADP cana-5942 112 22 the the DET cana-5942 112 23 above above ADJ cana-5942 112 24 boundary boundary ADJ cana-5942 112 25 conditions condition NOUN cana-5942 112 26 one one PRON cana-5942 112 27 can can AUX cana-5942 112 28 obtain obtain VERB cana-5942 112 29 the the DET cana-5942 112 30 following follow VERB cana-5942 112 31 system system NOUN cana-5942 112 32 of of ADP cana-5942 112 33 four four NUM cana-5942 112 34 simultaneous simultaneous ADJ cana-5942 112 35 equations equation NOUN cana-5942 112 36 βˆ‘ βˆ‘ PUNCT cana-5942 113 1 𝐴𝑖𝑗𝑍𝑗 𝐴𝑖𝑗𝑍𝑗 PROPN cana-5942 113 2 = = SYM cana-5942 113 3 𝐢𝑖 𝐢𝑖 PROPN cana-5942 113 4 , , PUNCT cana-5942 113 5 ( ( PUNCT cana-5942 113 6 𝑖 𝑖 SYM cana-5942 113 7 , , PUNCT cana-5942 113 8 𝑗 𝑗 NOUN cana-5942 113 9 = = NOUN cana-5942 113 10 1,2,3,4 1,2,3,4 NUM cana-5942 113 11 ) ) PUNCT cana-5942 113 12 , , PUNCT cana-5942 113 13 ( ( PUNCT cana-5942 113 14 28 28 NUM cana-5942 113 15 ) ) PUNCT cana-5942 113 16 where where SCONJ cana-5942 114 1 𝐴1𝑗 𝐴1𝑗 PROPN cana-5942 114 2 = = PUNCT cana-5942 115 1 [ [ X cana-5942 115 2 π‘Ž12 π‘Ž12 NOUN cana-5942 115 3 βˆ’ βˆ’ PROPN cana-5942 115 4 𝛿4 𝛿4 NOUN cana-5942 115 5 + + CCONJ cana-5942 115 6 π‘Ž13 π‘Ž13 VERB cana-5942 115 7 π‘π‘œπ‘ 2 π‘π‘œπ‘ 2 NOUN cana-5942 115 8 πœƒπ‘— πœƒπ‘— ADP cana-5942 116 1 + + CCONJ cana-5942 116 2 𝑏1 𝑏1 ADJ cana-5942 116 3 𝑏𝑗 𝑏𝑗 NOUN cana-5942 116 4 βˆ— βˆ— NOUN cana-5942 116 5 π‘˜π‘— π‘˜π‘— PROPN cana-5942 116 6 2 2 NUM cana-5942 116 7 + + NUM cana-5942 116 8 π‘Ž14 π‘Ž14 NOUN cana-5942 116 9 π‘Žπ‘— π‘Žπ‘— ADP cana-5942 116 10 βˆ— βˆ— NOUN cana-5942 116 11 π‘˜π‘— π‘˜π‘— PROPN cana-5942 117 1 2 2 NUM cana-5942 117 2 ] ] X cana-5942 117 3 π‘˜π‘— π‘˜π‘— PROPN cana-5942 117 4 2 2 NUM cana-5942 117 5 π‘˜1 π‘˜1 PROPN cana-5942 117 6 2 2 NUM cana-5942 117 7 , , PUNCT cana-5942 117 8 𝐴2𝑗 𝐴2𝑗 PROPN cana-5942 117 9 = = PUNCT cana-5942 117 10 𝑠𝑖𝑛 𝑠𝑖𝑛 ADJ cana-5942 117 11 2 2 NUM cana-5942 117 12 πœƒπ‘— πœƒπ‘— NOUN cana-5942 117 13 π‘˜π‘— π‘˜π‘— PROPN cana-5942 117 14 2 2 NUM cana-5942 117 15 π‘˜1 π‘˜1 PROPN cana-5942 117 16 2 2 NUM cana-5942 117 17 , , PUNCT cana-5942 117 18 𝐴3𝑗 𝐴3𝑗 PROPN cana-5942 117 19 = = PUNCT cana-5942 117 20 π‘Žπ‘— π‘Žπ‘— ADP cana-5942 117 21 βˆ— βˆ— NOUN cana-5942 117 22 π‘π‘œπ‘  π‘π‘œπ‘  NOUN cana-5942 117 23 πœƒπ‘— πœƒπ‘— VERB cana-5942 118 1 π‘˜π‘— π‘˜π‘— INTJ cana-5942 118 2 π‘˜π‘— π‘˜π‘— PROPN cana-5942 118 3 2 2 NUM cana-5942 118 4 π‘˜1 π‘˜1 PROPN cana-5942 118 5 2 2 NUM cana-5942 118 6 , , PUNCT cana-5942 118 7 𝐴4𝑗 𝐴4𝑗 PROPN cana-5942 118 8 = = SYM cana-5942 118 9 𝑏𝑗 𝑏𝑗 PROPN cana-5942 118 10 βˆ— βˆ— NOUN cana-5942 118 11 π‘π‘œπ‘  π‘π‘œπ‘  NOUN cana-5942 118 12 πœƒπ‘— πœƒπ‘— VERB cana-5942 119 1 π‘˜π‘— π‘˜π‘— INTJ cana-5942 119 2 π‘˜π‘— π‘˜π‘— PROPN cana-5942 119 3 2 2 NUM cana-5942 119 4 π‘˜1 π‘˜1 PROPN cana-5942 119 5 2 2 NUM cana-5942 119 6 . . PUNCT cana-5942 120 1 ( ( PUNCT cana-5942 120 2 𝑗 𝑗 NOUN cana-5942 120 3 = = NOUN cana-5942 120 4 1,2,3 1,2,3 NUM cana-5942 120 5 ) ) PUNCT cana-5942 120 6 . . PUNCT cana-5942 121 1 𝐴14 𝐴14 PROPN cana-5942 121 2 = = PRON cana-5942 121 3 π‘Ž13 π‘Ž13 VERB cana-5942 121 4 𝑠𝑖𝑛 𝑠𝑖𝑛 PROPN cana-5942 121 5 πœƒ4 πœƒ4 PROPN cana-5942 121 6 π‘π‘œπ‘  π‘π‘œπ‘  NOUN cana-5942 121 7 πœƒ4 πœƒ4 PROPN cana-5942 121 8 π‘˜4 π‘˜4 PROPN cana-5942 121 9 2 2 NUM cana-5942 121 10 π‘˜1 π‘˜1 PROPN cana-5942 121 11 2 2 NUM cana-5942 121 12 , , PUNCT cana-5942 121 13 𝐴24 𝐴24 VERB cana-5942 121 14 = = SYM cana-5942 121 15 βˆ’ βˆ’ NOUN cana-5942 121 16 π‘π‘œπ‘  π‘π‘œπ‘  NOUN cana-5942 121 17 2 2 NUM cana-5942 121 18 πœƒ4 πœƒ4 PROPN cana-5942 121 19 π‘˜4 π‘˜4 PROPN cana-5942 121 20 2 2 NUM cana-5942 121 21 π‘˜1 π‘˜1 PROPN cana-5942 121 22 2 2 NUM cana-5942 121 23 , , PUNCT cana-5942 121 24 𝐴34 𝐴34 PROPN cana-5942 121 25 = = PUNCT cana-5942 121 26 0 0 NUM cana-5942 121 27 , , PUNCT cana-5942 121 28 𝐴44 𝐴44 PROPN cana-5942 121 29 = = SYM cana-5942 121 30 0 0 NUM cana-5942 121 31 , , PUNCT cana-5942 121 32 communications communication NOUN cana-5942 121 33 on on ADP cana-5942 121 34 applied apply VERB cana-5942 121 35 nonlinear nonlinear ADJ cana-5942 121 36 analysis analysis NOUN cana-5942 121 37 issn issn NOUN cana-5942 121 38 : : PUNCT cana-5942 121 39 1074 1074 NUM cana-5942 121 40 - - PUNCT cana-5942 121 41 133x 133x NUM cana-5942 121 42 vol vol NOUN cana-5942 121 43 31 31 NUM cana-5942 121 44 no no NOUN cana-5942 121 45 . . PUNCT cana-5942 122 1 8s 8s PROPN cana-5942 122 2 ( ( PUNCT cana-5942 122 3 2024 2024 NUM cana-5942 122 4 ) ) PUNCT cana-5942 122 5 1135 1135 NUM cana-5942 122 6 https://internationalpubls.com https://internationalpubls.com X cana-5942 122 7 π‘Ž12 π‘Ž12 NOUN cana-5942 122 8 = = SYM cana-5942 122 9 βˆ’πœ”π›Ώ2 βˆ’πœ”π›Ώ2 NOUN cana-5942 122 10 , , PUNCT cana-5942 122 11 π‘Ž13 π‘Ž13 NOUN cana-5942 122 12 = = SYM cana-5942 122 13 βˆ’2𝛿3 βˆ’2𝛿3 NOUN cana-5942 122 14 , , PUNCT cana-5942 122 15 π‘Ž14 π‘Ž14 NOUN cana-5942 122 16 = = SYM cana-5942 122 17 πœ„πœ”π›½00 πœ„πœ”π›½00 NOUN cana-5942 122 18 , , PUNCT cana-5942 122 19 𝐢1 𝐢1 NOUN cana-5942 122 20 = = SYM cana-5942 122 21 βˆ’π΄11 βˆ’π΄11 PROPN cana-5942 122 22 , , PUNCT cana-5942 122 23 𝐢2 𝐢2 X cana-5942 122 24 = = SYM cana-5942 122 25 𝐴21 𝐴21 PROPN cana-5942 122 26 , , PUNCT cana-5942 122 27 𝐢3 𝐢3 PROPN cana-5942 122 28 = = SYM cana-5942 122 29 𝐴31 𝐴31 PROPN cana-5942 122 30 , , PUNCT cana-5942 122 31 𝐢4 𝐢4 NOUN cana-5942 122 32 = = SYM cana-5942 122 33 𝐴41 𝐴41 PROPN cana-5942 122 34 , , PUNCT cana-5942 122 35 𝑍1 𝑍1 X cana-5942 122 36 = = SYM cana-5942 122 37 𝐴1 𝐴1 PROPN cana-5942 122 38 𝐴0 𝐴0 PROPN cana-5942 122 39 , , PUNCT cana-5942 122 40 𝑍2 𝑍2 PROPN cana-5942 122 41 = = PUNCT cana-5942 122 42 𝐴2 𝐴2 PROPN cana-5942 122 43 𝐴0 𝐴0 PROPN cana-5942 122 44 , , PUNCT cana-5942 122 45 𝑍3 𝑍3 PROPN cana-5942 122 46 = = SYM cana-5942 122 47 𝐴3 𝐴3 PROPN cana-5942 122 48 𝐴0 𝐴0 PROPN cana-5942 122 49 , , PUNCT cana-5942 122 50 𝑍4 𝑍4 PROPN cana-5942 122 51 = = PUNCT cana-5942 122 52 𝐡1 𝐡1 PROPN cana-5942 122 53 𝐴0 𝐴0 PROPN cana-5942 122 54 . . PUNCT cana-5942 123 1 here here ADV cana-5942 123 2 , , PUNCT cana-5942 123 3 𝑍𝑖 𝑍𝑖 PROPN cana-5942 123 4 ( ( PUNCT cana-5942 123 5 𝑖 𝑖 NOUN cana-5942 123 6 = = NOUN cana-5942 123 7 1,2,3,4 1,2,3,4 NUM cana-5942 123 8 ) ) PUNCT cana-5942 123 9 are be AUX cana-5942 123 10 the the DET cana-5942 123 11 reflection reflection NOUN cana-5942 123 12 coefficients coefficient NOUN cana-5942 123 13 for for ADP cana-5942 123 14 the the DET cana-5942 123 15 incidence incidence NOUN cana-5942 123 16 of of ADP cana-5942 123 17 coupled couple VERB cana-5942 123 18 longitudinal longitudinal ADJ cana-5942 123 19 wave wave NOUN cana-5942 123 20 travelling travel VERB cana-5942 123 21 with with ADP cana-5942 123 22 speed speed NOUN cana-5942 123 23 v1 v1 NOUN cana-5942 123 24 . . PUNCT cana-5942 124 1 numerical numerical ADJ cana-5942 124 2 results result NOUN cana-5942 124 3 and and CCONJ cana-5942 124 4 discussion discussion NOUN cana-5942 124 5 following follow VERB cana-5942 124 6 dhaliwal dhaliwal PROPN cana-5942 124 7 and and CCONJ cana-5942 124 8 singh singh PROPN cana-5942 124 9 ( ( PUNCT cana-5942 124 10 1980 1980 NUM cana-5942 124 11 ) ) PUNCT cana-5942 124 12 , , PUNCT cana-5942 124 13 we we PRON cana-5942 124 14 consider consider VERB cana-5942 124 15 an an DET cana-5942 124 16 example example NOUN cana-5942 124 17 where where SCONJ cana-5942 124 18 magnesium magnesium NOUN cana-5942 124 19 crystal crystal NOUN cana-5942 124 20 like like ADP cana-5942 124 21 material material NOUN cana-5942 124 22 is be AUX cana-5942 124 23 modeled model VERB cana-5942 124 24 as as ADP cana-5942 124 25 an an DET cana-5942 124 26 isotropic isotropic ADJ cana-5942 124 27 thermoelastic thermoelastic ADJ cana-5942 124 28 solid solid ADJ cana-5942 124 29 with with ADP cana-5942 124 30 voids voids NOUN cana-5942 124 31 for for ADP cana-5942 124 32 numerical numerical ADJ cana-5942 124 33 computations computation NOUN cana-5942 124 34 . . PUNCT cana-5942 125 1 for for ADP cana-5942 125 2 this this DET cana-5942 125 3 test test NOUN cana-5942 125 4 material material NOUN cana-5942 125 5 , , PUNCT cana-5942 125 6 elastic elastic ADJ cana-5942 125 7 and and CCONJ cana-5942 125 8 thermal thermal ADJ cana-5942 125 9 constants constant NOUN cana-5942 125 10 are be AUX cana-5942 125 11 πœ† πœ† DET cana-5942 125 12 = = SYM cana-5942 125 13 2.17 2.17 NUM cana-5942 125 14 Γ— Γ— NOUN cana-5942 125 15 1010π‘π‘šβˆ’2 1010π‘π‘šβˆ’2 NUM cana-5942 125 16 , , PUNCT cana-5942 125 17     SPACE cana-5942 125 18 πœ‡ πœ‡ ADP cana-5942 125 19 = = NOUN cana-5942 125 20 3.278 3.278 NUM cana-5942 125 21 Γ— Γ— NOUN cana-5942 125 22 1010π‘π‘šβˆ’2 1010π‘π‘šβˆ’2 NUM cana-5942 125 23 , , PUNCT cana-5942 125 24 𝛽 𝛽 NOUN cana-5942 125 25 = = SYM cana-5942 125 26 2.68 2.68 NUM cana-5942 125 27 Γ— Γ— NOUN cana-5942 125 28 106 106 NUM cana-5942 125 29 π‘π‘šβˆ’2 π‘π‘šβˆ’2 NOUN cana-5942 125 30 π‘‘π‘’π‘”π‘Ÿπ‘’π‘’βˆ’1 π‘‘π‘’π‘”π‘Ÿπ‘’π‘’βˆ’1 PRON cana-5942 125 31     SPACE cana-5942 125 32 , , PUNCT cana-5942 125 33     SPACE cana-5942 125 34 𝑇0 𝑇0 NOUN cana-5942 125 35 = = SYM cana-5942 125 36 298𝐾 298𝐾 NUM cana-5942 125 37 , , PUNCT cana-5942 125 38 𝐾 𝐾 NOUN cana-5942 125 39 = = PROPN cana-5942 125 40 1.7 1.7 NUM cana-5942 125 41 Γ— Γ— NOUN cana-5942 125 42 102π‘Šπ‘šβˆ’1 102π‘Šπ‘šβˆ’1 NUM cana-5942 125 43 π‘‘π‘’π‘”π‘Ÿπ‘’π‘’βˆ’1 π‘‘π‘’π‘”π‘Ÿπ‘’π‘’βˆ’1 NOUN cana-5942 125 44 , , PUNCT cana-5942 125 45     SPACE cana-5942 125 46 𝑐𝑒 𝑐𝑒 X cana-5942 125 47 = = NOUN cana-5942 125 48 1.04 1.04 NUM cana-5942 125 49 Γ— Γ— NOUN cana-5942 125 50 103π½πΎπ‘”βˆ’1 103π½πΎπ‘”βˆ’1 NUM cana-5942 125 51 π‘‘π‘’π‘”π‘Ÿπ‘’π‘’βˆ’1 π‘‘π‘’π‘”π‘Ÿπ‘’π‘’βˆ’1 NOUN cana-5942 125 52 , , PUNCT cana-5942 125 53     SPACE cana-5942 125 54 𝜌 𝜌 X cana-5942 125 55 = = SYM cana-5942 125 56 1.74 1.74 NUM cana-5942 125 57 Γ— Γ— NOUN cana-5942 125 58 103πΎπ‘”π‘šβˆ’3 103πΎπ‘”π‘šβˆ’3 NUM cana-5942 125 59 . . PUNCT cana-5942 126 1 void void ADJ cana-5942 126 2 parameters parameter NOUN cana-5942 126 3 are be AUX cana-5942 126 4 given give VERB cana-5942 126 5 by by ADP cana-5942 126 6 𝛼 𝛼 NOUN cana-5942 126 7 = = NOUN cana-5942 126 8 3.688 3.688 NUM cana-5942 126 9 Γ— Γ— NOUN cana-5942 126 10 10βˆ’5𝑁 10βˆ’5𝑁 NUM cana-5942 126 11 , , PUNCT cana-5942 126 12         SPACE cana-5942 126 13 πœ‰1 πœ‰1 NOUN cana-5942 126 14 = = PROPN cana-5942 126 15 1.475 1.475 NUM cana-5942 126 16 Γ— Γ— PROPN cana-5942 126 17 1010π‘π‘šβˆ’2 1010π‘π‘šβˆ’2 PROPN cana-5942 126 18 , , PUNCT cana-5942 126 19       SPACE cana-5942 126 20 πœ’ πœ’ X cana-5942 126 21 = = SYM cana-5942 126 22 1.753 1.753 NUM cana-5942 126 23 Γ— Γ— NOUN cana-5942 126 24 10βˆ’15π‘š2 10βˆ’15π‘š2 NUM cana-5942 126 25 , , PUNCT cana-5942 126 26 𝑏 𝑏 PROPN cana-5942 126 27 = = SYM cana-5942 126 28 1.13849 1.13849 NUM cana-5942 126 29 Γ— Γ— NOUN cana-5942 126 30 1010π‘π‘šβˆ’2 1010π‘π‘šβˆ’2 NUM cana-5942 126 31 , , PUNCT cana-5942 126 32         SPACE cana-5942 126 33 π‘š π‘š NOUN cana-5942 126 34 = = SYM cana-5942 126 35 2 2 NUM cana-5942 126 36 Γ— Γ— PROPN cana-5942 126 37 106π‘π‘šβˆ’2 106π‘π‘šβˆ’2 NUM cana-5942 126 38 π‘‘π‘’π‘”π‘Ÿπ‘’π‘’βˆ’1 π‘‘π‘’π‘”π‘Ÿπ‘’π‘’βˆ’1 NOUN cana-5942 126 39 . . PUNCT cana-5942 127 1 other other ADJ cana-5942 127 2 constants constant NOUN cana-5942 127 3 involved involve VERB cana-5942 127 4 in in ADP cana-5942 127 5 the the DET cana-5942 127 6 problem problem NOUN cana-5942 127 7 are be AUX cana-5942 127 8 : : PUNCT cana-5942 127 9 𝜏𝜐 𝜏𝜐 INTJ cana-5942 127 10 = = NOUN cana-5942 127 11 0.1 0.1 NUM cana-5942 127 12 , , PUNCT cana-5942 127 13     SPACE cana-5942 127 14 πœπ‘ž πœπ‘ž NOUN cana-5942 127 15 = = NUM cana-5942 127 16 0.2 0.2 NUM cana-5942 127 17 , , PUNCT cana-5942 127 18     SPACE cana-5942 127 19 πœπ‘‡ πœπ‘‡ VERB cana-5942 127 20 = = PUNCT cana-5942 127 21 0.15 0.15 NUM cana-5942 127 22 , , PUNCT cana-5942 127 23     SPACE cana-5942 127 24 πœ” πœ” ADP cana-5942 127 25 = = NUM cana-5942 127 26 3.5 3.5 NUM cana-5942 127 27 Γ— Γ— NOUN cana-5942 127 28 1012 1012 NUM cana-5942 127 29 . . PUNCT cana-5942 128 1 to to PART cana-5942 128 2 discuss discuss VERB cana-5942 128 3 the the DET cana-5942 128 4 nature nature NOUN cana-5942 128 5 of of ADP cana-5942 128 6 dependence dependence NOUN cana-5942 128 7 of of ADP cana-5942 128 8 reflection reflection NOUN cana-5942 128 9 coefficients coefficient NOUN cana-5942 128 10 on on ADP cana-5942 128 11 the the DET cana-5942 128 12 angle angle NOUN cana-5942 128 13 of of ADP cana-5942 128 14 incidence incidence NOUN cana-5942 128 15 , , PUNCT cana-5942 128 16 we we PRON cana-5942 128 17 have have AUX cana-5942 128 18 computed compute VERB cana-5942 128 19 their their PRON cana-5942 128 20 expressions expression NOUN cana-5942 128 21 using use VERB cana-5942 128 22 matlab matlab PROPN cana-5942 128 23 software software NOUN cana-5942 128 24 . . PUNCT cana-5942 129 1 all all DET cana-5942 129 2 the the DET cana-5942 129 3 relative relative ADJ cana-5942 129 4 amplitudes amplitude NOUN cana-5942 129 5 are be AUX cana-5942 129 6 found find VERB cana-5942 129 7 to to PART cana-5942 129 8 be be AUX cana-5942 129 9 complex complex ADJ cana-5942 129 10 valued value VERB cana-5942 129 11 . . PUNCT cana-5942 130 1 all all DET cana-5942 130 2 the the DET cana-5942 130 3 figures figure NOUN cana-5942 130 4 have have AUX cana-5942 130 5 been be AUX cana-5942 130 6 taken take VERB cana-5942 130 7 in in ADP cana-5942 130 8 the the DET cana-5942 130 9 context context NOUN cana-5942 130 10 of of ADP cana-5942 130 11 thermoelastic thermoelastic ADJ cana-5942 130 12 theory theory NOUN cana-5942 130 13 based base VERB cana-5942 130 14 on on ADP cana-5942 130 15 : : PUNCT cana-5942 130 16 ( ( PUNCT cana-5942 130 17 i i NOUN cana-5942 130 18 ) ) PUNCT cana-5942 130 19 three three NUM cana-5942 130 20 - - PUNCT cana-5942 130 21 phase phase NOUN cana-5942 130 22 - - PUNCT cana-5942 130 23 lag lag NOUN cana-5942 130 24 model model NOUN cana-5942 130 25 with with ADP cana-5942 130 26 voids voids NOUN cana-5942 130 27 ( ( PUNCT cana-5942 130 28 3plv 3plv NUM cana-5942 130 29 ) ) PUNCT cana-5942 130 30 ( ( PUNCT cana-5942 130 31 ii ii NOUN cana-5942 130 32 ) ) PUNCT cana-5942 130 33 gn gn PROPN cana-5942 130 34 - - PUNCT cana-5942 130 35 iii iii PROPN cana-5942 130 36 model model NOUN cana-5942 130 37 with with ADP cana-5942 130 38 voids voids NOUN cana-5942 130 39 ( ( PUNCT cana-5942 130 40 gn3v gn3v PROPN cana-5942 130 41 ) ) PUNCT cana-5942 130 42 ( ( PUNCT cana-5942 130 43 iii iii NOUN cana-5942 130 44 ) ) PUNCT cana-5942 130 45 three three NUM cana-5942 130 46 - - PUNCT cana-5942 130 47 phase phase NOUN cana-5942 130 48 - - PUNCT cana-5942 130 49 lag lag NOUN cana-5942 130 50 model model NOUN cana-5942 130 51 without without ADP cana-5942 130 52 voids voids NOUN cana-5942 130 53 ( ( PUNCT cana-5942 130 54 3plwv 3plwv NUM cana-5942 130 55 ) ) PUNCT cana-5942 130 56 . . PUNCT cana-5942 131 1 figures figure NOUN cana-5942 131 2 2 2 NUM cana-5942 131 3 - - SYM cana-5942 131 4 5 5 NUM cana-5942 131 5 are be AUX cana-5942 131 6 drawn draw VERB cana-5942 131 7 for for ADP cana-5942 131 8 incidence incidence NOUN cana-5942 131 9 of of ADP cana-5942 131 10 a a DET cana-5942 131 11 p p NOUN cana-5942 131 12 - - PUNCT cana-5942 131 13 wave wave NOUN cana-5942 131 14 of of ADP cana-5942 131 15 speed speed NOUN cana-5942 131 16 𝑉1 𝑉1 NOUN cana-5942 131 17 . . PUNCT cana-5942 132 1 considered consider VERB cana-5942 132 2 range range NOUN cana-5942 132 3 for for ADP cana-5942 132 4 angle angle NOUN cana-5942 132 5 of of ADP cana-5942 132 6 incidence incidence NOUN cana-5942 132 7 is be AUX cana-5942 132 8 0∘ 0∘ VERB cana-5942 132 9 ≀ ≀ NUM cana-5942 132 10 πœƒ πœƒ NUM cana-5942 132 11 ≀ ≀ NUM cana-5942 132 12 90∘. 90∘. NUM cana-5942 132 13 figure figure NOUN cana-5942 132 14 2 2 NUM cana-5942 132 15 shows show VERB cana-5942 132 16 the the DET cana-5942 132 17 variations variation NOUN cana-5942 132 18 in in ADP cana-5942 132 19 the the DET cana-5942 132 20 absolute absolute ADJ cana-5942 132 21 values value NOUN cana-5942 132 22 of of ADP cana-5942 132 23 reflection reflection NOUN cana-5942 132 24 coefficient coefficient NOUN cana-5942 132 25 𝑍1 𝑍1 PROPN cana-5942 132 26 . . PUNCT cana-5942 133 1 it it PRON cana-5942 133 2 is be AUX cana-5942 133 3 clear clear ADJ cana-5942 133 4 that that SCONJ cana-5942 133 5 |𝑍1|has |𝑍1|ha VERB cana-5942 133 6 value value NOUN cana-5942 133 7 almost almost ADV cana-5942 133 8 equal equal ADJ cana-5942 133 9 to to ADP cana-5942 133 10 unity unity NOUN cana-5942 133 11 during during ADP cana-5942 133 12 the the DET cana-5942 133 13 whole whole ADJ cana-5942 133 14 range range NOUN cana-5942 133 15 of of ADP cana-5942 133 16 incidence incidence NOUN cana-5942 133 17 for for ADP cana-5942 133 18 all all DET cana-5942 133 19 the the DET cana-5942 133 20 three three NUM cana-5942 133 21 models model NOUN cana-5942 133 22 considered consider VERB cana-5942 133 23 . . PUNCT cana-5942 134 1 despite despite SCONJ cana-5942 134 2 of of ADP cana-5942 134 3 this this PRON cana-5942 134 4 , , PUNCT cana-5942 134 5 difference difference NOUN cana-5942 134 6 in in ADP cana-5942 134 7 trends trend NOUN cana-5942 134 8 of of ADP cana-5942 134 9 variations variation NOUN cana-5942 134 10 of of ADP cana-5942 134 11 |𝑍1| |𝑍1| PROPN cana-5942 134 12 for for ADP cana-5942 134 13 3plv 3plv NUM cana-5942 134 14 , , PUNCT cana-5942 134 15 gn3v gn3v PROPN cana-5942 134 16 and and CCONJ cana-5942 134 17 3plwv 3plwv NUM cana-5942 134 18 models model NOUN cana-5942 134 19 is be AUX cana-5942 134 20 easily easily ADV cana-5942 134 21 noticeable noticeable ADJ cana-5942 134 22 from from ADP cana-5942 134 23 the the DET cana-5942 134 24 graph graph NOUN cana-5942 134 25 . . PUNCT cana-5942 135 1 presence presence NOUN cana-5942 135 2 of of ADP cana-5942 135 3 voids voids NOUN cana-5942 135 4 increases increase VERB cana-5942 135 5 the the DET cana-5942 135 6 values value NOUN cana-5942 135 7 of of ADP cana-5942 135 8 |𝑍1|in |𝑍1|in PRON cana-5942 135 9 the the DET cana-5942 135 10 entire entire ADJ cana-5942 135 11 range range NOUN cana-5942 135 12 of of ADP cana-5942 135 13 angle angle NOUN cana-5942 135 14 of of ADP cana-5942 135 15 incidence incidence NOUN cana-5942 135 16 . . PUNCT cana-5942 136 1 absence absence NOUN cana-5942 136 2 of of ADP cana-5942 136 3 three three NUM cana-5942 136 4 phase phase NOUN cana-5942 136 5 lags lag VERB cana-5942 136 6 increases increase VERB cana-5942 136 7 the the DET cana-5942 136 8 values value NOUN cana-5942 136 9 till till SCONJ cana-5942 136 10 πœƒ πœƒ NOUN cana-5942 136 11 = = SYM cana-5942 136 12 45∘and 45∘and NOUN cana-5942 136 13 after after SCONJ cana-5942 136 14 that that PRON cana-5942 136 15 causes cause VERB cana-5942 136 16 a a DET cana-5942 136 17 decrement decrement NOUN cana-5942 136 18 in in ADP cana-5942 136 19 the the DET cana-5942 136 20 values value NOUN cana-5942 136 21 . . PUNCT cana-5942 137 1 communications communication NOUN cana-5942 137 2 on on ADP cana-5942 137 3 applied apply VERB cana-5942 137 4 nonlinear nonlinear ADJ cana-5942 137 5 analysis analysis NOUN cana-5942 137 6 issn issn NOUN cana-5942 137 7 : : PUNCT cana-5942 137 8 1074 1074 NUM cana-5942 137 9 - - PUNCT cana-5942 137 10 133x 133x NUM cana-5942 137 11 vol vol NOUN cana-5942 137 12 31 31 NUM cana-5942 137 13 no no NOUN cana-5942 137 14 . . PUNCT cana-5942 138 1 8s 8s PROPN cana-5942 138 2 ( ( PUNCT cana-5942 138 3 2024 2024 NUM cana-5942 138 4 ) ) PUNCT cana-5942 138 5 1136 1136 NUM cana-5942 138 6 https://internationalpubls.com https://internationalpubls.com X cana-5942 139 1 in in ADP cana-5942 139 2 figure figure NOUN cana-5942 139 3 3 3 NUM cana-5942 139 4 , , PUNCT cana-5942 139 5 the the DET cana-5942 139 6 reflection reflection NOUN cana-5942 139 7 coefficients coefficient NOUN cana-5942 139 8 |𝑍2| |𝑍2| X cana-5942 139 9 decrease decrease NOUN cana-5942 139 10 with with ADP cana-5942 139 11 increasing increase VERB cana-5942 139 12 angle angle NOUN cana-5942 139 13 of of ADP cana-5942 139 14 incidence incidence NOUN cana-5942 139 15 , , PUNCT cana-5942 139 16 which which PRON cana-5942 139 17 are be AUX cana-5942 139 18 affected affect VERB cana-5942 139 19 due due ADJ cana-5942 139 20 to to PART cana-5942 139 21 void void VERB cana-5942 139 22 parameters parameter NOUN cana-5942 139 23 and and CCONJ cana-5942 139 24 relaxation relaxation NOUN cana-5942 139 25 times time NOUN cana-5942 139 26 . . PUNCT cana-5942 140 1 in in ADP cana-5942 140 2 the the DET cana-5942 140 3 absence absence NOUN cana-5942 140 4 of of ADP cana-5942 140 5 voids voids NOUN cana-5942 140 6 and and CCONJ cana-5942 140 7 three three NUM cana-5942 140 8 relaxation relaxation NOUN cana-5942 140 9 times time NOUN cana-5942 140 10 πœπ‘ž,𝜏𝜐andπœπ‘‡ πœπ‘ž,𝜏𝜐andπœπ‘‡ PROPN cana-5942 140 11 , , PUNCT cana-5942 140 12 the the DET cana-5942 140 13 values value NOUN cana-5942 140 14 of of ADP cana-5942 140 15 |𝑍2| |𝑍2| X cana-5942 140 16 rise rise VERB cana-5942 140 17 at at ADP cana-5942 140 18 each each DET cana-5942 140 19 angle angle NOUN cana-5942 140 20 of of ADP cana-5942 140 21 incidence incidence NOUN cana-5942 140 22 . . PUNCT cana-5942 141 1 variations variation NOUN cana-5942 141 2 in in ADP cana-5942 141 3 the the DET cana-5942 141 4 values value NOUN cana-5942 141 5 of of ADP cana-5942 141 6 |𝑍3| |𝑍3| NOUN cana-5942 141 7 against against ADP cana-5942 141 8 the the DET cana-5942 141 9 angle angle NOUN cana-5942 141 10 of of ADP cana-5942 141 11 incidence incidence NOUN cana-5942 141 12 are be AUX cana-5942 141 13 depicted depict VERB cana-5942 141 14 in in ADP cana-5942 141 15 the the DET cana-5942 141 16 figure figure NOUN cana-5942 141 17 4 4 NUM cana-5942 141 18 . . NOUN cana-5942 141 19 magnitude magnitude NOUN cana-5942 141 20 of of ADP cana-5942 141 21 𝑍3 𝑍3 PROPN cana-5942 141 22 is be AUX cana-5942 141 23 very very ADV cana-5942 141 24 small small ADJ cana-5942 141 25 for for ADP cana-5942 141 26 all all DET cana-5942 141 27 the the DET cana-5942 141 28 three three NUM cana-5942 141 29 cases case NOUN cana-5942 141 30 . . PUNCT cana-5942 142 1 also also ADV cana-5942 142 2 pattern pattern NOUN cana-5942 142 3 of of ADP cana-5942 142 4 variations variation NOUN cana-5942 142 5 in in ADP cana-5942 142 6 the the DET cana-5942 142 7 graph graph NOUN cana-5942 142 8 is be AUX cana-5942 142 9 similar similar ADJ cana-5942 142 10 to to ADP cana-5942 142 11 that that PRON cana-5942 142 12 for for ADP cana-5942 142 13 |𝑍2| |𝑍2| X cana-5942 142 14 . . PUNCT cana-5942 143 1 figure figure NOUN cana-5942 143 2 5 5 NUM cana-5942 143 3 characterizes characterize VERB cana-5942 143 4 the the DET cana-5942 143 5 behaviour behaviour NOUN cana-5942 143 6 of of ADP cana-5942 143 7 |𝑍4|with |𝑍4|with PROPN cana-5942 143 8 increasing increase VERB cana-5942 143 9 angle angle NOUN cana-5942 143 10 of of ADP cana-5942 143 11 incidence incidence NOUN cana-5942 143 12 . . PUNCT cana-5942 144 1 its its PRON cana-5942 144 2 trend trend NOUN cana-5942 144 3 of of ADP cana-5942 144 4 variations variation NOUN cana-5942 144 5 is be AUX cana-5942 144 6 different different ADJ cana-5942 144 7 from from ADP cana-5942 144 8 the the DET cana-5942 144 9 remaining remain VERB cana-5942 144 10 reflection reflection NOUN cana-5942 144 11 coefficients coefficient NOUN cana-5942 144 12 . . PUNCT cana-5942 145 1 |𝑍4| |𝑍4| X cana-5942 145 2 starts start VERB cana-5942 145 3 from from ADP cana-5942 145 4 zero zero NUM cana-5942 145 5 value value NOUN cana-5942 145 6 , , PUNCT cana-5942 145 7 increases increase NOUN cana-5942 145 8 till till SCONJ cana-5942 145 9 it it PRON cana-5942 145 10 attains attain VERB cana-5942 145 11 its its PRON cana-5942 145 12 maximum maximum ADJ cana-5942 145 13 value value NOUN cana-5942 145 14 at at ADP cana-5942 145 15 πœƒ πœƒ NOUN cana-5942 145 16 = = SYM cana-5942 145 17 45∘and 45∘and NOUN cana-5942 145 18 then then ADV cana-5942 145 19 decreases decrease VERB cana-5942 145 20 smoothly smoothly ADV cana-5942 145 21 to to ADP cana-5942 145 22 zero zero NUM cana-5942 145 23 . . PUNCT cana-5942 146 1 absence absence NOUN cana-5942 146 2 of of ADP cana-5942 146 3 relaxation relaxation NOUN cana-5942 146 4 times time NOUN cana-5942 146 5 ( ( PUNCT cana-5942 146 6 three three NUM cana-5942 146 7 phase phase NOUN cana-5942 146 8 lags lag VERB cana-5942 146 9 ) ) PUNCT cana-5942 146 10 causes cause VERB cana-5942 146 11 an an DET cana-5942 146 12 increment increment NOUN cana-5942 146 13 in in ADP cana-5942 146 14 the the DET cana-5942 146 15 numerical numerical ADJ cana-5942 146 16 values value NOUN cana-5942 146 17 of of ADP cana-5942 146 18 𝑍4 𝑍4 NOUN cana-5942 146 19 while while SCONJ cana-5942 146 20 absence absence NOUN cana-5942 146 21 of of ADP cana-5942 146 22 voids voids NOUN cana-5942 146 23 decreases decrease VERB cana-5942 146 24 the the DET cana-5942 146 25 numerical numerical ADJ cana-5942 146 26 values value NOUN cana-5942 146 27 . . PUNCT cana-5942 147 1 concluding conclude VERB cana-5942 147 2 remarks remark VERB cana-5942 147 3 the the DET cana-5942 147 4 facts fact NOUN cana-5942 147 5 extracted extract VERB cana-5942 147 6 from from ADP cana-5942 147 7 our our PRON cana-5942 147 8 study study NOUN cana-5942 147 9 can can AUX cana-5942 147 10 be be AUX cana-5942 147 11 concluded conclude VERB cana-5942 147 12 as as ADP cana-5942 147 13 : : PUNCT cana-5942 147 14 ( ( PUNCT cana-5942 147 15 i i NOUN cana-5942 147 16 ) ) PUNCT cana-5942 147 17 it it PRON cana-5942 147 18 can can AUX cana-5942 147 19 be be AUX cana-5942 147 20 checked check VERB cana-5942 147 21 out out ADP cana-5942 147 22 from from ADP cana-5942 147 23 the the DET cana-5942 147 24 calculatory calculatory ADJ cana-5942 147 25 part part NOUN cana-5942 147 26 that that SCONJ cana-5942 147 27 amplitude amplitude NOUN cana-5942 147 28 ratios ratio NOUN cana-5942 147 29 are be AUX cana-5942 147 30 independent independent ADJ cana-5942 147 31 of of ADP cana-5942 147 32 the the DET cana-5942 147 33 wavelength wavelength NOUN cana-5942 147 34 of of ADP cana-5942 147 35 the the DET cana-5942 147 36 incident incident NOUN cana-5942 147 37 wave wave NOUN cana-5942 147 38 but but CCONJ cana-5942 147 39 depend depend VERB cana-5942 147 40 only only ADV cana-5942 147 41 upon upon SCONJ cana-5942 147 42 angle angle NOUN cana-5942 147 43 of of ADP cana-5942 147 44 incidence incidence NOUN cana-5942 147 45 and and CCONJ cana-5942 147 46 wave wave NOUN cana-5942 147 47 numbers number NOUN cana-5942 147 48 of of ADP cana-5942 147 49 the the DET cana-5942 147 50 incident incident NOUN cana-5942 147 51 wave wave NOUN cana-5942 147 52 . . PUNCT cana-5942 148 1 ( ( PUNCT cana-5942 148 2 ii ii NOUN cana-5942 148 3 ) ) PUNCT cana-5942 148 4 at at ADP cana-5942 148 5 grazing grazing NOUN cana-5942 148 6 incidence incidence NOUN cana-5942 148 7 ( ( PUNCT cana-5942 148 8 πœƒ πœƒ NOUN cana-5942 148 9 = = PUNCT cana-5942 148 10 90∘ 90∘ NOUN cana-5942 148 11 ) ) PUNCT cana-5942 148 12 of of ADP cana-5942 148 13 longitudinal longitudinal ADJ cana-5942 148 14 wave wave NOUN cana-5942 148 15 of of ADP cana-5942 148 16 speed speed NOUN cana-5942 148 17 𝑉1 𝑉1 NOUN cana-5942 148 18 , , PUNCT cana-5942 148 19 no no DET cana-5942 148 20 other other ADJ cana-5942 148 21 reflected reflect VERB cana-5942 148 22 wave wave NOUN cana-5942 148 23 appears appear VERB cana-5942 148 24 except except SCONJ cana-5942 148 25 the the DET cana-5942 148 26 longitudinal longitudinal ADJ cana-5942 148 27 wave wave NOUN cana-5942 148 28 of of ADP cana-5942 148 29 the the DET cana-5942 148 30 same same ADJ cana-5942 148 31 amplitude amplitude NOUN cana-5942 148 32 as as ADP cana-5942 148 33 that that PRON cana-5942 148 34 of of ADP cana-5942 148 35 incident incident NOUN cana-5942 148 36 wave wave NOUN cana-5942 148 37 . . PUNCT cana-5942 149 1 in in ADP cana-5942 149 2 case case NOUN cana-5942 149 3 of of ADP cana-5942 149 4 normal normal ADJ cana-5942 149 5 incidence incidence NOUN cana-5942 149 6 ( ( PUNCT cana-5942 149 7 πœƒ πœƒ NOUN cana-5942 149 8 = = NOUN cana-5942 149 9 0∘ 0∘ NUM cana-5942 149 10 ) ) PUNCT cana-5942 149 11 , , PUNCT cana-5942 149 12 the the DET cana-5942 149 13 reflected reflect VERB cana-5942 149 14 longitudinal longitudinal ADJ cana-5942 149 15 wave wave NOUN cana-5942 149 16 having have VERB cana-5942 149 17 maximum maximum ADJ cana-5942 149 18 amplitude amplitude NOUN cana-5942 149 19 is be AUX cana-5942 149 20 the the DET cana-5942 149 21 wave wave NOUN cana-5942 149 22 of of ADP cana-5942 149 23 amplitude amplitude NOUN cana-5942 149 24 equal equal ADJ cana-5942 149 25 to to ADP cana-5942 149 26 that that PRON cana-5942 149 27 associated associate VERB cana-5942 149 28 with with ADP cana-5942 149 29 incident incident NOUN cana-5942 149 30 p p NOUN cana-5942 149 31 - - PUNCT cana-5942 149 32 wave wave NOUN cana-5942 149 33 . . PUNCT cana-5942 150 1 thus thus ADV cana-5942 150 2 at at ADP cana-5942 150 3 normal normal ADJ cana-5942 150 4 and and CCONJ cana-5942 150 5 grazing grazing NOUN cana-5942 150 6 incidence incidence NOUN cana-5942 150 7 , , PUNCT cana-5942 150 8 it it PRON cana-5942 150 9 appears appear VERB cana-5942 150 10 that that SCONJ cana-5942 150 11 incident incident NOUN cana-5942 150 12 p p NOUN cana-5942 150 13 - - PUNCT cana-5942 150 14 wave wave NOUN cana-5942 150 15 is be AUX cana-5942 150 16 reflected reflect VERB cana-5942 150 17 as as ADP cana-5942 150 18 a a DET cana-5942 150 19 p p NOUN cana-5942 150 20 - - PUNCT cana-5942 150 21 wave wave NOUN cana-5942 150 22 . . PUNCT cana-5942 151 1 ( ( PUNCT cana-5942 151 2 iv iv X cana-5942 151 3 ) ) PUNCT cana-5942 151 4 voids voids NOUN cana-5942 151 5 and and CCONJ cana-5942 151 6 three three NUM cana-5942 151 7 phase phase NOUN cana-5942 151 8 lag lag NOUN cana-5942 151 9 parameters parameter NOUN cana-5942 151 10 are be AUX cana-5942 151 11 having have VERB cana-5942 151 12 a a DET cana-5942 151 13 pronounced pronounced ADJ cana-5942 151 14 effect effect NOUN cana-5942 151 15 on on ADP cana-5942 151 16 reflection reflection NOUN cana-5942 151 17 coefficients coefficient NOUN cana-5942 151 18 . . PUNCT cana-5942 152 1 references reference NOUN cana-5942 152 2 1 1 NUM cana-5942 152 3 . . PUNCT cana-5942 152 4 abo abo PROPN cana-5942 152 5 - - PUNCT cana-5942 152 6 dahab dahab PROPN cana-5942 152 7 , , PUNCT cana-5942 152 8 s.m s.m PROPN cana-5942 152 9 . . PROPN cana-5942 152 10 , , PUNCT cana-5942 152 11 abd abd PROPN cana-5942 152 12 - - PUNCT cana-5942 152 13 alla alla NOUN cana-5942 152 14 , , PUNCT cana-5942 152 15 a.m. a.m. NOUN cana-5942 152 16 and and CCONJ cana-5942 152 17 mahmoud mahmoud PROPN cana-5942 152 18 , , PUNCT cana-5942 152 19 s.r s.r PROPN cana-5942 152 20 . . PROPN cana-5942 152 21 , , PUNCT cana-5942 152 22 effects effect NOUN cana-5942 152 23 of of ADP cana-5942 152 24 voids voids NOUN cana-5942 152 25 and and CCONJ cana-5942 152 26 rotation rotation NOUN cana-5942 152 27 on on ADP cana-5942 152 28 plane plane NOUN cana-5942 152 29 waves wave NOUN cana-5942 152 30 in in ADP cana-5942 152 31 generalized generalized ADJ cana-5942 152 32 thermoelasticity thermoelasticity NOUN cana-5942 152 33 , , PUNCT cana-5942 152 34 j. j. PROPN cana-5942 152 35 mech mech PROPN cana-5942 152 36 . . PUNCT cana-5942 153 1 sci sci PROPN cana-5942 153 2 . . PUNCT cana-5942 153 3 tech tech PROPN cana-5942 153 4 . . PUNCT cana-5942 153 5 , , PUNCT cana-5942 153 6 vol vol NOUN cana-5942 153 7 . . PROPN cana-5942 154 1 27 27 NUM cana-5942 154 2 , , PUNCT cana-5942 154 3 pp pp ADJ cana-5942 154 4 . . PUNCT cana-5942 155 1 36073614 36073614 NUM cana-5942 155 2 , , PUNCT cana-5942 155 3 2013 2013 NUM cana-5942 155 4 2 2 NUM cana-5942 155 5 . . PUNCT cana-5942 155 6 cowin cowin PROPN cana-5942 155 7 , , PUNCT cana-5942 155 8 s.c s.c PROPN cana-5942 155 9 . . PROPN cana-5942 155 10 and and CCONJ cana-5942 155 11 nunziato nunziato PROPN cana-5942 155 12 , , PUNCT cana-5942 155 13 j.w j.w PROPN cana-5942 155 14 . . PROPN cana-5942 155 15 , , PUNCT cana-5942 155 16 linear linear PROPN cana-5942 155 17 theory theory NOUN cana-5942 155 18 of of ADP cana-5942 155 19 elastic elastic ADJ cana-5942 155 20 materials material NOUN cana-5942 155 21 with with ADP cana-5942 155 22 voids void NOUN cana-5942 155 23 , , PUNCT cana-5942 155 24 j. j. PROPN cana-5942 155 25 elasticity elasticity PROPN cana-5942 155 26 , , PUNCT cana-5942 155 27 vol vol NOUN cana-5942 155 28 . . PROPN cana-5942 155 29 13 13 NUM cana-5942 155 30 , , PUNCT cana-5942 155 31 pp pp ADJ cana-5942 155 32 . . PUNCT cana-5942 156 1 125 125 NUM cana-5942 156 2 - - SYM cana-5942 156 3 147 147 NUM cana-5942 156 4 , , PUNCT cana-5942 156 5 1983 1983 NUM cana-5942 156 6 3 3 NUM cana-5942 157 1 . . PUNCT cana-5942 157 2 das das PROPN cana-5942 157 3 , , PUNCT cana-5942 157 4 p. p. NOUN cana-5942 157 5 and and CCONJ cana-5942 157 6 kanoria kanoria PROPN cana-5942 157 7 , , PUNCT cana-5942 157 8 m. m. NOUN cana-5942 157 9 , , PUNCT cana-5942 157 10 magneto magneto NOUN cana-5942 157 11 - - PUNCT cana-5942 157 12 thermo thermo NOUN cana-5942 157 13 - - PUNCT cana-5942 157 14 elastic elastic ADJ cana-5942 157 15 response response NOUN cana-5942 157 16 in in ADP cana-5942 157 17 a a DET cana-5942 157 18 perfectly perfectly ADV cana-5942 157 19 conducting conduct VERB cana-5942 157 20 medium medium NOUN cana-5942 157 21 with with ADP cana-5942 157 22 three three NUM cana-5942 157 23 - - PUNCT cana-5942 157 24 phase phase NOUN cana-5942 157 25 - - PUNCT cana-5942 157 26 lag lag NOUN cana-5942 157 27 effect effect NOUN cana-5942 157 28 , , PUNCT cana-5942 157 29 acta acta PROPN cana-5942 157 30 mech mech PROPN cana-5942 157 31 . . PUNCT cana-5942 157 32 , , PUNCT cana-5942 158 1 vol vol NOUN cana-5942 158 2 . . PROPN cana-5942 158 3 223 223 NUM cana-5942 158 4 , , PUNCT cana-5942 158 5 pp pp ADJ cana-5942 158 6 . . PUNCT cana-5942 159 1 811 811 NUM cana-5942 159 2 - - SYM cana-5942 159 3 828 828 NUM cana-5942 159 4 , , PUNCT cana-5942 159 5 2012 2012 NUM cana-5942 159 6 . . PUNCT cana-5942 160 1 4 4 X cana-5942 160 2 . . X cana-5942 160 3 dhaliwal dhaliwal PROPN cana-5942 160 4 , , PUNCT cana-5942 160 5 r.s r.s PROPN cana-5942 160 6 . . PROPN cana-5942 160 7 and and CCONJ cana-5942 160 8 singh singh PROPN cana-5942 160 9 , , PUNCT cana-5942 160 10 a. a. NOUN cana-5942 160 11 , , PUNCT cana-5942 160 12 dynamic dynamic ADJ cana-5942 160 13 coupled couple VERB cana-5942 160 14 thermoelasticity thermoelasticity NOUN cana-5942 160 15 , , PUNCT cana-5942 160 16 hindustan hindustan PROPN cana-5942 160 17 publ publ PROPN cana-5942 160 18 . . PUNCT cana-5942 161 1 corp corp PROPN cana-5942 161 2 . . PROPN cana-5942 161 3 , , PUNCT cana-5942 161 4 new new PROPN cana-5942 161 5 delhi delhi PROPN cana-5942 161 6 . . PUNCT cana-5942 162 1 p. p. NOUN cana-5942 162 2 726 726 NUM cana-5942 162 3 , , PUNCT cana-5942 162 4 1980 1980 NUM cana-5942 162 5 . . PUNCT cana-5942 163 1 5 5 NUM cana-5942 163 2 . . X cana-5942 163 3 iesan iesan PROPN cana-5942 163 4 , , PUNCT cana-5942 163 5 d. d. PROPN cana-5942 163 6 , , PUNCT cana-5942 163 7 a a DET cana-5942 163 8 theory theory NOUN cana-5942 163 9 of of ADP cana-5942 163 10 thermoelastic thermoelastic ADJ cana-5942 163 11 materials material NOUN cana-5942 163 12 with with ADP cana-5942 163 13 voids voids NOUN cana-5942 163 14 , , PUNCT cana-5942 163 15 acta acta PROPN cana-5942 163 16 mech mech PROPN cana-5942 163 17 . . PUNCT cana-5942 163 18 , , PUNCT cana-5942 163 19 vol vol NOUN cana-5942 163 20 . . PROPN cana-5942 163 21 60 60 NUM cana-5942 163 22 , , PUNCT cana-5942 163 23 pp pp ADJ cana-5942 163 24 . . PUNCT cana-5942 164 1 6789 6789 NUM cana-5942 164 2 , , PUNCT cana-5942 164 3 1986 1986 NUM cana-5942 164 4 . . PUNCT cana-5942 165 1 6 6 NUM cana-5942 165 2 . . X cana-5942 165 3 kundu kundu NOUN cana-5942 165 4 , , PUNCT cana-5942 165 5 s. s. PROPN cana-5942 165 6 , , PUNCT cana-5942 165 7 kalkal kalkal PROPN cana-5942 165 8 , , PUNCT cana-5942 165 9 k.k k.k PROPN cana-5942 165 10 . . PROPN cana-5942 165 11 , , PUNCT cana-5942 165 12 sangwan sangwan PROPN cana-5942 165 13 , , PUNCT cana-5942 165 14 m. m. NOUN cana-5942 165 15 and and CCONJ cana-5942 165 16 sheoran sheoran PROPN cana-5942 165 17 , , PUNCT cana-5942 165 18 d. d. PROPN cana-5942 165 19 , , PUNCT cana-5942 165 20 two two NUM cana-5942 165 21 - - PUNCT cana-5942 165 22 dimensional dimensional ADJ cana-5942 165 23 deformations deformation NOUN cana-5942 165 24 in in ADP cana-5942 165 25 an an DET cana-5942 165 26 initially initially ADV cana-5942 165 27 stressed stress VERB cana-5942 165 28 nonlocal nonlocal ADJ cana-5942 165 29 micropolar micropolar ADJ cana-5942 165 30 thermoelastic thermoelastic ADJ cana-5942 165 31 porous porous ADJ cana-5942 165 32 medium medium NOUN cana-5942 165 33 subjected subject VERB cana-5942 165 34 to to ADP cana-5942 165 35 a a DET cana-5942 165 36 moving move VERB cana-5942 165 37 thermal thermal ADJ cana-5942 165 38 load load NOUN cana-5942 165 39 , , PUNCT cana-5942 165 40 international international ADJ cana-5942 165 41 journal journal NOUN cana-5942 165 42 of of ADP cana-5942 165 43 numerical numerical ADJ cana-5942 165 44 methods method NOUN cana-5942 165 45 for for ADP cana-5942 165 46 heat heat NOUN cana-5942 165 47 & & CCONJ cana-5942 165 48 fluid fluid ADJ cana-5942 165 49 flow flow NOUN cana-5942 165 50 , , PUNCT cana-5942 165 51 vol vol NOUN cana-5942 165 52 . . PROPN cana-5942 166 1 33 33 NUM cana-5942 166 2 no no NOUN cana-5942 166 3 . . NOUN cana-5942 167 1 3 3 NUM cana-5942 167 2 , , PUNCT cana-5942 167 3 pp pp ADJ cana-5942 167 4 . . PUNCT cana-5942 168 1 1116 1116 NUM cana-5942 168 2 - - SYM cana-5942 168 3 1143.(2023 1143.(2023 NUM cana-5942 168 4 ) ) PUNCT cana-5942 168 5 . . PUNCT cana-5942 169 1 https://www.emerald.com/insight/search?q=sandeep%20kundu https://www.emerald.com/insight/search?q=sandeep%20kundu NOUN cana-5942 169 2 https://www.emerald.com/insight/search?q=kapil%20kumar%20kalkal https://www.emerald.com/insight/search?q=kapil%20kumar%20kalkal NOUN cana-5942 169 3 https://www.emerald.com/insight/search?q=monika%20sangwan https://www.emerald.com/insight/search?q=monika%20sangwan PROPN cana-5942 169 4 https://www.emerald.com/insight/search?q=devender%20sheoran https://www.emerald.com/insight/search?q=devender%20sheoran PROPN cana-5942 169 5 https://www.emerald.com/insight/publication/issn/0961-5539 https://www.emerald.com/insight/publication/issn/0961-5539 PROPN cana-5942 169 6 https://www.emerald.com/insight/publication/issn/0961-5539 https://www.emerald.com/insight/publication/issn/0961-5539 PROPN cana-5942 169 7 communications communication NOUN cana-5942 169 8 on on ADP cana-5942 169 9 applied apply VERB cana-5942 169 10 nonlinear nonlinear ADJ cana-5942 169 11 analysis analysis NOUN cana-5942 169 12 issn issn NOUN cana-5942 169 13 : : PUNCT cana-5942 169 14 1074 1074 NUM cana-5942 169 15 - - PUNCT cana-5942 169 16 133x 133x NUM cana-5942 169 17 vol vol NOUN cana-5942 169 18 31 31 NUM cana-5942 169 19 no no NOUN cana-5942 169 20 . . PUNCT cana-5942 170 1 8s 8s PROPN cana-5942 170 2 ( ( PUNCT cana-5942 170 3 2024 2024 NUM cana-5942 170 4 ) ) PUNCT cana-5942 170 5 1137 1137 NUM cana-5942 170 6 https://internationalpubls.com https://internationalpubls.com X cana-5942 170 7 7 7 X cana-5942 170 8 . . X cana-5942 170 9 malik malik PROPN cana-5942 170 10 , , PUNCT cana-5942 170 11 s. s. PROPN cana-5942 170 12 , , PUNCT cana-5942 170 13 gupta gupta PROPN cana-5942 170 14 , , PUNCT cana-5942 170 15 d. d. PROPN cana-5942 170 16 , , PUNCT cana-5942 170 17 kumar kumar PROPN cana-5942 170 18 , , PUNCT cana-5942 170 19 k. k. PROPN cana-5942 170 20 , , PUNCT cana-5942 170 21 sharma sharma PROPN cana-5942 170 22 , , PUNCT cana-5942 170 23 r.k r.k PROPN cana-5942 170 24 . . PROPN cana-5942 170 25 , , PUNCT cana-5942 170 26 plane plane NOUN cana-5942 170 27 wave wave NOUN cana-5942 170 28 propagation propagation NOUN cana-5942 170 29 and and CCONJ cana-5942 170 30 fundamental fundamental ADJ cana-5942 170 31 solution solution NOUN cana-5942 170 32 in in ADP cana-5942 170 33 functionally functionally ADV cana-5942 170 34 graded grade VERB cana-5942 170 35 couple couple NOUN cana-5942 170 36 stress stress NOUN cana-5942 170 37 micropolar micropolar ADJ cana-5942 170 38 thermoelastic thermoelastic ADJ cana-5942 170 39 solid solid ADJ cana-5942 170 40 with with ADP cana-5942 170 41 diffusion diffusion NOUN cana-5942 170 42 and and CCONJ cana-5942 170 43 voids voids NOUN cana-5942 170 44 , , PUNCT cana-5942 170 45 wave wave NOUN cana-5942 170 46 rand rand NOUN cana-5942 170 47 . . PUNCT cana-5942 171 1 compl compl PROPN cana-5942 171 2 . . PUNCT cana-5942 171 3 media medium NOUN cana-5942 171 4 . . PUNCT cana-5942 172 1 doi doi NOUN cana-5942 172 2 : : PUNCT cana-5942 172 3 10.1080/17455030.2022.2155331 10.1080/17455030.2022.2155331 NUM cana-5942 172 4 , , PUNCT cana-5942 172 5 2022 2022 NUM cana-5942 172 6 . . PUNCT cana-5942 173 1 8 8 NUM cana-5942 173 2 . . X cana-5942 173 3 mukhopdhyay mukhopdhyay NOUN cana-5942 173 4 , , PUNCT cana-5942 173 5 s. s. PROPN cana-5942 173 6 and and CCONJ cana-5942 173 7 kumar kumar PROPN cana-5942 173 8 , , PUNCT cana-5942 173 9 r. r. PROPN cana-5942 173 10 , , PUNCT cana-5942 173 11 analysis analysis NOUN cana-5942 173 12 of of ADP cana-5942 173 13 phase phase NOUN cana-5942 173 14 - - PUNCT cana-5942 173 15 lag lag NOUN cana-5942 173 16 effects effect NOUN cana-5942 173 17 on on ADP cana-5942 173 18 wave wave NOUN cana-5942 173 19 propagation propagation NOUN cana-5942 173 20 in in ADP cana-5942 173 21 a a DET cana-5942 173 22 thick thick ADJ cana-5942 173 23 plate plate NOUN cana-5942 173 24 under under ADP cana-5942 173 25 axisymmetric axisymmetric ADJ cana-5942 173 26 temperature temperature NOUN cana-5942 173 27 distribution distribution NOUN cana-5942 173 28 , , PUNCT cana-5942 173 29 acta acta PROPN cana-5942 173 30 mech mech PROPN cana-5942 173 31 . . PUNCT cana-5942 173 32 , , PUNCT cana-5942 173 33 vol vol NOUN cana-5942 173 34 . . NOUN cana-5942 173 35 210 210 NUM cana-5942 173 36 , , PUNCT cana-5942 173 37 pp pp ADJ cana-5942 173 38 . . PUNCT cana-5942 174 1 331 331 NUM cana-5942 174 2 - - SYM cana-5942 174 3 344 344 NUM cana-5942 174 4 , , PUNCT cana-5942 174 5 2010 2010 NUM cana-5942 174 6 . . PUNCT cana-5942 175 1 9 9 X cana-5942 175 2 . . X cana-5942 175 3 nunziato nunziato PROPN cana-5942 175 4 , , PUNCT cana-5942 175 5 j.w j.w PROPN cana-5942 175 6 . . PROPN cana-5942 175 7 and and CCONJ cana-5942 175 8 cowin cowin PROPN cana-5942 175 9 , , PUNCT cana-5942 175 10 s.c s.c PROPN cana-5942 175 11 . . PROPN cana-5942 175 12 , , PUNCT cana-5942 175 13 a a DET cana-5942 175 14 non non ADJ cana-5942 175 15 - - ADJ cana-5942 175 16 linear linear ADJ cana-5942 175 17 theory theory NOUN cana-5942 175 18 of of ADP cana-5942 175 19 elastic elastic ADJ cana-5942 175 20 materials material NOUN cana-5942 175 21 with with ADP cana-5942 175 22 voids voids NOUN cana-5942 175 23 , , PUNCT cana-5942 175 24 arch arch NOUN cana-5942 175 25 . . PUNCT cana-5942 176 1 ration ration NOUN cana-5942 176 2 . . PUNCT cana-5942 177 1 mech mech PROPN cana-5942 177 2 . . PUNCT cana-5942 178 1 anal anal PROPN cana-5942 178 2 . . PUNCT cana-5942 178 3 , , PUNCT cana-5942 178 4 vol vol NOUN cana-5942 178 5 . . PROPN cana-5942 178 6 12 12 NUM cana-5942 178 7 , , PUNCT cana-5942 179 1 pp pp ADJ cana-5942 179 2 . . PUNCT cana-5942 180 1 175 175 NUM cana-5942 180 2 - - SYM cana-5942 180 3 201 201 NUM cana-5942 180 4 , , PUNCT cana-5942 180 5 1979 1979 NUM cana-5942 180 6 . . PUNCT cana-5942 181 1 10 10 NUM cana-5942 181 2 . . X cana-5942 181 3 quintanilla quintanilla PROPN cana-5942 181 4 , , PUNCT cana-5942 181 5 r. r. PROPN cana-5942 181 6 and and CCONJ cana-5942 181 7 racke racke PROPN cana-5942 181 8 , , PUNCT cana-5942 181 9 r. r. PROPN cana-5942 181 10 , , PUNCT cana-5942 181 11 a a DET cana-5942 181 12 note note NOUN cana-5942 181 13 on on ADP cana-5942 181 14 stability stability NOUN cana-5942 181 15 in in ADP cana-5942 181 16 dual dual ADJ cana-5942 181 17 - - PUNCT cana-5942 181 18 phase phase NOUN cana-5942 181 19 - - PUNCT cana-5942 181 20 lag lag NOUN cana-5942 181 21 heat heat NOUN cana-5942 181 22 conduction conduction NOUN cana-5942 181 23 , , PUNCT cana-5942 181 24 int int NOUN cana-5942 181 25 . . PUNCT cana-5942 182 1 j. j. PROPN cana-5942 182 2 heat heat PROPN cana-5942 182 3 mass mass PROPN cana-5942 182 4 transf transf PROPN cana-5942 182 5 . . PUNCT cana-5942 183 1 , , PUNCT cana-5942 183 2 vol vol NOUN cana-5942 183 3 . . PROPN cana-5942 183 4 49 49 NUM cana-5942 183 5 , , PUNCT cana-5942 183 6 pp pp ADJ cana-5942 183 7 . . PUNCT cana-5942 184 1 1209 1209 NUM cana-5942 184 2 - - SYM cana-5942 184 3 1213 1213 NUM cana-5942 184 4 , , PUNCT cana-5942 184 5 2006 2006 NUM cana-5942 184 6 . . PUNCT cana-5942 185 1 11 11 NUM cana-5942 185 2 . . PUNCT cana-5942 186 1 roychoudhuri roychoudhuri PROPN cana-5942 186 2 , , PUNCT cana-5942 186 3 s.k s.k PROPN cana-5942 186 4 . . PROPN cana-5942 186 5 , , PUNCT cana-5942 186 6 on on ADP cana-5942 186 7 a a DET cana-5942 186 8 thermoelastic thermoelastic ADJ cana-5942 186 9 three three NUM cana-5942 186 10 - - PUNCT cana-5942 186 11 phase phase NOUN cana-5942 186 12 - - PUNCT cana-5942 186 13 lag lag NOUN cana-5942 186 14 model model NOUN cana-5942 186 15 , , PUNCT cana-5942 186 16 j. j. PROPN cana-5942 186 17 therm therm PROPN cana-5942 186 18 . . PUNCT cana-5942 187 1 stress stress NOUN cana-5942 187 2 . . PUNCT cana-5942 188 1 , , PUNCT cana-5942 188 2 vol vol NOUN cana-5942 188 3 . . PROPN cana-5942 188 4 30 30 NUM cana-5942 188 5 , , PUNCT cana-5942 189 1 pp pp ADJ cana-5942 189 2 . . PUNCT cana-5942 190 1 231 231 NUM cana-5942 190 2 - - SYM cana-5942 190 3 238 238 NUM cana-5942 190 4 , , PUNCT cana-5942 190 5 2007 2007 NUM cana-5942 190 6 . . PUNCT cana-5942 191 1 12 12 NUM cana-5942 191 2 . . PUNCT cana-5942 191 3 sharma sharma PROPN cana-5942 191 4 , , PUNCT cana-5942 191 5 k. k. PROPN cana-5942 191 6 and and CCONJ cana-5942 191 7 kumar kumar PROPN cana-5942 191 8 , , PUNCT cana-5942 191 9 p. p. NOUN cana-5942 191 10 , , PUNCT cana-5942 191 11 propagation propagation NOUN cana-5942 191 12 of of ADP cana-5942 191 13 plane plane NOUN cana-5942 191 14 waves wave NOUN cana-5942 191 15 and and CCONJ cana-5942 191 16 fundamental fundamental ADJ cana-5942 191 17 solution solution NOUN cana-5942 191 18 in in ADP cana-5942 191 19 thermoviscoelastic thermoviscoelastic ADJ cana-5942 191 20 medium medium NOUN cana-5942 191 21 with with ADP cana-5942 191 22 voids voids NOUN cana-5942 191 23 , , PUNCT cana-5942 191 24 j. j. PROPN cana-5942 191 25 therm therm PROPN cana-5942 191 26 . . PUNCT cana-5942 192 1 stress stress NOUN cana-5942 192 2 . . PUNCT cana-5942 193 1 , , PUNCT cana-5942 193 2 vol vol NOUN cana-5942 193 3 . . PROPN cana-5942 194 1 36 36 NUM cana-5942 194 2 , , PUNCT cana-5942 194 3 pp pp ADJ cana-5942 194 4 . . PUNCT cana-5942 195 1 94 94 NUM cana-5942 195 2 - - SYM cana-5942 195 3 111 111 NUM cana-5942 195 4 , , PUNCT cana-5942 195 5 2013 2013 NUM cana-5942 195 6 . . PUNCT cana-5942 196 1 13 13 NUM cana-5942 196 2 . . X cana-5942 197 1 singh singh PROPN cana-5942 197 2 , , PUNCT cana-5942 197 3 j. j. PROPN cana-5942 197 4 and and CCONJ cana-5942 197 5 tomar tomar PROPN cana-5942 197 6 , , PUNCT cana-5942 197 7 s.k s.k PROPN cana-5942 197 8 . . PROPN cana-5942 197 9 , , PUNCT cana-5942 197 10 plane plane NOUN cana-5942 197 11 waves wave NOUN cana-5942 197 12 in in ADP cana-5942 197 13 thermoelastic thermoelastic ADJ cana-5942 197 14 materials material NOUN cana-5942 197 15 with with ADP cana-5942 197 16 voids voids NOUN cana-5942 197 17 , , PUNCT cana-5942 197 18 mech mech NOUN cana-5942 197 19 . . PUNCT cana-5942 197 20 mate mate NOUN cana-5942 197 21 . . PUNCT cana-5942 197 22 , , PUNCT cana-5942 197 23 vol vol NOUN cana-5942 197 24 . . PROPN cana-5942 197 25 39 39 NUM cana-5942 197 26 , , PUNCT cana-5942 197 27 pp pp ADJ cana-5942 197 28 . . PUNCT cana-5942 198 1 932 932 NUM cana-5942 198 2 - - SYM cana-5942 198 3 940 940 NUM cana-5942 198 4 , , PUNCT cana-5942 198 5 2007 2007 NUM cana-5942 198 6 . . PUNCT cana-5942 199 1 14 14 NUM cana-5942 199 2 . . X cana-5942 200 1 tzou tzou PROPN cana-5942 200 2 , , PUNCT cana-5942 200 3 d.y d.y PROPN cana-5942 200 4 . . PROPN cana-5942 200 5 , , PUNCT cana-5942 200 6 a a DET cana-5942 200 7 unified unified ADJ cana-5942 200 8 field field NOUN cana-5942 200 9 approach approach NOUN cana-5942 200 10 for for ADP cana-5942 200 11 heat heat NOUN cana-5942 200 12 conduction conduction NOUN cana-5942 200 13 from from ADP cana-5942 200 14 macro macro ADJ cana-5942 200 15 to to ADP cana-5942 200 16 micro micro NOUN cana-5942 200 17 scales scale NOUN cana-5942 200 18 , , PUNCT cana-5942 200 19 asmej asmej NOUN cana-5942 200 20 . . PUNCT cana-5942 201 1 heat heat NOUN cana-5942 201 2 . . PUNCT cana-5942 202 1 transf transf PROPN cana-5942 202 2 . . PUNCT cana-5942 202 3 , , PUNCT cana-5942 202 4 vol vol NOUN cana-5942 202 5 . . PROPN cana-5942 202 6 117 117 NUM cana-5942 202 7 , , PUNCT cana-5942 202 8 pp pp ADJ cana-5942 202 9 . . PUNCT cana-5942 203 1 8 8 NUM cana-5942 203 2 - - SYM cana-5942 203 3 16 16 NUM cana-5942 203 4 , , PUNCT cana-5942 203 5 1995 1995 NUM cana-5942 203 6 . . PUNCT cana-5942 204 1 figure figure NOUN cana-5942 204 2 captions caption NOUN cana-5942 204 3 : : PUNCT cana-5942 204 4 fig fig NOUN cana-5942 204 5 . . PUNCT cana-5942 205 1 2 2 NUM cana-5942 205 2 variation variation NOUN cana-5942 205 3 of of ADP cana-5942 205 4 the the DET cana-5942 205 5 modulus modulus NOUN cana-5942 205 6 of of ADP cana-5942 205 7 reflection reflection NOUN cana-5942 205 8 coefficient coefficient NOUN cana-5942 205 9 z1 z1 VERB cana-5942 205 10 with with ADP cana-5942 205 11 the the DET cana-5942 205 12 angle angle NOUN cana-5942 205 13 of of ADP cana-5942 205 14 incidence incidence NOUN cana-5942 205 15 of of ADP cana-5942 205 16 coupled couple VERB cana-5942 205 17 longitudinal longitudinal ADJ cana-5942 205 18 wave wave NOUN cana-5942 205 19 with with ADP cana-5942 205 20 speed speed NOUN cana-5942 205 21 v1 v1 NOUN cana-5942 205 22 fig fig NOUN cana-5942 205 23 . . PUNCT cana-5942 206 1 3 3 NUM cana-5942 206 2 variation variation NOUN cana-5942 206 3 of of ADP cana-5942 206 4 the the DET cana-5942 206 5 modulus modulus NOUN cana-5942 206 6 of of ADP cana-5942 206 7 reflection reflection NOUN cana-5942 206 8 coefficient coefficient NOUN cana-5942 206 9 z2 z2 PROPN cana-5942 206 10 with with ADP cana-5942 206 11 the the DET cana-5942 206 12 angle angle NOUN cana-5942 206 13 of of ADP cana-5942 206 14 incidence incidence NOUN cana-5942 206 15 of of ADP cana-5942 206 16 coupled couple VERB cana-5942 206 17 longitudinal longitudinal ADJ cana-5942 206 18 wave wave NOUN cana-5942 206 19 with with ADP cana-5942 206 20 speed speed NOUN cana-5942 206 21 v1 v1 NOUN cana-5942 206 22 . . PUNCT cana-5942 207 1 fig fig NOUN cana-5942 207 2 . . PUNCT cana-5942 208 1 4 4 NUM cana-5942 208 2 variation variation NOUN cana-5942 208 3 of of ADP cana-5942 208 4 the the DET cana-5942 208 5 modulus modulus NOUN cana-5942 208 6 of of ADP cana-5942 208 7 reflection reflection NOUN cana-5942 208 8 coefficient coefficient NOUN cana-5942 208 9 z3 z3 PROPN cana-5942 208 10 with with ADP cana-5942 208 11 the the DET cana-5942 208 12 angle angle NOUN cana-5942 208 13 of of ADP cana-5942 208 14 incidence incidence NOUN cana-5942 208 15 of of ADP cana-5942 208 16 coupled couple VERB cana-5942 208 17 longitudinal longitudinal ADJ cana-5942 208 18 wave wave NOUN cana-5942 208 19 with with ADP cana-5942 208 20 speed speed NOUN cana-5942 208 21 v1 v1 NOUN cana-5942 208 22 . . PUNCT cana-5942 209 1 fig fig NOUN cana-5942 209 2 . . PUNCT cana-5942 210 1 5 5 NUM cana-5942 210 2 variation variation NOUN cana-5942 210 3 of of ADP cana-5942 210 4 the the DET cana-5942 210 5 modulus modulus NOUN cana-5942 210 6 of of ADP cana-5942 210 7 reflection reflection NOUN cana-5942 210 8 coefficient coefficient NOUN cana-5942 210 9 z4 z4 PROPN cana-5942 210 10 with with ADP cana-5942 210 11 the the DET cana-5942 210 12 angle angle NOUN cana-5942 210 13 of of ADP cana-5942 210 14 incidence incidence NOUN cana-5942 210 15 of of ADP cana-5942 210 16 coupled couple VERB cana-5942 210 17 longitudinal longitudinal ADJ cana-5942 210 18 wave wave NOUN cana-5942 210 19 with with ADP cana-5942 210 20 speed speed NOUN cana-5942 210 21 v1 v1 NOUN cana-5942 210 22 . . PUNCT cana-5942 211 1 https://doi.org/10.1080/17455030.2022.2155331 https://doi.org/10.1080/17455030.2022.2155331 ADJ cana-5942 211 2 communications communication NOUN cana-5942 211 3 on on ADP cana-5942 211 4 applied apply VERB cana-5942 211 5 nonlinear nonlinear ADJ cana-5942 211 6 analysis analysis NOUN cana-5942 211 7 issn issn NOUN cana-5942 211 8 : : PUNCT cana-5942 211 9 1074 1074 NUM cana-5942 211 10 - - PUNCT cana-5942 211 11 133x 133x NUM cana-5942 211 12 vol vol NOUN cana-5942 211 13 31 31 NUM cana-5942 211 14 no no NOUN cana-5942 211 15 . . PUNCT cana-5942 212 1 8s 8s PROPN cana-5942 212 2 ( ( PUNCT cana-5942 212 3 2024 2024 NUM cana-5942 212 4 ) ) PUNCT cana-5942 212 5 1138 1138 NUM cana-5942 212 6 https://internationalpubls.com https://internationalpubls.com X