I'roc. lndian'~Acad. Sci., Vol. 88 A, Part Ill, Number 2, Mat 1979, pp. 133-146, 9 printed in India.

On

propagation

and attenuation of Love waves

A R BANGHAR Seismology Section, Bhabha Atomic Research Centre, Bombay 400 085 MS received 22 April 1978; revised 10 August 1978 Abstract. The period equation for Love waves is derived for a layered medium, which is composed of a compressible, viscous liquid layer sandwiched between homogeneous, isotropic, elastic solid layer and homogeneous, isotropic half space. In general, the period equation will admit complex roots and hence Love waves will be dispersive and attenuated for this type of model. The period equation is discussed in the limiting case when thickness //. and coefficient of viscosity, t12, of the liquid layer tend to zero so as to maintain the ratio P -- H J q ~ = a constant. Numerical values for phase velocity, group velocity, quahty factor (Q) and displacement in the elastic layer and half space have been computed as a function of the frequency for first and second modes for various values 3f the parameter P. It is shown that Love waves are not attenuated when P = 0 and oo. The computed values of Q for first and second modes indicate that when P:~ 0 or co the value of Q attains minimum value as a function of dimensionless angular frequency. Keywords. Love waves; period equation; viscous compressible liquid; equation.

I.

wave

Introduction

Viscous liquids like elastic solids can transmit [5] both compressional and shear waves. Most liquids are viscous, even though the viscosity may be small. For example, water has coefficient of viscosity 0.0178 in CGS units. Whenever a wave propagates ia a viscous fluid, the amplitude of the wave attertuates with increasirtg distance from a reference point. Hence by introducing a viscous liquid layer of finite thickness irt between homogeneous, isotropic elastic solid and homogeneous elastic half space, it is possible to study the dispersion and attenuation o f surface waves. Banghar e t a l [3] have studied the propagation of Rayleigh waves in a system composed of viscous compressible liquid layer sandwiched, between two homogeneous isotropic solid half spaces. The object of this paper is to study the propagation of Love waves in a system composed of a compressible viscous, liquid layer sandwiched betweer~ homogeneous, elastic Layer and homogeneous isotropic elastic, half space. The period equation for Love waves for the above mentioned model is derived and discussed for a particular limiting case. 133

P. (A)---4

A R Banghar

134 2.

Theory

The waves studied propagate in a stratified medium composed of a viscous compressible liquid layer of finite thickness H 2 sandwiched between homogeneous isotropic solid layer and elastic half space. Figure 1 shows the geometry of the system under consideration. Let (x,y, z) be a Cartesian co-ordinate system such that x and y are horizontal while z is taken positive downwards. We shall assume the wave propagation to be two-dimensional (i.e., wave propagating only in x - z plane). It is also assumed that all variables of state are independent of y.

3.

Basic equations

Isotropic homogeneous elastic solid. The equations of motion (in the absence of body forces) can be written as

p(~2S/~t'2)=(,~ + ~ ) V |

+lzV2S,

(1)

where S = (u, v, w) is a displacement vector, E)=V

9

(2)

S,

V " = (~-'/~x ~) + ~'/~z.',

(3)

p is mass density and 2 and/z are Lam6 constants.

For Love waves

S = (o,v, o),

(4)

|

(5)

Hence, the ectuatiort of motion for Lava waves reduces to p ( ~ . / ~ t ~) = ~ V ~ v.

(6) Z =0

~X Eloshc sohd layer Z

P'I, Pl, fil

Z=HI V,scous

hqu=dlayer

~2' P2 ii

i

,,

Z=HI+H 2

Eloshc sohd half space Figure 1. Diagram showing the geometry of the problem under investigation.

Propagation and attenuation o f Love waves

135

Equation (6) can be written as ~2 v / a t ~ = f12 V 2 v ,

(7)

where fl = (lz/p)t is the x elocity of shear waves in an elastic solid. Let/zl, Pl, fll represent rigidity, dertsity and shear wave velocity in a solid layer, while/13, pa, fls represent corresponding quantities in elastic half space such that

pdJ' = u~.

(8)

Displacements v~ and v 3 in solid layer and half space will satisfy ~: vl/~t" = fl~ V " vl,

(9)

and b%,s/i~t 2 = ,B| V " v3.

4.

(10)

Viscous, compressible liquid

The equations of motion (in the absence of body forces) for viscous compressible liquid [5] call be writtert as p2(bV/M)=--Vp +-~V(V.V)+n~V

~v,

(11)

where P2 is density of the liquid, t/2 the coefficient of viscosity and p is the overpressure, and V = (Us, 112, Ws) is the velocity vector. For Love waves,

and

V = (0, V2, 0),

(12)

27 . V = O.

(13)

Therefore (11) reduces to a single equation, which can be written as Ps (~ V2/bt) = ri2 V 2 Vs.

(14)

It may b~ mentioned here that the constancy of overpressurc follows from (11). This constant overpressure is taken as zero. .'. p = 0.

05)

Hence, compressibility of the liquid does not affect the propagation of Love waves. Equation (14) can also be represented as

~vd~t =

v V ~ ~,

where v = qz/Ps is the coefficient of kinematic viscosity,

(16)

A R Banghar

136

5.

Solutions of wave equations

We assume that time depondance is given by a factor exp (iwt), hence (9), (10) and (16) can be represented by (,: e ,, k~,)v~ = O,

(17)

~ ~ k ~ ) v~ = o,

and where

08)

(, ~ -~- k~,) V,,.= 0,

(19)

2 o 2 k#~ = oo-/fl~,

(20)

k ~ --~ o f-t,~T>,

(21)

k~, = - - icolv,

(22)

and e ) = 2nf is the angular frequoncy. Sciaticas of (17), (19) and (18), for the geometry under consideration, are written as

v~ = [A ~o~ vxz + B

sm vlz] t~xp [i (~ot

-- kx)],

V.2 = i~o [C1 cos v,,z -r- D sin v.,z] exp [i (cot -- kx)], and where

va = E exp (-- Va z) exp [i (wt -- kx)]

(23) (24)

(25)

v[ = k~, -- k ~,

v] = k 2 -- kS,, Real v3 > 0, arid

C = co~k,

(26)

and k is a horizontal wave lJumber, C is horizontal phase velocity and A, B, C1, D and E are constants to be determined. It may be pointed out that while writirtg the solution of wave equatiort in the elastic half space, we have takert into consideration Sommerfeld [7] cortditiort. Further we remark, that the form of ~ given by (24) was chosert so that constartt~ A, B, G , D and E have the same dimensions.

6.

Boundary conditions and period equation

Tile boundary conditions [4] are the vanishing of normal and tangential stresses at the free surface z = 0, and continuity [2] of velocities and normal and tangerttial

137

Propagation and attenuation o f Love waves

stresses at the interface z = Ht and z =/-/1 + H,,. Following file ~mtation used by Ewing et al [4] for normal and tangerttial stresses, the boundary conditions are P,,=0

and

(27)

at z = 0 ,

[e,,la = [P~,12

"] at

icova = V2

d

(28)

/41,

z

[P,,]2 = [P,,]a'] V2 = io~va] .I at z =/-/1 + Hz

(29)

In the present case we have P.

and P,, are identically zero,

and P,u----/t (a~/az) for elastic solids while P~ = ~/(aVJ?z) for viscous fluid. Using (27), (28), (29) and (30), we obtain

(30)

ltl v~B -----O,

-- #~v, A sin v, H1 + lt, Vl Bcos vail, = ico~12 [-- c~v,. sinv~ Hj + Dv.~

cos v2/-/d, A cos v t H a + B sin vlH~ = Ca cos v2H1 + D sin v~H1,

i~o~l~[ - Cl vz sift v~ (1-11 + H2) + Dv~ cos v2 (//1 + H~)] = I~av.~E x exp [ - v/3 (1-11+ t/~)l, Cx cos v~ (/-/i +/-/2) + D sinv~(H1 + H~) = E exp [-- va (Ha + H~)]. (31) Equafiort (31)are five homogeneous equations. In order to have values of A, B, Ca, D artd E different from zero, we obtairt /tlva

0

#j vl

0 ~ - e -~a (H~+H~)

0

0

0

- - # 1 v~

io9~/~ v e

- - icon/z v~

cos vail1

sin vlHa

sin v2H 1

cos v~/-/1

0

sirt vlH1

cos vlH1

-- cos v~H1

-- sin v~//1

0

0

0

0

0

-- iogrl2Vz io)rl~ v,~ sin v2 (/-/1 + H~) cos v~ (/-/1 + H 0

0

cos v2 (Ha + H~) sift v_.(/-/1 + H.,) -- 1 (32)

Expanding ttte determinant represented by (32), we obtain io9v~12 [lll v~ tart villi

-

-

llzV3]

+ ~ l v t # a v a t a r t vxH1 -- co2 ~1~ v~] tart v~H2 = O.

(33)

A R Bang~mr

138

Equation (33) represents the period equation for Love waves in a layered medium composed of a compressible, viscous, liquid layer, sandwiched between homogoneous, isotropic solid layer and homogorteous, isotropic, solid half space. In general (33) will admit complex roots. Using (23), (24), (25) and (31), it can be shown that vl -----A cos vxz exp [i(oot -- kx)], va = X exp [va (Hx q- H~. -- z) q- i(o~t -- kx) cos hHx cos vzH2]

and

-1 F ilaxVl + 1 1 -+- a,~/awz tan vxHl tan vsH.[ - - j "!.

7.

(34)

Discussion

We shall discuss the period (33) in the following limiting eases. Thickness (H2), and coefficient of viscosity (r/s) of ~he compressible liquid layer tend to zero separately in suGh a way such that the ratio of thickness to coefficient of viscosity is finite, i.e., H z =~ 0, r/z "-' 0, such that

P = H~/r/2 = a constant. With those approximations, (33) reduces to [1 -

(ipmv~)/o,] tan vlH1 = (/~vs/~ v3.

(35)

With the same approximation the displacements in the elastic layer and elastic half space, given by (34), reduce to ~t -----A cos vx z oxp [i (ot --

and

~

kx)],

,~ ei?~- ~ ( z - ~,)+ ~(~ot-kx)l ~ +,~eco tan ,~H~

COS v i e 1.

(36) It may be pointed out that for P --- 0 (thickness of liquid layer becomes zero for a finite coefficient of viscosity) equation (35) gives

tan vdtl = (mv~/~ vO.

(37)

which is the period equation [4] for Love waves for a case whoa a homogeneous isotropic elastic solid layer is in welded contact with homogeneous, isotropic half space. Using (35) numerical values of phase and group velocity for first and second modes of Love waves as a function of 7 ~- 2fill~ill for the case P3/Pl ---- 1.0, and /t;//t~ = 1.25, were computed and are shown in figaro 2. When P toads to infinity (i.e,, coo~cie'tt of viscosity toads to zero for a finite thickness of the liquid layer), eqtiation (35) reduces to tan v~rl -- o.

(38)

Propagation and attenuation of Low' waves

139

1-15

/Firs) mode

I\

110 U

',

/3)

\

c

\

1.0

__

C

Fhrst mode"

'\c

u_A \

\\

II ~

Second mode --''j'4~

/9

0 1

~

z/

1

I

1.0

10 0

y

=

2 f-H 1

100.0

-

P1 Figure 2. Variation of non-dimemiona! phase--(C/,81) and g r o u p - (U/,81) velocity as a f u n o t i o n o f non-dimensionalfrequency for first and second modes of Love waves. Equation (38) is the condition of existence of Love waves [6] in a homogeneous, isotropic solid layer overlying an inviscid fluid layer 1"4]. It may be pointed out that no Love wave energy is coupled to the lower half space, because a perfect fluid does not transmit Love waves. Using (38) numerical values ofphaseand group velocity for first and second modes of Love waves as a function of frequency have been calculated and are shown in figure 3. It may be noted, that in this case Love modes show inverse dispersion and the group velocity varies from 0 at long periods to fl~ at short periods. Displacement v 1 and ~3 given by (36) were calculated for first and second modes for some values of P as a function of depth and are shown in figures 7 and 8. It is interestio.g to rtote that for P about 20, elastic solid layer decouples from th0 rest of the system under consideration. Figure 8 also shows that the node in displacemeat for second mode is dependant on the value of P. For all values of P ~ 0 and P ~ co (35) will have complex roots. We shall choose the complex values of k and C in such a way such that 09 is real. We introduce hero quality factor Q, which is defined as 1/Q = 2Uk*/k,C [1] where k, and k* are real and imaginary parts of a complex wave number k while U and C are respectively the real parts of the complex group velocity and phase velocity. Equztion (34) was solv0d numerically for phase velocity, group velocity and Q for

140

A R Banghal

5.0

40 Second mode

U

B1 30 C

C

/31 20

F~rst mode

1.0

/",,~'~"~ I

i/

0

c~

,

\

~1 II ! I

Second mode

First mode

1-0

1'0 0 2fH I 7

=

Figure 3. Variation of non-dimensional phase--(C/fll) and group--(U/fll)as a function of non-dimensional frequency for first and second modes of Love waves

various values of P, as a function of the artgular frequency for first and second modes o f Love waves, assuming pa/pl = 1.0 and pa//tl ---- I .25. Figures 4, 5 and 6 ~how the variation of Q for first and second modes of love waves as a function of non-dimensional angular frequency for various values o f P. It may be poio.ted out that the value of Q for a given P first decreases, attains a minimum value artd then ira.creases as a function of the angular frequency. Table 1 shows the values of the real part of the phase and group velocity for first m o d e as a function of angular frequency for various values of P. It is noted from this table, that the value of group velocity is more than that o f phase velocity for certain frequencies and for some P ' s ,

Propagation and attenuation of Love waves

o

o,o.1

.~o

"7.. 0

~ C~

I

0 ~-

0 ~

_

141

~

0 ~"

I

0

0

0 0 0 0

0 0 0 ~

.o

_

I

......

I

0 0 0

o

0 0

(2'

o "T 0

o uO

0

0

0

-~

.~ ~Ta 0

0

9~ ,

(

f

0 0 0 0 -09

0 0 0 0

._

~

~

I 0 0 0 "-

Q

0 0 ~-

0

.~.~

142

A R Banghar

- ---

I

Firsi mode Second mode

I I

100000 /

t

I

/ / / /

I

,../

(b

10000

I

Q

/ /I/

1000

I I00

//

I0

~0 0

Y1 =

IC0 0

~) H I

,%

Figure 6. Comparison of Q for first and secovdmode of Love waves for some values of the parameter P.

Propagation and attenuation of Love waves

143

Displocement 0

1"0

-T

0.5

#

1-0

1.5 z

'7 P : 0 x 2 O 5 ,~ 10 20

HI

2-0

2.5

3.0

-!

coil1

P~

-

6.1

Figure 7. Variation o f displacement for first mode of Love waves as a function o f depth for some values o f the parameter P and coH~/fl~ = 6 . 1 .

A R Banghar

144

O [splacemenl 0

-1.0

10

0

0-5

1.0 z

1.5

2.0

2 " 5

P=O 2 5 10 20

x 0 A B

--

el 3.0

,

_

m

Figure 8. Variation of displacement for second mode of Love waves as a function of depth for some values of the parameter P and oHa/fll = 12.02.

8.

Conclusions

The period equation for Love waves is derived for a layered system composed o f a compressible liquid layer sandwiched between homogeneous, isotropic, elastic solid layer and elastic half space. The period equation is solved for the case when thickness and coefficient of viscosity of the liquid layer tend to zero such that their ratio is held constartt (P = H2/r12 ---- a cortstant). It is shown that the case P = 0 corresportds to the case of solid elastic layer in welded corttact with half space. When P tends to infinity, it correspo~.ds to the case in which the overlying elastic solid layer is not coupled to half space. It is inferred that for any layered model composed of a compressible, viscous liquid layer overlain by homogeneous isotropic elastic layers and underlain by homogeneous isotropic half space, Love waves will not only be dispersive but also are subject to attenuation as they propagate. This problem is of great relevance to compute the response o f structures due to Love waves generated by crustal earthquakes.

Propagation and attenuation of Love waves

145

~

8

H

j

9

o

8~

II

L ~r~

~'~

0 ~,

~

~

~'~

r

II

q~ ~

II

146

A R Banghar

Acknowledgements The author is thankful to Dr. G S Murty and the late Dr T G Varghese for useful discussion during the course of this investigation.

References [1] [2] 13] [4]

Anderson D L 1967 Geophys. d. R. Astron. See. 14 135 Aoki H 1973 d. Phys. Earth 21 97 Banghar A R, Murty G S and Raghavaoharyulu I V V 1976 J. Aeoust. Soc. Am. 60 1071 Ewing W M, Jardetzky W S and Press F 1957 Elastic waves in a layered media (New York: MoGraw-Hill Book Co., Inc.) [5] Lamb H 1932 Hydrodynamics (New York : Dover Publication) [6] Press F and Ewing M 1951 Trans. Am. Geophys. Union 32 673 [7] Sommerfeld A 1949 Partial differential equations in physics (New York: Academio Press)