A sludy on lhe displal~emenl compollenls 

of Hayleigh waye (*) 

K. G. BIlA'!'!.\. - {'" '"l ,r. ('. ])nAKOl'OlTLOR ('"'"'") 

Ri{'C'nlto l'l l }'('hhra io 1071 

St:)DLlltT. - 'l'Il!' i'.llld)' ha" heCll ClllTi(>.,1 ont to O])I<C1TC tIIe cJl'cct" 
of l'Oi;;~OIl mt io on I he di;;pb('cmCllt ('OJll)lOIlCll ts of Hayleigll "\l'an's in clasti(' 
mc,lia. Tll g-encral 1llc prohlem or ~llr!a('e wave;; lJave hel'n sLndied hy 
JIIan~' alll!wr". 'l'IJC allthor~ Iwve adopled arhitral)' valnes or l'oi""on ndio 
in onicI' lo ;;tlld.v tI,e vllrililions in 1111" ratio of Illc ,li~plat'cmCllt (·oJllponl'ld.~ 
on tlw ;;l11'[a('e and inside thl' ml',lill. Thc variat-ioll.~ of 1h(\ IlOriwll l al an(1 
vcrli('al aJllplitu,]P>< of ~llrfa('c wavc ,li.~pla(·cJllent with depIlI il' a JIIatler 
of ,!l;reat intel'e"t l,otll in Sei;;Jllolog-.I" and l<:nl'lhquake Eng-illel'ring. 

Hr ,ISSU:<ITO. - (Illc><to stwlio (, ~1[1,to c.~CI-.'1tito allo "ropo ,li o><"eryare 
g-1i efl'eHi ,lei l'apporto ,li Poi'~''',n Sllll(, "oJllponellti ,leg-li I<poslaJllellti delle 
onde di Hayleig"11 in JIIe:o::o:i ela~li('i. In g-enerale il prohleJlla delle onde "nper-
fkiali ì' "talo ><tlHliato ,la molti :mtori. (:Ii autol'i llallllO a,lo!tllto 
yalori arhifrari per il l'apporto ,li Pois.~(l1l in JIIo,lo ,la I<twlilll'nc lr variaziolli 
;;ullC' ('OJIIIHIIJCllti deg-li ;; postamenti ri s pctto alla I<upel'fkic c nell ' jll!enlO 
dei JIIe:o:zi. Le nll'iaziolli (Ielle ampie:o:ze ori:o:wntali l' verti('ali (Iello i'.]lo sta· 
mento dell'olllia ~11perfi('iall' ri"pet-lo alla pl'OfOliditit, sono di noj.(\volr inte· 
rp;;sC' "ia Ilei' la :';i"m(,I('g'ia ('he IJf'r l'lng'l'gneria "ismi('a. 

1. - l:VI'll,OLlI'('TIO-"\. 

(*) Intcruntional In"t ill111' o[ :-\l'i;;Jllolog-~' alld l<:al'l!J'Illakc EIIg-inecrin!!; 
'l'ok.vo. ,rapan. 

(U) :-\('I)(lOl or Heseardl allOl Trainillg- in l-:'1l'1h(pwke Eng-ineel'illg, 
Univer~i!y or Hoorkee. [llOlia. 

(***) :-\l'isJllolog'i('al Lahorntory, . .\..thclI" l ' niverl<ity. Grl'e(·(·,. 



K. !<. BlI.\T!A - .1. C. l>Jt.·\K()I>O\Jr.".~ 

11"\1"\, showc(l tlw.t tll(' fl'l'p SlIl"l":Wl' or an cla.4i(· hOIllO).!l'IWO\l.~ lu:dl'l·ia.l 

(·H.Il ~lIppOl't H. (li~tlll"hH.lI(·(·. il1\c oh'ill).! (li~phwP1I1(,llt~ iII a Yel·ti(·l~l pialle 

;~IOll).! tll!' (lil·p(·tioIl or pl"opa.).!atioll iIl tll!' [01'111 or Ho I·PÌ!·o).!l·a(Lp (·llip.~(' 

alHL ])l·opa).!atp with a· \'(·]o(·ito\" whi("]t \\'a~ about O.D tillH'.~ the ~IH'H,I' 

wa\'(' ,"(·1()(·iI.\' ill tltat llH'di\llll. 

,J('Jl'l"l''y.~ an(L StolH'il\~' H.IHllat('I' Ewill).!: alu] P]'{',~~ gl'('aH," (\xjH11l(lc(l 
tlt(· tlt('m',\' to ('0'"('1' (·a .. ~e~ or la~·(']'(·(l ~(,1I1i·illlillitl' llH'(lia., '['lw I"cslIlt.~ 

ilu1kl~te tltat iII H,Il hnt tll!' ~i1l1pl(·~t ('lh~{' a,~~\lllw(l h," },01'(1 Ha~-l(·ig'h 

tllP~p wayp~ ~lIjT(,l'('(L (Li~]){,l"~ion. 'l'hl' ,~\lI'fl~("C wan'~ ('01l1])(1,~ill.!! the 

Ita,\'lpi).!h aluL l,oyp wa·n·,~ are (Li,~tillg'\li~h(\(l fl'ol11 l' alu1 N [)()(l.\· wa.vl·.~ 

in la·in).! 1I101'{' 01' l('~,~ ('olllilj(\(l to tlw ~\ll"r~We, '['h\l~ tllt'il' alllplit\l(ll' 

(lt·('I"(%~('." YN," mpi(ll," with (lepth lwlO\\' tlw slIl'fa('(· . 

. , 

.\.~~1I1l1{' a ,~illlpil\ hH.I·lllOlli(" pbIlP \nwp j,mill ll'l~,"('l1ing' in tlwa' 

(lil"edioIl ~\ldt tilat 

1) '['li/' (li,~t\ll"hl~IH'e i,~ in(lep('lldpllt or .1/ ('oo]'(lillah' 

11) It iIl('n'H.s{'~ mpidl.\· witlt (Li~Ì1~I)(,(' z fl"OllI UH' fn'e ,~\ll'rlwe. 

?{ Illn/hm 

'l'he [olIowill):{ ,,:vmhol.~ hav!) 11eell 11dopt.ell [ot' ll~e iII thiH 11n1l('r; 
E ela~tidt~" 
Il C'oetli!'lent of rij!idi t y 
!1 ilensiL~- o[ lIIellia 
l, Lame';; !'onHtaJlt 
<]J ~ealar 1l1JtC'lltial of di;;plaC'ellient 
'l' ('1'"'1',,,'1'.) yedor potellt ial o[ d j,'plat'eJlIC'nl 
L W:lve \ength 
U {'OlIl\H·e;;.;;ional wave ye\o('ity 

" RhC'ar wave .elo!'it:r 
c pha.~e vcloC'itr 
() l'lI1)i!'al (lilatat.ioJl 
r:r POi;;SOIl',; rat.io 
w all!:{ulat' freqllelU'Y 
k pal'amct.er [or waTe ef]lwtioJl 
i"j (.i":,..l"y,F,) .~ hml.," fon:c~ 
U,\",IV = dii>plaeelllellt iJl J:,!J an,1 z dircdioJls l'e;;pediycly 
F" Rayleig-h wave \-clOl'ity 
T rate of dencase oi alliplitwlc 
') l'atio of ;;.hC'ar \l'a'-e velodly to l'IJJllpl'{'~siollal \l'aye yelodi)" 
~ ratio o[ phasc vclo('ity to 8heal' ,un-e ,"C'lol'ity" 



{'()]IHidel' the [{' . .1/ plalW to be H·~ bonlHlarr 1\o1\(1 t,he ;; nxis llo~iti-ve 
tO\\'H.I'd t.lle intl\rior of thc Holi(l. J~et. C l', alHl 11' he tllf' (li~plH'('('nwnt~ 
in .1', H alld ;; dil'edioll~ re~pC(,ti"\·{'I.'·, 

'l'IL(' (li~plH'('omp,lIt Ìllll(·t.jOll {}_ t (l', l", 1\') i~ dofill('.(l (3) H.,~ 

Sill('o diHphwement i~ i))(!{'llC)Hlcnt. of H Wl', lmve 

r 
a(/I a'1-'~ 

\ 
- a;1' a;;; 

a(f'l alV, . 
" I " 

a,1' 

a([J aVI ~ 
IV 

" a.r 

[:: l 

[,1] 

whcre J. H,]](l fI H·re t,he Lame's (·on~t-a.nt~ a.))(l a·I'(' g-iY!'ll 

al',' 
;l1)(1,1f 

~ (1 -1- a) (1 --'- rr) (l - a) 

o i,~ t,lw (·,nlli(·a.l (lila.t,H,tion and ÒII i~ ('rOlH'(']wr (!{'lta 1.(', 

=1fori=j 
1)/1 " 

" OfOl'i ,z'oj 

G is the POi~~Oll l'l1,t.io all(l E i~ the Hl()(llll(!~ 01 f'h~j.i(·i1.y. 
Bqnation 01 illotioll i~ ~ivell :l~ 

wlwl'O Dr i~ the M'\'(md (ll\l'jvM,ivl\ 01' n, with J'('~I)(,(·t t,o time, !l i,~ tllf' 
(l(,lI~it.., H.II(l 

!' '-' -' 

1"l'om [IJ, [~:r, [::)[ alld [II 

a (l'u) 
'j) j 

.. af) 
!} Vi - (I, + Il) - --;- ,II V" 0, 

, "' 



Bill o 

a,ntl W(, ()ht;:~in 

tI:. (l. 1\ !I .\T 1.\ - .1. e. l'J\.\I\ O \·ÒU J.oS 

'! n = (). + Il) V f) -;-ft V~.II. 

l' c -c 
\ il x 

, r il II' \ 
...J- = V. f) = V 2(/> 

il)" ilzl 

'! j) = (i. -+- .Il) v (V · q») -I- /' y2 ( D). [H] 

Rllhstit.ut.iug fur 11 lmd flo l' :~ml;i ll~ I"c:~1 :~I\( I i ll lHog i llary part>l Wc uht:~i n 

,,·he.t·c 

Su l llt.iult u( f'jJ 

whCt·c 

I V~ ,p = 
l tF '/J 

I /1- Ò I. ~ 

) I iI~ y'J, 
V ~ l/Il 

I 
,,~ a I ~ 

I ~ r 
\1 ' , -

b' a t~ 

.. Ii .~. :! /I 
I{= 

" alltl I " 
( '1~1I ho w rjLt c lI ;\ .. ..; 

) 

m =_ .·1 '-'xI' [ = q:: -:- i (1·,1 - /;.1.)1 
'p = IJ oxp [± .~:: ~ i ((.,1 -'- }.-;1")1 
1- - o cxp [ ± .~z -+- -i ((.,1 - k'f)l 

a llli 

,., 
a,wl 

[7] 

[S] 

Wl 

[ W .I 

R i lll'C IOI" ill1l 'fa("{~ W:~\,j!S tl ,o 1Il0tiOil hl-\(·.Qm('I.~ llog1igihlc n.t ; ~ dist.all(,C 
of a f(\\\' \\':we lcugt.!ts fl"tlm thc f,'oo SIIl"f:t,(·() i\nlu Lioll [9] tak~ I.he funll 

) 

(/) = AI' Xp [-fF ...... i (l'lt · -k.r)l 
III = IJ \lXp [- .~;; + -j ("11 - k..l· )] 
V = C (l XP [ . . ilZ + i (1"/ - 1.-.r)] 

jll'()yillHI! C < /J < Il. 

[11 1 



'rlw ~ii.(-ll ha,~ heCll MI l'I1OS(1) SO that potcllti:~! he('()mc~ ZCI'O a,~ ::: 
telHh to illfillit,\', 

'l'o ohtn-i)) i'·OllHtallt~ ..:1, H, C, tlw 1!())!lU!n.I')' ('Oluliti01)~ ;~l'e: ali 
~tre~sl'~ lH\l~t Y:~lli~h at hp(' I\ul'fal'c i,e, ::: = O 

::: =0, [1:! J 

FI'Olll [:{I, [11] a!l(l [I:!J 

) 

C - " 
A (:! i'1k) - l:i (k~ + s~) -- () 
A [(i, + :! l') q~ - },k~J -1- N (:! pis) = () 

[13J 

b 
Il = u 

F!'Olll thi,~ Wl' SCl\ that (1)e l'oot of ~ i~ ZO!'O lmt this :..dYi>-~ phu,w. 
yd(wity {'(Limi to Z('!'O alu! 1)(\1)(,{' i~ of 110t \l~l\. 

'flwl'ofm'c 

.\!~O rl'Olll eq. [1-1.1 it is ohYiou~ that tlle eClImtio\! llO\ds ).!"OOl! \\"11('1) 

() <IJ <!J < fl. 

b 
il = 

ti 

lli~plaI'ClllCllt~ U : ~1ll1 IV are 

[ alli1 li" 
à<jJ a'P 

+ ax oZ 

11" ha~ hel'1) !'l'p!u.(,('(l hy VI he·('·:mso othl'l' two i'OmIHl\lf'-1)t.~ or Il' (lo 
llot ('ome i11 lo l'lay, 

FI'Olll [I Ll a.lul [I Cii we I!et (I) 

U = -..:1 L..:1kie- -gE -- Nsc" J cxp [i (ud - k.r)J. 



il-t l" (i, 1l1lA'f]A - .1. <:. IlRAKOPOULO;; 

'l'akillg olll.\" ma,l jml't 

a,II(1 Himila.r1\' [W] 

Il' --AI[ t- q,- ~ ,- t- O< cos(U)t-/;.I') l 
., k' l 

1;2 + I;~ 

'l'he mt-l' at whi(']t t.hl' l~mplitnde or tlw diHpla'('('JIl('llj, l~lollg t.lw 
(Iil'p('/,ioll or pI'OpH,gn.tioll dCI'nmms with (Iept,h (IO]lOlldH npoll t 110 Iad:lll' 

[Iii 

. \1:-;0 the mte l~t. whir',h t.he amplitnde 01' thc lllot.iou ill lI. (lil'o(,tiOll 
llol'mal lo lIw ~nrfH,ce de('l'eaHe~ wii:h dcp1h (Io]l{)]ul:-; npOll t.lw fl~(·tlli' 

[181 

" 

", 

l<'ig. I - Coellil'ÌCllt Hi aera.y of allljllihltie in lite (lirediull or pTupag-alillll 
u[ Hayleigh \\"a\'e ill ela~tie lllc(li;t [or aiITNcllt Ynlu('.~ of POiHROl\'S rali\). 



~()\\-

a nd 

" -'" 1.: = 
C 

= ~i: whe rn J~ i" "he \favo k ng l. h n{ H<~yl(;'igh w.wc. 

" 
o- o 

\ 
" 

~ , o l S 
~ , o J O 

o • O" 
• '" 

\ 
~ - o le 

" . . ~, 00() . -' 
" 
" 

" '\ 
" 

~ 
o " " " " " " U~--

F ig_ :2 - Cno/lic·-ient. Cl f ,Ioca)- CI I :l llI p litwlc iII t-Ile ,li l"ellt·iu ll llf'-rPclI.lit,ulll r lo 
pn' 111l~!C t i nlL {l f RIl,\'I~ i gh waye in cla.~ t i e UI (."(lill fn r ,I ilfcre n t. "ailwJ; IlE 

l'ui"sOIl '~ ratio 

t:ìlllmt.i t.ntin l:-' [3] ill [li] :md [18J wo .L!ol 

-r" =- oX !, I -:!:t "';1 -1}~ , 



:JU 

all ~1 

[ :! OJ 

(~ee Fi~ . 1 a mi :!) . 
:\g:\in l et (." " au d lv. he th(> di >l p hwelllell t "umpol\cll t>l iII tlw :~lLd 

pCl"p tmdi ("nbl" t o th c dil"(I(·t.ioll of ]ll"op:tgatioll l'e~ptwtiy(\ly at t.lw f]"( .' l~ 

~ ll IL\f"c z = fI. 
'J'ltcu 

Th e ll 

. 1 ' ';1- ''''' ';]- '' 1 l; ~= Ak 1 - ~ --=--.;: -

[I 

W 'I 

"2 1[.~ 
"~9' _ - . .. o-o. 

f,;~ -I- ~1 

~llbs ti t.util\~ 101" q. If alll i f,; II-t! ub t.nin 

[ :!.] j 

[ :!· ~. l 

[e3J 

[:l5 J 



U 
\l' - ./1 - 'I = !~ 

., -.I;~ - , 
[27J 

EI/lmti llllil Hl, :2n, :W , :':7 , ~g lwe l'lo t tetl fil" diffel't\nt yal uo:; or Pojssnn 
l'M io. 'l'J w plo Ul a· .. e o b taill e d nO ll diII1Bll.<;i ol1a.l. Sile Fi~. (],:.:, ;~ , 4: ,"» ) . 

.. 
" 

[" 
~----------------------==~~ 

i~. 3 _ R:1 li u fl{ rlllIl,Jj t mle ill th e d inw-l io u (I[ Jlmp:1~a.ti,m al ( ho .. urlncl\ 
(U~) a ll fl i !\ t,llll l!l(\( tia ( U) [ m' fli ffll nlllL Yilhlt!~ of Pui .'ls( )JI ',~ 1"nti u. 



K. G. 1I1iATIA - J. C. DRA1\oPOUI,OS 

;{. - ('O,\T·LDUO.'\S. 

l"ig'. 1 ;~Il(L Fig'.:! l'CIIl'llf;ellt the ('ocftì('ient of dway (lf amplitude 
ill thc di!"C(,.tioll aliti pCI'IIClHLiellla.1' to thc (Lil'c(·t.ioll of ])J'opag'lt,iù)JI, 
Fig. li indieatcs a eompa,n.tivc ~tlldy of t.hc.~c t.wo ('ocffi('icllt.~ lt,Il(L 

,. ~ 
" 

, " \\\ ,. 
" \ 

\ 

" \ 

" \\ 

~::~:: " 
" <1. o lO 

-'::"::::-":: '1 .8 20 
~a . I O 

a ·O OO 
O O " " " " l /L _ 

Fig. 4 - Ratio of alllplitulle in tllC rlired.ion perpendieular to propagation 
at the ;;urfare (IV,,) and in the Tne,lia (W) fol' difIel'ent values of Pois~on's ratio. 

revcals tlmt tlte ('ocftì('icnt ('ol'J'esponding to the (lispla('cmcnt ('OHl])(II!ellt 
in the dil'ct'tiOll or pI'OII;~g'ation f;how a ('lmng-e in the ph;~He llelH to 



Pi/.!:. Ci - Ratio of amplitwle ill thc (lil'CdjOll (U) all(l pCl'pelHliclllar t.u t.ll(' 
,1iJ'ediull (W) of pl'O]Ja)..!;atioll foJ' (lifferellt. valllcH of l'oif'.,<.;OIl'H ratio, 

" " " 

Fig. ti - Coetlidellt of (le{'a~- of (lispla{'cllOent in lite (1irediull of prop:l-
gat.ion (~·x) arul in thc ,lirf'.t'tion l'erpeJl(lio-nlar to propagat.ion (T,). 



100 K. ,;. IlIIATIA - .r. C. llRAKO!'OVLOS 

the d('pth :: = O :!.fo wlwl"p H.S tlw OtIWI' l'Ollll'OIHmt ILm'ays g-l·allua.]].\" 
al\(l hav(\ H. t('llILpIH'Y to hel·OIl\P. a,\\"lllptOtjl' :òt:: _= 1 .HL aluL ol!wl1l"!b. 

I t is Oh,q'TY('ll from Fig.;, that \\"ith tlw illl'l"('a~{l il! tlw l'oi~sol! mtio 
of th(' media. tlw di,~pl:w('ment l'omponent ill the llil"l'diol! o[ tllP jll"O-

pagl~tion l~lso inerea.s{>.s :md tlw nmximllm Yl~llle lies \\"lwn Ilepth is 

bet\\"{\{>.n 0.37 l~ll(l ilAri tillles the \WIoVP leng-th or the Hayh>.iidl waye. 

"\1.~o [01' ali tlw l'ois.~oll l"atio Iff tlw lll('(lil1 thi.~ ('ollìpol\(>nt ehange~ 
phl1.~p at tlw t[ppth jJl'twpell 1l.1;") to il.:!.il tilll(J~ the w;t.Ye length. 

" 

" 
" , , 
O> , 

• , 
• , 

-O> 

-" 

-,. 

" \ " 06 08 ljL _ ... . lO \ 2 Vi" , 

Fig. ""i - [;/(10, II"jl!'" ilJHI U/II" verS\l;; t/L. 

()n tIlP (·olltl"a.l".\" the displa.(·.rnwnt ('ollìponrnt in the (linwtiol! 

PI'l"I)('IHlil·nla.r t.o pl"opagl~tion or waye does not sho\\" l~II.\· pha;·w l·ha.II)!(' . 

..\lso Uw amplit\lIL(\ il!l'n\:t-~ps \\'hrn Ilepth jjrs 1wtwren 0.1 to il.:!. tillì('." 
the wa.V{\ l('ngth l~nlL tlWl! it ILep]"(;l1SI\S l"apit[l~' an(l hr·I·.o]Jl{'S a~~'lllptoj-.i(' 

at 11 t[(\pth or l.:! tillH'S th{' \\"l~\·e [l'llg-th (';l'p Fig. J). 
,-

'l'lw g-"alli{\nt or de(·a..'" or.-.. (SI'I' Fig. :")) ilH·'·el1.q'." with inl'n'l~se 
" 

in tlw Yl~llle of l'oi~sol! l"lttio and IIp to d{>,pth ,- il. il I,. 'l'lwl! IlpI'1"P:'.S(\'; 



;\ STUO)" 01' T II E DlSl'LACE~IE};'f COMPO!'lE!'ITS Ol,' RAYLEiGii WAVE lOl 

" with in('rease in POif!SOIl l'atio. .\180 the ,alue , j 
lV 

is illdepeJldellt, 

of t1l8 rOi~SOIl mtio wIwn dept1l is nea l' to 0.5 t.imes t.lte wa,o !f'llgth. 

U U I\' 
l<'ig. 'j shows eomparative st.ud.\· of TI'. 1 ::lolld com· 

,- Li. Il'. 
ponellts of tlte (lisplaeement. 

4. - RmrAHK. 

The study e::m fUl'tltel' lw extel!(l('d fol' the y,lol'iation in tlte modulus 
of ela~titity and the density of tlw media. 

'l'Iw al1thol's ,we tlmllkfl1! to t,he stall of tomput.el' sedùHl of Ill· 
tel'lIatiolla.l Illf!tit'l1te of Seislllo!OI!Y alld Hal'thql1ake ]!;ngineering 'rokyo, 
Ja.pan fol' tlwil' tÙllcly (·oOlJGI'f[·tÙHl. 

REli'ERENCES 

(I) BIIATIA K. G., SOlllr A spccls 01 sU1Iace II'nve propa(fntlon in Rlaslic alld 
_J /l.I'IIIN/.iC .lI edin, .I Ji.<sfrl.nliOI' for Jl (jstel' rh'gn'cin Rm·thqunkc Engi1lccring, 
Uni\cl'~ity of Rool'l\Oo India, (1!J68). 

(2) HULLEN IL E., Ali In/rod/lctiol, lo 7'heory 01 Scismology "Camb. Univo 
Press ", (l !J(j3). 

(J) EWI!'IG .:'II., .JARJ)El'ZKY ìY. lLllIl l''. P.RESS, E/astic TVal'es in Laycnd _Uedia 
:\TcCraw·TTill Book Company, (1!J57).