UNDER WAVES

Journalof^AppliedMathematicsandStochasticAnalysis 4, Number1, Spring1991,71-82

SURFACE WAVES IN HIGHER ORDER VISCO-ELASTIC MEDIA UNDER THE INFLUENCE OF GRAVITY*

AnimeshMukherjee

Indian Institute

of

^Mechanics

of

^Continua

201,ManiktalaMainRoad, Suite

No.

42 Calcutta- 700054,

West

BengalIndia

P

2.

Sengupta Department of

Mathematics

University

of

^Kalyani

Kalyani,

West

Bengal,India LokenathDebnath

Department of

Mathematics University

of

^{CentralFlorida}

Orlando,Florida32816,

USA ABSTRACT

BaseduponBiot’s[1965]theory of initialstressesof hydrostaticnatureproduced by the effect of gravity, a study is madeofsurfacewaves in higher order visco-elasdc mediaunder the influence ofgravity. ^Theequationfor thewavevelocity ofStonely waves in thepresenceofviscousand gravitational effectsisobtained. This isfollowed byparticular cases of surface waves including Rayleigh waves^andLovewaves inthe presenceofviscous andgravity effects. In all cases thewave-velocity equations ^are found tobe inperfect agreement with the corresponding classical results when the effects of gravityandviscosity are neglected.

Keywords: SurfaceWavesinElastic Media,Rayleigh Waves, StonelyWaves,Effect ofViscosity andGravity.

AMSsubject classification: 73D20, 73D25.

*

Received: December 1989; Revised: April 1990

Printedin theU.S.A. (C)1991 The Society of Applied Mathematics,ModelingandSimulation 71

72 ANIMESHMUKHERJEE, P.R. SENGUPTA AND LOKENATHDEBNATH

1.

INTRODUCTION

Love

(1911) hasstudiedtheeffects of gravityonvariouswave

problems

and shown that the velocity^ofRayleigh waves increases significantly due^to the gravitational field when the

wavelength

of the waves is large. Subsequently, Biot

(1965)

has

developed

a mathematical theory of initial stresses toinvestigate the effects of gravity on Rayleigh waves in an incompressible medium assuming that gravity generates an initial stress hydrostatic in nature. Based upon Biot’s theory,

Sengupta

^{et al}

(1974-1987)

^has investigatedthe effect ofgravityonsomeproblems ofelasticwaves andvibrations,and on the propagation of waves in an elastic layer. The effect of viscosity was not considered inthese studies.

Themain

purpose

^{of this paperisto} study surface^wavesinhigherordervisco-elastic solidunder theinfluenceof gravity. Theequationfor thewave velocity of Stonely waves in the

presence

of viscous and gravitational effects is derived. This is followed by particular casesofsurfacewaves including Rayleigh wavesand

Love

waves.

It

is shown that in all cases the wave-velocity equations are in excellent agreement with the corresponding classical results [Bulen

(1965)]

^{when the}effects of viscosity and gravity are neglected.

2.

BASIC EQUATIONS OF MOTION IN A VISCO-ELASTIC MEDIUM.

We

consider two

homogeneous

semi-infinite visco-elastic media,

M

and

M2

ⁱⁿ

contact with each other

(M2

being above

M1)

^{along a common horizontal plane}

boundary.

We

choosetherectangularCartesian coordinate system with theorigin^atany pointon theplane boundary andthe z-axis normalto

M

_1.

We

assumethe disturbance is confinedtotheneighborhoodof the commonboundary and examine thepossibilityofa kindofwavetravelinginthepositive x direction.

We

further assume thatatany^instant oftime all the particular^{in aline}

parallel

^tothey-axis have

equal

displacements, thatis, allpartial derivativeswithrespectto y ^vanish.

Introducing thedisplacement^{vector u} (u,v,w) atanypoint (x,y,z) attime t, it is convenientto separate^thepurelydilatafional and

purely

rotational disturbances associated withthe components ^u and w byintroducing the^twodisplacementpotentials ^and

Surface ^{WavesinItigher}Order Hsco-Elastic Media under theinfluenceof^{Gravity 73}

whichare functionsof x,z and t in the form (2.1.ab) u

= x ^z,

^w

^{= t}

^z

⁺ x

From

theseresults, itfollows that

(2.2ab)

V2 ⁼

^ux

+ Vy +

^w

z-- ^A, VZ ⁼

^{wx u}z

In

standard notations (Bullen

[1965]),

the component v is associated with purely distortional waves, and the quantities

, xg

^{and v} are associatedwith

P

waves,

SV

waves, and SH waves respectively. The dynamical equations of motion for three-dimensionalwave problemunderthe influence ofgravity are

11 ^{C12 )U13}

^)W

2U

aX ⁺ 0y ⁺ aZ ^{+Pg"’x =} POt ²

(2.4/

0’21

Ox + 30"22 00"23

^)W

32V Oy-+ ^gz +pg’"

^:

^p It2II

(2.5)

)0’31 )(3’32 (3’33

(U )V) ^{),2 .W}

0x ⁺ 0y ⁺ :Oz

^Pg

"x ⁺

"" ^{= p’’3t}

²

where

ij

^{are the stress} components,

p

is thedensity of the medium and g is the acceleration due togravity.

According to Voigt’s definition, the stress-strain relations in a higher-order visco-elasticmedium are

(2.6) crij = iij %k--- ^A +

2

g eij Ot

^k

where

0, go

^and

1, gl, ,2, i.i.2,

^)t

n, I.t",

are respectively

Lame’s

elastic

74 ANIMESHMUKHERJEE, P.R. SENGUPTA AND LOKENATtIDEBNATII

constants and theeffectofviscosityconstants,respectively.

We

substitute (2.6)intoequations

(2.3)

(2.5), andassumethat all partial derivatives withrespect^to yvanish. Thisleads us toobtai’nthe following dynamical equations of motion of ageneralvisco-elastic solid under the influenceofgravity.

(2.7)

(.k ₊ l.l.k) k x ^{A + I.t}

^k

^{V2u+pg "aTx W = p 2U}

(2.8) ^k

O2V

k=Ol.t ^{V2v =}

¹⁹

^)t2

(2.9)

(gvk+gk) ⁸

^k

^8A

k--O

k

V

²

Ou w

+

_l.t

w-pg-" ⁼

⁹

dt²

k=O

Finally, equations (2.7)

(2.9)

^can be expressed in terms of the displacement potentials ^and

q

in theform

(2.10)

Ccj 7 ^{V2j +}

^g

⁼ ’at

²

(2.11) ^k

= O

^k

V2itj- _g--- ⁼

(2.12)

Ik=

⁰

^{k2 k} I

⁼

^2(v)j

wherethe suffixes j 1, 2 have been usedtodesignate quantities for the^media

M

and

M

₂ respectively ^and

ka

⁽

⁺

^{2 ) kz}

^l.tj

(2.7ab)

ctj ⁼

PJ

^and

^13j =---,pj

^(k

⁼

^{0,1,2, n)}

Surface ^{WavesinHigher}Order Visco-ElasticMediaunder theInfluence of^{Gravity 75}

3. TIlE

SOLUON OF THE PROBLEM.

In

order^tosolve

(2.10)-(2.12)

forthe medium

M

_x,we write downthesolutions in the form

(3.1)

=

^F(z)

exp

^{{io(x ct)}}

(3.2)

xg ⁼

G(z) exp {io(x

ct)}

(3.3) (v)t ^{= H(z)}

^{exp {i0(x}

^et)}

Substituting for and

11/1

^intothe relations (2.10) and(2.11) weobtain

(3.4)

(d2

²

^+h

²

1 ^F+

_(-1ⁱ

^gonG _{)k (icoc)k}

^{kz =0}

1

k=O

(3.5)

^+R

²¹

3 ^G

^{I1 igolF =0}

k=0

where

22 22

2 ^(0C 2 2 OC 2

hi=

n ^{Co and}

Rx=

n ^-03

E ^{(-l)k (icOc)k OCl} k2 ^{E (-l)k (iOc)k}

k=O k=O

From

equations (3.4)

(3.5)

^{wefind that}

F

and

G

satisfy the ordinary differential equation

(3.7)

+ PlC ⁺

^ql

^{{F,G} =}

⁰

where

76 ANIMESttMUKtIERJEE,P.R.SENGUPTAANDLOKENA’I DEBNATIt

(3.8ab) Pl +

^o,

=

_{2 Plql}

=

4- with

2

cog

2 2

Asolution for

F

from the equation (3.7) is

(3.10)

F =

A exp

(icopz) + ^B

₁ exp

(ioxtz) ^{+ L}

1exp

(-ioz)

+ N1

^exp

^(-icoqlz)

where

A, B1, L

1, and

N

^{are constants.}

For

surface wave solutions,

F

tends to zero atlarge distances from the boundary.

This requirement is fulfilledprovidedthe real partof the argument of the exponential functionisnegative.

In

viewof this condition, theconstants

L1

^and

N

^{in thesolution}

(3.10) for

F

mustvanish in the lower medium

M

_1. Then the solutionfor in

M

is (3.11)

= {A ^exp (icoPlZ) ^{+ B}

1

exp (icoqlz)} exp

{ico(x- ct)}

Similarly, wecanfindthe solution for

t

^and

v

(3.12)

/1 = {C1 ^exp (icoPlZ) ^{+ D}

^exp

^{(icoqlz)} exp}

{ico(x-ct)}

(3.13) (v)

=

exp {io

(sz ⁺

^x

^ct)}

where

c2

Sl

⁼

Pln

1

)k )k

^k

Z(-1

^(icoc

^g

k=O

1/2

Surface ^WavesinItigher Order Visco.ElasticMediaunder theInfluence of^{Gravity 77}

withapositive imaginary part, and

C

^and

D

are constants.

It

thenfollowsfrom(3.4) that

C

^and

D

^{are relatedto}

A

^and

^B

1

through

C

⁼

nA

^and

D rB

^where

(3.14)

(-

l)k (ic0c)k ₁

2 2 k=0

n₁

=

^(C0

p

^{h )}

icog

)k

^k

(-1 (ic0c)^k

2 2

h21)

^{k 0}

r

=

(CO

q

icog

A

similar argument enables usto findthe solutions in theuppermedium

M2.

We

next formulate the two boundary conditions which must be satisfied for the presentproblem:

I.

The components of displacementattheboundary surfacebetween themedia

M

and

M2

^{mustbecontinuousatallpointsand times.}

II.

Thestresscomponents 3, r32, ^and

33

^{mustalso becontinuous atallpointsand}

times across theboundarysurface.

Usingtheboundarycondition

I,

from the values of

(

^and

gr

in thetwomedia, after use oftherelation

(2.lab)

ⁱⁿeach case, we obtain

(3.16) AI(1 _-nlP I) ⁺ B1(1

^qql)

⁼ A2(1 _{+ n2P} 2) ⁺ B2(1 ⁺

r2q

2)

(3.17)

E = E

₂ ^and

(3.18)

AI(P ⁺

ⁿ

1) ^{+ Bl(q +}

^r

I) ⁼ A2(-P2 ⁺

ⁿ

2) ^{+ B2(-q2 +}

^r

2)

The stress componentsin the visco-elastic media ofVoigt’stypearegivenby

78 ANIMESHMUKHERJEE, P.R. SENGUPTA AND LOKENATIIDEBNATH

(3.19)

(3.20)

(32)j ⁼ I.t ^Oz

^and

(3.21)

+

2

gj

)Z2 ⁺ 0x0z)

where, as before j = 1, 2 for the media

M, ^M

2.

Applying^{the second}boundaryconditiontoequation

(3.19)

(3.21),we obtain (3.22)

l.t{Al(nlP

^{* 2 2p}

n) ⁺ Bl(rq

²

2ql r)}

* 2 2 2

= {A2(n2Pe ⁺ 2P2- ^{n2) + B2(r}

^q2

^{+ 2q2 r2)}}

(3.23)

SlktlE =-s2k2E2

(3.24)

A {%1(1 ⁺

^{p )}

^{+ 2glPl(Pl +} nl)} ⁺ B{t,l(1 ⁺ q) ⁺ 21.tql(ql ⁺ rl)}

= A2{)(1 ^{+ p22) + 2l.t2P2(p}

2

n2)} ⁺ B2{,2(1 ⁺ q)+ 213,2q2(q

₂

r2)}

wherethe asterisksindicate thecomplex quantifies as

(3.25) 0

n

= O + E (-l)k(icoc)kO

^k

k=O

It

follows fromequations (3.17) and

(3.23)

^{that both}

E

and

F

^{vanish and hence}

there is no displacement in the y direction, that is, the is no transverse component ^of displacement. Thusno

SH

waves occurin thiscase.

By

eliminating the ^constants

A1, B1, A,

^and

B=

^fromequations (3.16),

(3.18),

Surface ^{WavesinHigherOrder}Visco.ElasticMediaunder theInfluence of^{Gravity 79}

(3.22),

and

(3.24),

we obtain the equationfor the wave velocity in determinant form

(3.26)

F l(P

i,

nl) ^F l(q

i,

nl) ^F:(pl, nl) Fl(qz Hi(pt, hi) ^Hi(qi, hi) H(p2, n) H(q, p)

where

(3.27)

F i(p,n) ⁼ gi(P

^{* 2n}

^{2p n),} F2(p,n) ⁼ 2(n

^{* np2}

^2p),

*

2)

^*

L(1 ^p2) 21a.2p(n ^p).

(3.28)

Hl(P,n) ⁼ -1,1(1 ⁺

^p

^2[.tlp(

^p

⁺

^{n) tg}2

= +

Theroots ofequation

(3.26)

determine the wavevelocityofsurfacewavepropagation along the commonboundarybetweentwovisco-elastic solidmediaof the

Voigt typeinthe

presence

ofagravitational field.

In

otherwords, this equation gives the wave velocity of Stonely waves inthe

presence

^ofviscous andgravity effects.

In

the absence of theseeffects, equation

(3.8)

reducesto thatfor the classical Stonely waves (Stonely,

1924).

Finally, wecan derive results forRayleigh waves and

Love

waves as special cases of^this analysis.

4.

RAYLEIGH WAVES.

In

thiscase, theuppermedium

M

₂ isreplaced by vacuumso that theplane boundary now becomes a freesurfaceof the lower medium

M.

Consequently,

A

₂

B

₂ 0 inequations

(3.22)

^and

(3.24),

^{and these}equations assume^theform (4.1)

Al(niP

²i

2Pl

ⁿ

^{I) + Bl(riq}

²

^2ql

^r)

⁼

⁰

(4.2)

AI{,I(1 ⁺ p) ⁺ 2[.tlPl(Pl ⁺ nl)

^}

⁺ B1{t.1(1 ⁺ q) ⁺ 2].tlql(ql ⁺ rl)}

^{= 0.}

80 ANIMESH MUKHE1LIEE, P.R. SENGUPTAAND LOKENATH DEBNATH

Elimination of the constants

A

^and

B

^{fromequations (4.1) and (4.2) yields the}

followingresult:

(4.3)

(nlP

²1

2Pl-

ⁿ

¹⁾

^{(1

^{+ ql)L1}

^{2 *}

^{+ 2].tlql(ql}

^*

^{+ rl)}}

2 * *

-(rlq

₁

2ql-

^r

1) ^{{(1 +} p) ^{X +} 2tlPx(Pl ⁺ nl)} ⁼

⁰

This is the required wave velocity equation of Rayleigh waves in a higher order visco-elastic solidmediumunder the influence ofgravity. When the effects of gravity and viscosity are ignored, this equation

(4.3)

^{reduces to}the correspondingclassicalresult for the Rayleigh waves (Bullen

(1965)).

5.

LOVE WAVES.

For

theexistenceof

Love

waves, we consider alayeredsemi-inf’mite mediumin which

M

₂ is boundedby two horizontalplane surfaces at a finite distance

H

apart, whilethe lower medium

M

remains infinite as before.

We

nowhavetodetermineonly thedisplacementcomponent v in the directionofthey-axis.

For

the medium

M

we proceed exactly as in thegeneral case, and thus (v

1)

^is given

by (3.1)

^with the imaginarypart of s positive.

However,

for the medium

M

₂

we mustretainthe full solution since thedisplacement^nolongerdiminisheswith

increasing distance from the common boundary of the two media.

Consequently,

we have

(5.1) (v)₂

= A’exp{ic.0(s2z ⁺

^{x ct)}}

B’exp{ieo(s:z +

^x

+

^ct)}

wheretheimaginarypartofthecomplex quantity

sa

^{is nownotpositive.}

Since the displacement component (v)₂ ^{and stress} component

3a

^{must be}

continuousacrosstheplane ofcontact,we have

(5.2ab) (v)

=

^(V)₂

((32)1 ⁼ (Cr3;Z)

^{on z}

⁼

⁰

Surface ^{Wavesinttigter}Order Hsco.Elastic Media under theInfluence of^{Gravity 81}

Itfollows from(3.13) and(5.1) combined with(5.2ab)that

= A’ B’ = I.t2s2(A’-

^B’)

(5.3ab) E ⁺ lSlE1

Eliminationof

E

^{betweenequations}

^(5.3ab)

^yields

(5.4)

A’(s21.t

2 $1

1) ^B’(s2g

2

+ $1.1.1)

Also makinguseoftheboundary conditionthat there is no stress across thefreesurface

(5.5) ((32):z ⁼

^{0 at z}

^=-H

we have from equation

(5.1)

(5.6)

A’ exp(-e.0s2H) ^{= B’} exp(ios2H)

Eliminating

A’

and

B’

between equations

(5.4)

^and

(5.6),

we obtain the result (5.7) _s2g

tan(0sH) + islg

₁

⁼

⁰

This is therequired wavevelocityfor

Love

waves in ahigherorder visco-elastic mediumunderthe influence ofgravity.

It

is seenfrom theequation^that

Love

wavesare notaffectedbythepresence ofagravitationalfield.

For perfectly

elasticmedia,

1

^{k =}

[.l.2k

^{= 0, (k 1,2, ...,} n), equation

(5.7)

reduces to the corresponding classical result(Bullen

(1965)).

6.

CLOSING REMARKS.

Thepresent study reveals that effects of viscosity^andgravityarereflectedin thewave velocity equations corresponding ^{to the} Stonely waves, Rayleigh waves, and

Love

waves.

So

the results ofthis analysis seem tobe useful in circumstances where these effects ^cannotbeneglected.

Some

specialcasesofthisstudy^havebeendiscussedby

82 ANIMESHMUKHERJEE,P.R. SENGUPTAAND LOKENATHDEBNATH

several authorsincluding

Sengupta

etal

(1974-

1987).

ACKNOWLEDGEMENT:

Thisworkwas partially supported by the

University

of CentralFlorida.

REFERENCES

[1] Biot,

M.A.,

Mechanics

of

IncrementalDeformation, John Wiley,

New

York

(1965)

[2] Bullen,

K.E., An

Introduction to the Theory

of

Seismology, Cambridge University

Press (1965).

[3]

Love, A.E.H., Some

Problems

of

Geodynamics,

Dover

Publications,

New

York

(1911).

[4]

Stonely, R., Proc. Roy. Soc.

^London

A

106:416-428

(1924)

[5]

Sengupta,

P.R and

De, S.N.,

Journ. Acous.

Soc.

Amer. 55: 919-921 (1974)

[6] Sengupta, P.R.

and Acharya,

D.,

Gerlands Beitr. Geophys. 87: 141-146

(1978)

[7]

Sengupta, P.R.,

and

Roy, S.K.,

Gerlands Beitr. Geophys. 92" 435-442

(1983)