Lamb and Love wave propagalion 

in an infinite micropolar elastic plate 

E. .Bosem (>!') 

Reeeive(1 on Febrnary 2w1, Hli:~ 

SU)I~IAln:. - This puper is eOlwerne,l with mon(lehl'Omatie W;lve 
propagation in an illfillit.e humogenmlUH lllicropolai' elnsti .. plnte ])()unde,l 
by two purnllcl free l)lunes. Two kiJl(1s of l)ropagation are ,lis('nsse,l; Lamh 
an,] Love waveR. \Ye fin,l t.hat a (Iisplaeement field (n" n" O) uwl n mino-
I"otution neld (O, 0, 'l'a) le(\(1;; to Lnmb';; waye;;, while a ,Iisplueellwnt fiel,l 
(O, 0, n,) an(1 a minol'obtion Del(1 (lo!> lo., O) 1ca(18 t.o Love'H wnves. 

RIA.~SUl>:·I'O. - In qnesto lavoro si tratta la l)rOpngallionc di on,le 
monoeromatielw in nn piatto elastieo, mirropolare, omog"elwo e,l illnllito, 
limitato ,In ,1ue piani pnralleli c liheri. Vengono ,liseusRi ,lne tipi (Ii pl"Opa-
guzione, qndlo ,li Lamll e quello ,Ii ].ove. :'ii trova ehe il ealllpo di sposta-
menti (1!'1' n" O) c,l il e:~mllO ,li mierorotUllioni (O, 0, 't'a) porta a,l on,le (Ii 
Li!mb, mentre nn eampo ,li spostamenti (0, O, u. J ) e,l Uli eumpo ,li mkro· 
l'otllllioni ('P" 'Pe, O) port.a n,l owlc ,li Loye. 

I::\T]UlDl:CTIO:-; 

OseilJn.tion.~ of an cla~tic plate, ·with stress-f)'{-l(; snrfa.('!'~, lmve 
been inve~i.igai.c(l by Raylcigh (l~ ), J,amò (") n·nll othcrs, aml more 
recently by PresC'ott (12), Gogohllllll-l ("), Sai.o (1.1) aml Bwin)..!.', .fn.rdci.zky 
iJ.ll11 Press ("l). 

In this VllJCr wc want to (leal ·with thl-l ~amc Vroblem in the frame-
·wark of thc thcary oI mierOllOlal' eJn.si.irity. 

(~) Istituto ,li Fisiea, Fnivt\1'8itit ,li Bolognu. 
nipartimento ,Ii i:)l'ienllc ,Iella TelTi!, Lniversitit ,li ,\neona (Haly). 



3-l-:! l,. BOSCHI 

'l'he linear Llu'or,\" of miero1JOlar ('laHtiC'ity has bCPll illtrodlU,(·a 
by Erillgl'n ('), rc~an (1") ha,~ (lcl'ivea the [uIHlamenbl equatiolls nsing 
invariatH'e c'01Hlitions umlel' SUl)('rllOsed righI hody motiom, 

.:\licl'opolal' clastieity ma.y givc intere,~Ling re,~llllH Khen ullilliea 
to geollh~-HicHI problelll~, \Ve have all'eady applicd it 1.0 Ilpl'ive an 
cxplic'it eX1Jr(\SHioll oI the body IOl'ce Hnd Lhe body coullie eqllivalenLs 
fol' seismie (LiHlocatiolls (1,2), 

Itunc EQCA1'lOXS 

'l'hl'oughOllt thi,~ 1HqH'1' \H\ em1Jloy the uHual in(Licial notatiollH, 
"\11 r<\1,.'1t!arity hnlOthcses on thc eOllHi(Lel'ea funetiol!H \\-ill be omittcll. 

\\Te consiacl' a reeLa,nglllal' Cal'LeHiall ft'ame O,t' k (k '--= ',2,3). 'l'he 
lJH,~ie (\quations in thc lincar thcOl'y of honlOgenco1lH amI allisotropie 
ela,~Lie ,~olillf; are (la): 

- the kinematiC' relaliollS: 

- the eqllatio1lH o[ motion: 

tll,l +]j'j = e iii 
-miCJ + f:IJk t} 1t + JL = Iii TI 

- the constitutive law,~: 

ti) = AllltI C~I + Bjj~j %kl 
mi} = Bklil CkJ + GO t i %kl 

[1.a] 

Il.bJ 

[2.a] 

[2.bJ 

[3.»J 

[3.b] 

III t-he abovc cquaLiollS 'Hl havc med thc follo\\'ing llotations: 
1t., the ('01111)()]Hmts oI tlte (liflphl('Pl11ent vectol'; !Pl, Lite eOI111)OllCnL,~ 
of the lllierOl'otaLioll veetol'; Ci} aIHL %Ih thc killematic eharactel'i,~ti(l,~ 
of Lhc ,~train; t}l, the COlllj)(mell1:H of t-he stre,~s tenwl'j 1nJI, the (l011111O-
nellts oI tlte eouplc HtresH tensol'; 1<'J, Lhc eomvollents oI the bOlly IOl'eej 
_MI, Llw comllOnents of Llw body COUlllci (!1 the mass densitYi AlJItI, 



343 

B"I·', 0,,1:1, I", t lre Chanl{!lAwisti c constant.I; {lf t he mllwl"i al ; fjJ l:, t!w 
nlt.l::rml,ting t~~IlSOI"; a· COIlUllrL (IOllot.es partùl,I <1 cl" ivntioll wit.h ros}lcct 
to ~pn,ce variab lcs, amI a supcrpo~Nl doL p a rtial dcrivn.tion with reSl) Cct 
to the ti m e t . FlU·thel"mol'c , wc hav e: 

A/ikl = A . I / j 

00.' = Ol:l/j 

I" = I j/ 

, 
) 

[4] 

Jf the bo(l.y is homogf.l! HWl\S , isotropie a ll d l'.cnLros)"mlllct ric, wc 
hnvo: 

A l i t I = À Ilo (hl + (p - ;-; ) b lk bl l + f.l bu b, t 
Bjj~1 = U 

CIJ~I = a bj) bkr + r /l/t /lil + {J /lu /ljJr 

anl1 eql1aLiolis {3] are a lteretl to: 

t'i = À eu /l" + (p. + xl CII + Il CII 
mll = a ~H Oli + Y i1:H + P XII 

[5] 

whel"e l., Il, x , a, y a·mI p r\1"e materia l const.an ts satis!,rillg llic foll owing 
i lleq unlities: 

3À+2.u + ;-; ~ O 

I 21l +r.~ O ;< ~ O 
3a + /J + i' ~O 

[G] 

\ - y ~{J ~y 
y ;. U 

whi ch are the necessal".y r ~\lII I; uflicient cOlH1itiollS lol' tlic internaI 
CIICl"gl' tu be n on-negative. Furt·lrermnrc wc cali n lso write: 

III =.T (j" [7 ] 

\\"hcl"t~ I is another m a teri ;11 cOllstan L 



E. BOHCIIl 

][ cquatiom [5] alHl [7J are flubstitute<l into equationfl [1], we get.; 

(i. -+- /1) l' !.!i + (ft + y.) Ui.JJ + % [;li 1; rp~.) + E, = IJlli, [.':I.a.] 
(H + (1) (f!J.H + y rpl.!! + Y.[;fj k nk.! -:! % rl' l + J[I = I ij)1 [Kh] 

Lpt u~ IlOW H}leeiali;.;e OUI' HnHlysis to t.ll(' ense in whieh; 

Bquntiom [H] lenl! to: 

(p + %) V-l' , + (I. ~- lt) (',l -1- %1f~.2 -- IJ iII 
(/1 + r.) 1/-112 i- (i. ··i II) ('t~ - % ({:l.l = (}lIl 
ì' p'rp:l + r. (11l.1 - 101.') -:.! % rp~ = I i"l 
y V~rpl + (a 1- (3) e,l _.:.! % f{'l + %1.:.., = 1 ipl 
Y [7"rp2 -;.. (a + fJ) e)2 -:! r. fp2 - X U3.l = 1 Tè. 
(p + r.) V : nJ + Y. (TU - 9'1.2) =!! il3 

whe!"c we haye intwl!llcCI! the [olloll"ln.!! 1l0lntionfl: 

e = (fl.\ + rp2.2 
V 2 = ì)!/ì).rl" + ì)~/ì);r2~ 

[H.a ] 

[D.h] 

The t.wu syHtelllH uI eqllati()n,~ [V.a] nJl(! [!I. h] are ilu!epen<lent.. The 
f!ystem [D.a] I!esel'ibes the dis}llaeemcnt !ipla (1.], Il") O) an<l the miel'u-
l"oLation li pII] (O, 0, fr~ ). 'l'he Hy.~t.em [H.b] (leflcribefl thc di~]lIacellwllt 
ti eli! (O) 0, nJ) nlH! the mic!"o!"otatioll tiel<l (rr'l) (P2) O). Om lluqlOse iH 
IO show tlmt the system [H.a] le~Hlfl to lI"aVeH o[ Lnlllh klll<l) while the 
syslem [H.h] lcnds IO II"nve~ of l~oye kin<l. 

I~.UIH W..iV.E.<;. 

'l'hl' eom})()Il{'llls of the Ht.reHH and COll]lh~ ~11'('f!H ltmf!Ol' <letenniIH,a 
by t.he aispIacmncnl fieli! (lh)U2, O) allI! by the micr()]'oLatiOll fielil 
(O, O, rpJ) are: 



LDIR A~ll LOV,; WAVE l'ROl'AGNr!l):\, E're. 

tll =...c (:3 Il + r.) 111,1 + i. e 
/22 = (2 Il + ;.:') !t2.2 + i. e 
',,3 = i. C 
tl~ = Il (11:<.1 +11I.~) + % !tu - r. 9a 
I~l -- f1 (101,: ...!.. U~ , l ) + y. 111,: -I ~ 'J'3 
tla -- I:" '-- h3 = /32 ,....., U 
11113 = Y (F".I 
11131 = f3 !P3,1 
/1/:3 = Y r~ .~ 
'/nJ2 =. f3 !P~.2 
'/nll = BI·z: = m·a3 = 'l!h2 = '/J'l'21 = O 

3.J-5 

[ IO ) 

I~pt ns llOW assume t.lmt a 1ll00l()('hl'omatie wave propagat C'fI, nloll)! 
t1le ,T ! - llil'edioll, ili ali illfinite llli(']'opoLlI' pJa.~ti(· ]llHte. I~et '2H be 
t1le thiekness o[ t1lc ('OllSi(lp]'p(l plate. ::\lo]'C'oyc]' wc nSflnme that the 
following eonditiol1S: 

/" = hz -= /1113 = U, at ,Tl = ± 11 [11] 

shoul(l be sat.i~!lell. In othe]' w()]'(ls, the sul'faees o[ thc platc are RUp-
lloSe(l free of stl'esses alHl eouple .~tl'(>ssPs. 

Eqnation.~ [U.nJ aTI' snt·isfip(l by: 

pl'oviùell 

lh c-= (j),1 - lP,Z 

112 = (j),z + '1',1 

(iì + :3 P + r.) [72 (j~ - f! (j) = O 
(p, + r.) V2 Ip - f! lp - % g;~, = U 
[y 17 2 - '2 r. - I ò2 /òP] !f3 + r. 17 ~ !j-' = U 

[l:3J 

The fil'~t equnt ion [12J llef\(·l'ibC'.~ thc pl'opag,ltioll of 1ong-itlHtinal (li~­
jlbeement wayes with yt'loeit~·: 

i,+2 ,u+r. 
[1:3) 

The la~t two equations [l:! J ('11ll be rt'(luet'(l to tlll' fol1owillg [orm: 

{[y [.h _ :! r. - I ~ ~/W] [(II + r.) Vz - !!Ò2/~n -i- %! V~} (f1 = U [l·La] 
{ [y V2 - '2 % - I {l2/W] [(.u --;- r.) V~ - e ~~/N~J ...!.. r. ~ 17~} !fa = U [l·Lb] 



E. nOSCHI 

Let us assulllc solutiollS of thc fOl'lll: 

W (XI, X" t) = ([J* (X!) ei (k";2 ._--" wt) 
':1' (Xl, .7:" t) = I.p* (Xl ) ef (!eX, - wl) 

rp~ (ah, x" t) = rpa* (xI) ei {kf, - wl} 

The 1irst cqnatioll [12J ancl cquatiollS [HJ beeome: 

where t.he followillg llotntioll~ have ueeu introc1nccc1: 

b~ = 2 r.fy 

a2 = u 2 /[Y (f' + r.)] 
C2

2 = /tle 
C3~ = r./e 

C4
2 = ì'II 

\ 

( [15J 
) 

[lOJ 

Ip*=O 

[17.a.J 

- le') I ~,' ~O, 
[17.b] 

[18J 

Equations [liJ (ltlseribe the moc1if'tccl tranSVCl'flC waves. We seurch 
solutiollS oI equation8 [16] a!l(l [17J of the fOl'm: 

rp* = A i\illh (1) Xl ) + B c08h (1J xl) [19J 
1.jI* = C sinh (h xl) + D cOflh (7,;1 x'I) + R sinh (k2 xI) + F cosh (7,;. xJ) 
rpa* = G sinh (1'1 Xl ) + L cosh (h ,'(1) + l' sillh (1;2 xl) + (J eosh (le. xl) 



LAMU A~D LOYE \l'AVE PROrAGATIQ:s", lèTC, 

Wher(l 

1 ','1 ~~ = k~ + --, . :2 l b2 _ a~_0'~_d2 ± I ( (/.~ + 
I )I~('l - -1. a~ Vl~ -b~) . 

'1 = J,.z_ --
(

W' ) 

VI2 

347 

Let lL~ 111m Sll(leialize our st.u<ly to the ense or s~-I11Illet.rie vibra-
tions, Then the eomllonent. 1'~ or the (Usplne0ment veeto]', the COIll-
llonents tll aml t~2 or t.he ~tress t-ensor aml the eOIllponent "/1113 of the 
eoullIrl stress tensor IllUSt. be symmeb·ic with respeet. t.o the pIane 
Xl = O. 'l'his Irmds to: 

A=JJ=P=L=(J=O 

and tIle expressions [lOJ ar(l altere(l t-o: 

(jJ* = 13 eosIl (1J xl) 
IP* =-o O sinh (1:1 xl) + E Sillh (k~ Xl) 
rp3* = G Sillh (kl:et) + P Sillh (k2 xl) 

[~OJ 

Furthermore introdueing thcsc expl'e~sions into Nj\wt.ions [12J, "-e get.: 

G =rl O 

P =·hE 

whcre 

Thcrcforc rp3* can be writ.t.cn aH: 

[21J 

The boundary eomlitions [11J anl express(><l :1.';: 

13 [1)~ ( :! P ---i- À + 11) - À k~J eosh (1) H) - Ci k kt (~ p + 11) eosh (1.:1 H) 
- E i k k z (2 ft + 11) l'osh (1.2 H) = O, 

H i k 1) (2 ft ---1 11) ~inh (1) H) + (J 0'1 sinh (kt H) + E O'! sinh (k! H) = O, 
C kt TI cosh(k. II) + E k~ Tt eosh(k~ H) = 0, [22] 



I·~. 1I0"CIII 

wherc 

li = 1~2. 

Thc e1li1l"acteri~tie eqllation Or t.he sYf!t.em [~~J is: 

Igli (l/lI) 
t.g·h (k. H) 

(~/( -+- % + Al 1)2 - !.-2 A 
(2ft + %}2 1/ k 2 f,-1 ('2 - ,tl 

In the limit. oI t.he usual elastieit.~,) el]lUlt.ion [~:1J rednpeR to t.he 
peri olI wl'mtion Ior JJ:lmb "'aves (.l .H). Tlic llisenRHion oI the traseen-
denbl eqllation [2:1J presents gl'cat IIilIieultics and) t.herefore, onlr 
t.he aRymlltotic limitH Ior long and short "'aves are considerell. 

1.'or waves 10llg comparPl1 wHli tlw t.hiekness '211, t.he pl'olluets 
1/ H, h 11, /."2 H are so small tItat. tIte Ityperbolie rnnetiollS are rellhced 
hy theil' arl:,'11tlwnt.s and cqnation [23J t.akes tIte Iorm: 

('211 + %)~ 1 2 k~ (n - n) ... ~ [(2ft + % + À) 1)2 - fi2 Al . (0"1"t2 - 112n) 
[~../.J 

1"01" vcrr ShOIt waves, t.he quantitie~ 1JH, !."IH, /.",H n·re brge alHI 
tIte two ratios or hyperbolic tangent~, whieh ap]wH-J' in eqllatioll L~3J 
become unit.}', givillg: 

[~#+.+M~-··M~nh-~T'~ 
= ('2 ,IL -+- %)21 fi2 /."1 /."2 ('3 - TI) ['25j 

fn the eln-ssieal lilllit hot.li the equations [2../.J and 1'2;"5J reduce to 
t.lw COl"l"CSI)()J\(ling rclat.iom; foull1l in t.hc frame\l'ork of t.lie mmal el-
asticit.y ("). Tn partielùar, equation [~5J can be recogllizcll as tIte 
cham·ctel·ist.ic equation fOI' H.n.yleigh Wlrves in a micl"O})olar elastie 
llalf Hpacc. 

So far \l'e have discnsse([ the ease of sYllllllet.ric vibl"ation~. T,et 
us nO'\" Hpecialize onr st.nll} t.o tIte caf\C of allt.isynunetri(~ vihrat.ionf\. 
The cOlllponent. u! or the displacement veetor, tIte components tn 
all1I122 of tlm streHf\ tensor anll tlw eOlll])(ment 1n13 of the eouple st.ress 



'J.I f) 

temor are no\\' antiS,Ylilllwtrie ,\"it.h rCSl)Cet 10 the ]llnne Xl -= (l. Tlle8e 
eon(litioll8 l('aù to: 

B=C=E_G=]>=O 

am1 the eXIH'essions [19] he('ome: 

w* = A sinh (l/ '1'1) 
Ij'* = j) C'ORh (li. Xl) +}i' eosh U'2,'l'1) 

1f/'" = L e08h (kJ .'V I ) + () C'ORli (h ;t',) 
[2liJ 

~\S in n, pnwiously l'onsi(lere(l l'ase, b~' intro(lul'ing Uwsc cX]lmssions 

into equnt.ions [12], 'Hl get.: 

L =T,D 

Q=n}i' 

,dwl'e T I a1\(l n ha"c heen ahea<l,Y <lcfmc(l TlIm'dOl'B (f!~ * can no1\' 

lw \lTitten as: 

[27] 

The houlHlary l'01H1itions [lJ are ]Un\" eX]lresse(l as: 

A ['II" (2 Il, + % + l.) -I. l.:~] sin li (l] H) - f) i k kt (2 tt + %) sinh(k, Il)-
-Ili lo h (2 f.L + %) sinh (lo~ H) ~ 0, 

.... l i k 1] (2 ft --;- %) eosh ('q II) -+- DG, l'oRli (l" H) -! P (52 e08h (1.'2 H) = 0, 
f) Ti A'! sin h (A'! II) + P T2 h f!inll (h H) = 0, [28J 

l'he ellaraeteristie equntion oi the S'ystem [28J is: 

(_"':;"C' -;TC'"'''·';:- (52 TI kr \ , I ( II) - Lgl 11 ' tgh (h H) tgh (k~ II) l ' 
(2 tt + %)~ ;" 2 lJ~ k l h {TZ - TJ] 

(2 tt {- % + l,) 'I( - ". ~ l. [2!JJ 

fn j.jH) dasRieal limit equat.ion [29] H'{luees j.o thc eOrrCsl)olHling 

relation foun(l in the immcwork oi t.hc llsnal elaRtieity (1), 



F.. nO,H'1I1 

For wavcs long comparcll with the thickne108 '2lJ, retaining uv 
to the third tcrnL~ o[ the power expansion of the h.Ylwrbolie fUllctions, 
equatioll [29J become8: 

I 
alT . U2TI I 

(3 - ' I ll ") !.-I" (3 h" H2) h" (3 _.- 7.; 12 H2) J = 

_ (~ f.l + r.)2 (T 2 - Ti ) 
('2 l' + ~ + l,) 1)~ /,;2 J. [30J 

This is thc pcrioll eqw1,tion [or long llexllml waves. Disl)ersion OCClll"S 
for these waves, with l)hase yelocity decreasing t.o .zcro with increasing 
wayc length. 

In the classical limit, cqllation [30J rcdllces to the COrrCSl)onding 
relation found in the framcwork of the usual elasticity (8). 

]<'or wayc lcngth8 8IWlll comparcd to the thickm1810 'JlJ, equation 
[30] retIuces to equation ['25J. 

LOVE 'WAVES 

'l'he compollents of t.he 8tre108 amI coup1e stress tenSOl'8 !letcrmined 
by the dispbcement lieltl (O, 0, 1ta) alHI b.y the mierorntation fichI 
(!pl, !P2, O) are: 

tl3 = (11 + ~) 1t3, 1 + ~ !P2 
t:n = I-l U~.l - ~ !P2 

t' J = (I-l -t %) 1t3.: - % IPI 

tn = f11~3,2 + % !Pl 
mll = (f3 + y) f('1,1 + a e 
mn = (f3 + y) !P2,2 + a e 
tn33=ae 

ma = y f('2.1 + fi f(' 1 .2 
'J1h, = ì' 9)1.2 + f3!pu 

[.31J 

Let us now assmue that a monochromatic wave propagates, 
alnng thc x~ -Ilircct.ioll, in all infinite micropolar clastie plate, whosc 
thiekness is '2H. lIInrcovcr we a810ume that thc sut"faces of theplate 



Lum A:'D LOVE WAVE 1'1lOl'AGATIO"'. };TC. 

are free of st.l"esses allll eouple ~tl'eHS(,S. 1'his leadf\ to the followillg 
boundary eOIl(!iti()Il~: 

t.~ = 11111 = "IIh~ = 0, at Xl ± ll 

Equations [9.b] are sa.tisfied hy: 

lH'ovided: 

gh =A,1-F,2 

rp~ = A,: + F,l 

[(a + fi + y) [7~-2% -Iò2 j,W]A = O 
[y [72 -:3 % -I~~j~I~J F-% tt3 = ° 
[(,u -1- %) [7~ - f! ò2jòt:j 1t3 -1- % (72 r = O 

[33] 

The last two equations L33J, aft.er simple manil)ulations, ean be redueed 
to the following form: 

{[y [7~ - 2 % - I i'l~!i'll"] [(/t + %) [7~ -I] i'l~/i'lt2J + % (72} r = O 
{[y (72 -:3 % -Iò2jWJ [(/t + %) (72 -,!ò~jW] + % [7:} lt~ = O 

T>et us a,ssume solutions oi the iorm: 

A (x" X2, t) = ..1* I ~ ,\ ~'i;' ",,-fI)l) ,_o, y 

r (Xl, X2, t) =r* (Xl) ei ( I,; X2 - wt) 
'!I~ (.1.'1, X2, t) = 1~3* (:lh) ei (f~ x, - Wf) 

'l'hen the first. equatioll [33] and equations [34J heeome: 

, .. 2 , 

+ ":'A l ..1* = O 
ti!- I 

[3,LaJ 

[3U] 

[35] 

[36] 

_ko)!r. ~ O 
[37.a] 

1(
"0 ·(dO 
-- - k2 + a~ --
d.l.'l ~ ) d.1.'1 ~ -70"11 , , tI~* = O 

[37.hJ 



~:, I\n~('lll 

where 

w' 

\\'e f;eal'eh, as iu the {Jn'vious pase, wllll.ious of equalions 13li] a1l(1 

[37] or lhe ffJl'lll: 

whel'e 

L1'~' =.iI Sillh (~,r l ) + n eoslt (~,rl) 
F* = C sinh (1.-1 xl) + /) eosh (10 1 ;rJ) + 

+ p; siuh (b xd + p cosh (h ;:t'l) 
1t3 * = U sinh (l'1 xd + L eosh ('" ,Td -i· 

+ r f;illh (1.- 2 ;1'd + Q ('osh (h :l',) 

l~et Il,~ uow speeialize om' study to tlw ('.Hse or synullet.l'ie vihm·· 
t.ions. 'J'hen lIte POllllloupnt (f2 of t)l(' miel'ol'olalion YW'j:OI', t.he C'om· 
ponents mll a.Jl(1 m~~ or the eOllple stl'e~s teusoi' an!! lhe ('omllOlll'nl 
tl~ of lIte f;h'es8 tellsor must he symll1etl'iC' with l'espert to the pIane 
Xl ~ (I. 'J'hiR leadR lo: 

whel'e: 

(" 

..-1=D=P=L 

G = el () 

'II=],~. 

From the a~sllmplion thllt t.he delel'lniuant of lite system or the 
three homogeneous ewwtions, obtl1.ine!! giying tu! eXlllieH expression 
to the ])Ollllllllry tonditiollS F~:.!J, nllV;t. be zero, wc get tlw followiug 
pel'ifHl equation: 



L,nlll A:s"D LOVE WAVE PROPAGA'no:-.-, ~:TC. 3;)3 

t~h (l; 11 ) I . ( t ~h (/;~ 11 ) \ 
t.gh (1.:, H) = (l' + (3) Y. k- yz kt tg h (/;1 11 ) _ . y l h) + 

+ I la + ~ + y) ,e -" l ' I ( !l,I, -!le I, t":, I;'e 11
1
1) )' I. 

tg l (" ) 

, l-l . ; Iy + ~I' k' Il, I, - le Id [3DJ 
where: 

l'n = 
" J.. R , , I 1 ., 

~' 1/ = ,_o 

For wave Iengths HlllaU NJlnparell to the thiekneHH '2Il, equatioll 
[391 rmIllceH to: 

'~+~)~ C lhh-hh)-~h-~N 

. [(l' -+- fI) y. 1.: 2 - ((1 -I- fl + l') C ~ - (12 ""J 

It iH OlrViOIlH that we have ilO cOllntel'llart of eqllationfl l39J aJl(1 
[.W] ili eIa.~f1ieal rla~ti('ity. IIo\l'cver \l'e may eonrlude that ill a micro-
polar rla,<;tie mediulll wc Imve waves of Jjove type, while, in Uw elaH-
sieal framework, this is possible onlr in a layel'ell mellillin by illlposing 
some eOlHLit.ions on thc materiai eonsta.lltS. 

Let us 1I0W slJedalize Olll' stlUIy to the case of Hntisymmet.ric 
"ibl'ations. 'l'he pOlllllOncnt rp~ of the miC1'Orotatioll veetor, the com-
ponents 11/.11 and 11122 of the {'oujlle Stl'Cfl~ tewwr and the componellt 113 
of the stress tellsor are alltisymlllctl'ie with respl'eI- to thc pIane;1', = O. 

These eOIHIitionfl Icall t o: 

B=()=}iJ=(}=P=O 

L = ~ID 

Ci - ,e P 
all(l by the p1'Oeedlll'e, aiready employed in the preTiolls caHeH, \l'e 
get the following lJeriod eqlla.t.ion: 



l:. BOSCHI 

tgh (: H) 

tgll (l'III) 
I (y + fil (lo. I, - lo, Id lo' ( Il I (a ~ fi ~i y) C' - a lo ' I· 

I 
t.gh (lo. Il) 

. fIl h - y~ ti 19h (k~lI) + 

l I tgh ("'l II) + . (y ...!.. fi);,:~ H~ f;" , [J" . l (l' lI) 
:: t.g l " 1

)-1 
-. lo, Y' j ['l] 

For wave lellgths small ('omp~u·e(l with t.he t.hickness ':!.1I, cqu a -
hon [H] reaupeH lo equaj.jon [-l-OJ. 

CO-'"'CL1.:f-no)OO{ 

\Ye have stwli('(l t.lle I~amb ami I~ove \yavc lll·ollllgation in an 
infinite homogl'llcolls miel"Opolar ela.st.ip plate wit.h hee houlHlary 
sm·fa.c·es. "'e havc fOUlla that a diHplacement liel(1 (1/'1, u~, O) alld a 
miel"Ol"Otation tiel,1 (0, 0, op,,) leads lo waves which can be cOluù,lcrcd 
of Lamb's kin(l, \\"hile a, (lisplaccmcnt. fida (O, O,n3) an(l a mip.J'Ol'o-
t.at.ion fida ((Pl, V'e, O) leads to ,,·aves ,,·hich can bc ('(lllsidel'ed of Love's 
kina. 'rhcrcforc il- is possihle to have, in mipropohn mc,lia, I~ovc 
w,òve;; wit.hout. thc comlit.iollS rcquil·ed in t.he framc'\"ork of elaHf<ieal 
clastieily. 

T wish to tlmnk l'rof. 1\1. ('aputo for valuable aiSCURSions, cn-
eOllnl.gcmellt;; an(l support in thiH WOl-k. 

REl<'EREXCES 

(') Bù5CIlI E., 1972. - Body Force /Uui Body C'o!!plf 1~,/lIil'nlfllt~ for Sfi,~mie 
lJi.~locnlioll,~ in Jlic!"Opoln!" Jlfdin, to be published. 

(2) BOSCIlI E., H172. - Body Force, Body COllplr nl/l! 11l'al SOIl.Tec 7~lJlli!.'nlfnl~ 
for 81'iiNnir ])isl()()(1lio! .~ in JJie!"Opolar JJcdia, tu he publishCll. 

(l ) EHI~(;E~ A. C., lfHili. Limar 'J ' ltfO!"y 01 JJic!"opoln!" Eln~lieily. ",). 
.\lat.h. aut! .\leJ~h .. ', 15, (W9. 



LA~l]l AXI' LO\'~: 1\'AVJ-: 1'110\',\(:,1'1'10>;, ETC. 

(~) EI\"IXG \Y. :\1., .L\lWJ-:TZKY \\'. S., l'rn;~s F., Infi,. - '''lo",li(~ Jran'R in 
Layern! Jledill. X('W Yorl,. 

{' I 

{'I 
{'I 

{'I 

:-:iep, 

:-:ice, 

Sce, 

Ree, 

E'. g-. , ('1]. 

e.g., p(]. 

e.g- .• CI]. 

p.p:., el]. 

[(i.121, pa.!!;p 283, 

[G.151 an,] [(i-17J, 

[fi.ll], pag-I' 283, 

[U.211, pag-(' 28fi, 

or Rd. ,. 
png-c 283, or Hd. ,. 
01 Rel. ,. 
01 Hd. ,. 

(') G-OOOLAJ)ZE \'. (;-., In-1-7. - J)i~jJcr8ìon 01 Nn.yleiqh INn'f~ in a I,ayef. 
l'ul,1. Tnst. :Okisill. Aead. Sci. [T.R.:'..:"., 119, 27, 

(lO) Jll~AX n" I!HHJ. ,<,'a/' la 'l'héorif df lo T"'el'll/Oaa.~lirill J/ilTIl/wloiu 
('1!!ljilée. "C. Bt'IlI1. Aca,!. :)t:. 1'1Iris", 268, :;8. 

(") LA~11\ II., UliU. - On Jrlll'es i!l O!l l~IIINlic l'lale. ''l'rot". Bo~·. ~()('. (Loll-
11(11)", A, 93, 11-1-. 

(12) l'RllSCOl"l" .1., Ifl-1-2. - T~loslie Jral'e,· am! ri/!!"lIlif)!lN Ili 'J"hin .7:od~. "l'h il. 
:\Iog-."', 33. 703. 

(1:1) lL\1:Ll':wH, 188!). - Olio III" F/'e1' J"i/i)"{/Iioli~ Ili ali lnlinilr' l'lale oi Ho-
lIIoyen"ol!s isolro})il' Eh/Nlie J[III/I'I'. ·']'l'O('. 1,(I[1(lon \Iatl!. :-<01'.",20, 
22". 

(") ~A'T[) l'., Wfi1. !Slw/y Ili !SI(./,Ia('() Jrlln'~ li: J"e/odl!! 01 SIi/'fOCf ]l"lIl"fN 
/')"f)jiaqllll'f/ "!/)fJn Hla~lh' l'lale,·. '·Bull. Earlh. Hcs. Tnst. (']'ol,:\'())"', 
29, 223.