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Love type waves in a porous layer with irregular interface
0020-7225183 $3.00+ .W @ 1983 Pergamon Press Ltd.
Inf J Engng Sci Vol. 21. No. II, pp. 1295-1303, I983 Printed in Great Britain.
LOVE TYPE WAVES IN A POROUS LAYER WITH IRREGULAR INTERFACE A. CHATTOPADHYAY? and R. K. DE Department of Physics and Mathematics, Indian School of Mines, Dhanbad-826004,India (Communicated by G. A. MAUGIN) Abstract-The dispersion equation is derived relating to the frequency and the phase velocity of propagation of Love waves in a nondissipative liquid filled porous solid underlain by an isotropic and homogeneous half space. The rectangular irregularity in the interface between the upper porous layer and the lower semi-infinite medium with a source in it is studied herein. The modified dispersion equation of Mal and the standard dispersion equation of Love waves are deduced as particular cases. In the present study, the frequency equation is obtained by applying the method of perturbation and the phase velocity curves have been drawn for different irregularities by using the numerical parameteric values as suggested by Biot. I. INTRODUCTION THE PROPAGATION of
Love waves in layered media having different types of variation in crustal thickness has been discussed by several investigators, viz. Sato [l], De Noyer [2], Ma1 [3], Bhattacharya [4], Chattopadhyay [5], although none of them considered the layer as porous, Sato studied the abrupt change in the thickness for the propagation of surface waves in layered media. De Noyer [2] considered the irregularity of the boundary in the shape of sinusoidal curve. In a similar type of problem, Ma1 also considered the thickness increasing abruptly through a certain length of the path. Chattopadhyay [5] has shown the effect of viscosity and non-homogeneity by considering the same irregularity of thickness as Ma1 [3], Chattopadhyay and Kar [6] discussed the propagation of Love waves with rectangular irregularity at the interface, the layer being under initial compressive stress. Recently Chattopadhyay and De [7] studied the propagation of Love waves in an initially stressed visco-elastic layer of rectangular irregular interface with Voigt halfspace. The theory of the propagation of elastic waves in a statistically isotropic fluid-saturated porous solid has been discussed first by Biot [8]. By virtue of this theory, many authors viz., Deresiewicz[9], Bose[lO], Gardner[ll]] solved a number of problems. Deresiewicz studied the effect of boundaries on the propagation of elastic waves in a liquid-filled porous solid. Bose [lo] discussed the wave propagation in the marine sediments and water-saturated soils. In fluidsaturated porous cylinders, Gardner [l l] has shown the effect of extensional waves. In the present paper, the upper porous layer is assumed to be homogeneous and isotropic. The common boundary between the porous layer and the lower homogeneous and isotropic halfspace is considered as slightly curved in the form of a rectangle. The source of disturbance is taken in the lower semi-infinite medium below a certain depth of irregularity. The perturbation method as indicated by Eringen and Samuels [12] has been applied to find the frequency equation. The present analysis has studied the effect of porosity on the dispersion equation due to the irregularity of the common boundary. 2. MATHEMATICAL FORMULATION SOLUTION OF THE PROBLEM
AND
Consider a fluid-saturated porous layer of finite thickness (average) H overlying a semiinfinite medium. Both the layer and the medium are considered as isotropic and homogeneous. The upper surface of the porous layer is assumed to be free and horizontal. The irregularity in the common boundary is taken in the form of a rectangle of length 26 and depth h. The z-axis is taken vertically downward in the lower medium and passing through the source of disturbance tPresent address: Laboratoire de Mtcanique Thtorique Associt au Centre National de la Recherche Scientifique (CNRS), Universite Pierre-et-Marie-Curie, Tour 66, 4 Place Jussieu 75230Paris Cedex 05 France.
1296
A. CHATTOPADHYAY
POROUS
AND R. K. DE
T
LAYER
H
Fig. I. Model of the figure
S which is at a depth H + d from the upper surface of the porous layer such that d is greater than h. The x-axis is chosen parallel to the layer in the direction of wave propagation, the origin being taken at a depth H below the upper surface of the porous layer (Fig. 1). The equation of the common boundary between the upper layer and lower semi-infinite medium is assumed to be of the form z = ef(x) where E is small. The equations of motion without body forces, neglecting the viscosity of fluid, are [S]
aT
a2
etc. where a,,, r,,, . . . are the components of stress-tensor with the fluid pressure p by the relation - T = cup where cwis the porosity of the layer, u,, uy, u, are the components of the diplacement vectors of the solid and U,, UY,Uz are that of the fluid. The coefficients p,,, p,?, p2?are the mass codfficients related to the densities p, ps,pf of the layer, solid, fluid espectively given by, PI1 + PI2 = (1 -(YIP,, PI2 + P22 = LyPf
so that the mass density of the aggregate is p = PI1 + 2Pl2 f p22 = ps + 4Pf
- Pd.
Also these mass coefficients follow the following inequalities ~~~>0,~22>0,~12<0
and
P~~P~~-P?~>O.
The stress-strain relations for the fluid-saturated porous layer (isotropic) are given by a,, = 2Ne,, + Ae + QE, uYY= 2Ne,, + Ae + QE, a,, =
2Ne,, + Ae + QE,
rxy = Nexy,
= Neyl, r,, =
and T = Qe + RE, e = div u, E = div U
(2)
Love
type waves
in a porous
layer with irregular
interface
1297
where A and N correspond to the familiar Lame coefficients in the theory of elasticity, R is the measure of pressure required on the fluid to force a certain volume of the fluid into the aggregate while the total volume remains constant, and Q is of the nature of a coupling between the volume change of the solid and that of the fluid. Inserting (2) in (1) and using the conventional Love-wave conditions, viz., 4 =
0=
M,, II, =
v, (x, z, t) ,
u, = 0 = u:, u, =
(x,
2,
2,
t)
the only equation of motion for the y-component in absence of body forces is
where +, = Y(N/b) in which b=o,,-P:2
_ Pzr
If pz and pz are the rigidity and density of the lower semi-infinite medium which is isotropic and homogeneous, the only equation of motion without body forces is
where 02
=
v%dP2).
For a wave propagating along x increasing direction, it is assumed v = V(x,z)e’“‘,
(6)
By virtue of (6) the eqn (3) becomes (7) Also by (6), the eqn (5) takes the form
a%, a2v2 i2 ax’+~fjp2=0.
(8)
In order to find the solutions of the eqns (7) and (8), the following Fourier Transforms are used: V(x, z) eisx dx
so that
(9)
Using these transforms, the eqns (7) and (8) reduce to 2-
y&alv, =o
(10)
1298
A.CHAITOPADHYAYANDR.K.DE
and d2ii -&+-q2V2=o,
(11)
where 2 a2=[2-+
(12)
CN
and q2 = p-$
(13)
The appropriate solutions of the eqns (10) and (11) are
v,=A e-“* + B e“‘,
-H 6 z < d(x)
(14)
and
V2= C em”, z 2 ~j(x).
(15)
Hence, according to (9), the displacements V, and V2by virtue of (14) and (15) will be the form r
Vlk z) =
$ J_02[A eeax+ B e”‘] emi*’d[
(16)
and
V2k
2)
C
= k
e-4z
+ 2
e4z
e-itx
e-d
4
d[
I
(17)
where the second term in the integrand of V2has been introduced due to the contribution of the source. Since E is small, expanding A, B, C etc. in the ascending powers of E and retaining the term up to the first order of E the following approxiations are obtained,
(18) The boundary conditions suitable for the problem are
av,_ az -0 on
z=-H,
V, = V2 on z = cj(x),
av,
ah
av2
(19)
av2
1Nax+nNaz=1CL2ax+nCL2azonz=Ej(x), where I, 0, n are the direction cosines of the normal to boundary z = l j(x). The first two boundary conditions in (19) together with (18) yield A0 eaH - B0 emaH+ E(A, e OH-B,e-“H)=O
(20)
Love type waves in a porous layer with irregular interface
1299
and
=E
I-mm[A,,a - Boa - COq+ 2 edqdlf(x) em’*’d&
(21)
Introducing the Fourier Integral of f(x) defined by,
f(X)= & SO
1-1 f(q) eei9” dn,
that
(22) where
f(s) = f-1 f(x) eiv dx , the eqn (21) becomes, [Aoa - Boa - COq+ 2 ePd]f(n) e-ics+V’XdEdr,
Substituting .$+ q = 5 so that de = dc and replacing 5 by 5 on the r.h.s., the above equation reduces to (A,+B,,-Co)+e(A,+BI-CI)-~e~qd=e~,
(23)
where .?, =
& [_= [Aoa - Boa - Coq +2 e-qd]S’c-rl f(q) dv. x
(24)
For rectangular irregularity of the boundary z = l f(x), the direction cosines of the normal are
where
From the third boundary condition in (19), using (18) and proceeding as before, the similar result is deduced, as aN(Ao - B,) - C,,p2q + 2~~ eeqd+ l[aN(A, - B,) - qp2C1] = E$
(25)
1300
A.CHATTOPADHYAY ANDR.K.DE
where -
s2=%
lx
1-_x[{N(AotSd-p2
(C”+~e~qd)}&) 5=5-q
I
+ {a’N(A,, + B,) - ~~(q%, + 2q emqd)}
f(s) dn.
(26)
From (20), (23), (25), equating the absolute terms and the coefficients of E, the following relations are obtained, = 0, AleaH - B,emnH= 0, AoeaH- BOemaH A,,+B,,-Co=temqd, A1+B1-C1=SI,
aN(Ao - BO)- Co p2q = - 2~~ eeqd, aN(Ar - Br) - ct2q = 32. Solving the above equations, Ao, BO,A,, Br, C, are determined, viz.,
&=me
41*2 -aH e-qd,
A,
= (~26
-
s2)
e-aH
w?
4p7 B”=Doe
uH
-qd,
B,
&Z(I%
’
-
$2)
Wf)
e
euH ’
(27)
ceaH + ,-QH) _ i e-qd,
1 Cr = 0(5) aH(e -a” _ e”“)S, _ (ea” + e-““)S2 where D(t) = p2q(e”” + ePH) - aN(eeaH - eaH).
(28)
With the help of the relations in (27) the eqn (24) becomes,
ae G
((p2 - N) (eeaH - euH)}]c=i-q
f(q) dn.
(29)
In a similar way, the eqn (26) takes the form (eaH + e-O”) {(N - p2)571+ a2N - p2q2}]*=‘-’ f(n) dn .
(30)
Considering the effect on an abrupt increase in the thickness of the upper layer (porous) on the displacements in Love wave motion, the boundary between the upper layer and the lower half-space will have its equation in the form 2 = Ef(X) where Ef(X) = h, R - 6 < x < R +
= 0, (x + RI > 8,
s,
Love type waves in a porous layer with irregular interface
1301
R being the distance of the middle point of the rectangular irregularity from the origin. With h l=j$<
p2q&$2=2cL? x 7r
I K
UC& - N)(emaH - eaH)
-z
-(eaH
+ emaH){(N - p&q
+ a2N -
p2qz}) &]‘=‘-’f(q) dn (32)
where e-qd
8(5)=
-D(5)
[aq(p2- N)(emaH- eaH) - (eaH t emoH){(N - p&$q -t a"N - p2q2}]ei”*.
Applying the Willis [13] asymptotic formula to the integral in (32) and neglecting the terms containing the powers of l/S higher than the first for large values of S, [see Tranter 121 the integral on the r.h.s. of (32) becomes
I
‘[g(i-‘f)+g(i+‘))l~d$l~Zg(i)=?rg(i).
0
Consequently,
[aq(p2 - N)(eeaH - eaH)- (eaH+ e-“H)(a2N -
=y
f$.
p2q2)]
(33)
Thus, the displacement in the porous layer is obtained from (16) with the help of (18) and (33), as T 4&qd D(l) 1
=
=212I+
(1 + hg(&)eqd}[e-a(H+zf+en(H+r)]
4p,2e-qd
a(Hlz)+ ea(H+z)~e-i@
D(i){1 - hg(~)e4~ ]e-
e-ii”
df:
dc
(34)
Hence the value of the integral depends entirely on the contribution of the poles of the integrand, the dispersion equation for Love-type waves is given by D(5) - ~g(~)D(~) eqd = 0. Substituting the value of D(J) from (28) and g(c) from (33), the above dispersion equation takes BES Vol. 21 No. II--B
A. CHATTOPADHYAYAND R. K. DE
1302
the form tanh (oHI _ _ /+q + k(Nq’- p2q2) Nu + ~~q(~* - N) *
(35)
Replacing o by ck and 5 by k where c is the common wave velocity, the dispersion eqn (35) becomes, ?J(-$)-hk[g(l-$)+($l)] tan[kH ,/(-$-I)]= ~(~-~)+~k(~-*)~((~-~)(~-l))’
(36’
Putting 01= 0, ps = pl, N = pi, the eqn (36) reduces to ~~(l-~)-~k[~(~-~)~~-~]
tan[kH ,/($-I)]= d($-l)+hk(;-1),&&f+))
(37)
which is the frequency equation obtained by Ma1 [ 101. Substituting h = 0, in (37), the famous Love wave equation can be obtained. 3. DISCUSSION
AND CONCLUSION
The frequency eqn (36) expresses the fact that the phase velocity of Love waves depends upon the frequency, thickness, porosity of the layer and also on the perturbation parameter
1.725
1.665
1.655 (
kHFig. 2. Phase velocity curves vs kH.
Love type waves in a porous layer with irregular interface
1303
roots of the frequency eqn (36) are real when c > cN and c < pz. Thus the propagation of Love waves in a porous layer underlain by an isotropic medium having a rectanbular irregularity at the interface is possible provided
where cN is giVen by (4). For the purpose of graphical representation of phase velocity (c/cN). The following parametric values as suggested by Biot have been considered, T = O&y
= 0.0, f = 1.05,$ = 4.0 and H = 37.5 km.
when h/H = 0.0, 0.15 and 0.2, the values of kH for different values of C/CN have been calculated. The graphs in Fig. 2 show that the phase velocity (c/cN) decreases rapidly with small increase of kH for h/H = 0.15 and 0.2. But when h/H = 0.0, the rate of decreasing of corresponding phase velocity is less as compared with the previous one. Thus, the above analysis indicates that the rate of decrease of phase velocity (c/cN) is prominent for low frequency range when h/~ = 0.15, 0.2, whereas the corresponding rate of decrease is noted for high frequency range when huh = 0.0. Acknowledgemenf-One of the authors (R. K. De) is grateful to the authorities of the Indian School of Mines, Dhanbad for financial support. (Receiued 15 December 1982)
REFERENCES [I] Y. SATO, Brtfl. Ear&q. Res. Inst. 30. 101-120(1962). [2] J. DE NOYER, Bull. Seism. Sot. Am. Q(2), 227-235(1961). I31 A. K. MAL, Geof. Pure Appl 52; 59-68 (1%2). [4] J. BHATTACHARYA, Gerl. Beitr. Geophys. 71(6), 324-333(1962). [S] A. CHATTOPADHYAY, Bull. Cal. Mafh. Sot. 70, 303-316(1978). I61 A. CHATTOPADHYAY and B. K. KAR, Geoph. Res. Bull. 16, (1) 13-23 (1978). [7] A. CHATTOPADHYAY and R. K. DE, Rev. Roum. Sot. Techn. Met. Appl. Tome 26, (3), 449-460 (1981). 181M. A. BIOT, J. Acousf. Sot. Am. 28(a) 168-178,(b) 179-191(1956). [9] H. DERESIEWICZ, Butl. S&m. Sot. Am. 50(4), 599-607(1960). lOI S. K. BOSE, Geof. Purer e AppI 52-27 (1962). ,111G. H. F. GARDNER, .I. Acoust. Sot. Am., 34, (1962). 121 A. C. ERINGEN and C. J. SAMUEL& .I. Appl. Me&. 26491-498 (1959). 131C. J. TRANTER, Integral Transforms in Math. Phys. 63-67 (1966). l4] H. F. WILLIS, Phil. Msg. 39, 455-459 (1948).
Inf J Engng Sci Vol. 21. No. II, pp. 1295-1303, I983 Printed in Great Britain.
LOVE TYPE WAVES IN A POROUS LAYER WITH IRREGULAR INTERFACE A. CHATTOPADHYAY? and R. K. DE Department of Physics and Mathematics, Indian School of Mines, Dhanbad-826004,India (Communicated by G. A. MAUGIN) Abstract-The dispersion equation is derived relating to the frequency and the phase velocity of propagation of Love waves in a nondissipative liquid filled porous solid underlain by an isotropic and homogeneous half space. The rectangular irregularity in the interface between the upper porous layer and the lower semi-infinite medium with a source in it is studied herein. The modified dispersion equation of Mal and the standard dispersion equation of Love waves are deduced as particular cases. In the present study, the frequency equation is obtained by applying the method of perturbation and the phase velocity curves have been drawn for different irregularities by using the numerical parameteric values as suggested by Biot. I. INTRODUCTION THE PROPAGATION of
Love waves in layered media having different types of variation in crustal thickness has been discussed by several investigators, viz. Sato [l], De Noyer [2], Ma1 [3], Bhattacharya [4], Chattopadhyay [5], although none of them considered the layer as porous, Sato studied the abrupt change in the thickness for the propagation of surface waves in layered media. De Noyer [2] considered the irregularity of the boundary in the shape of sinusoidal curve. In a similar type of problem, Ma1 also considered the thickness increasing abruptly through a certain length of the path. Chattopadhyay [5] has shown the effect of viscosity and non-homogeneity by considering the same irregularity of thickness as Ma1 [3], Chattopadhyay and Kar [6] discussed the propagation of Love waves with rectangular irregularity at the interface, the layer being under initial compressive stress. Recently Chattopadhyay and De [7] studied the propagation of Love waves in an initially stressed visco-elastic layer of rectangular irregular interface with Voigt halfspace. The theory of the propagation of elastic waves in a statistically isotropic fluid-saturated porous solid has been discussed first by Biot [8]. By virtue of this theory, many authors viz., Deresiewicz[9], Bose[lO], Gardner[ll]] solved a number of problems. Deresiewicz studied the effect of boundaries on the propagation of elastic waves in a liquid-filled porous solid. Bose [lo] discussed the wave propagation in the marine sediments and water-saturated soils. In fluidsaturated porous cylinders, Gardner [l l] has shown the effect of extensional waves. In the present paper, the upper porous layer is assumed to be homogeneous and isotropic. The common boundary between the porous layer and the lower homogeneous and isotropic halfspace is considered as slightly curved in the form of a rectangle. The source of disturbance is taken in the lower semi-infinite medium below a certain depth of irregularity. The perturbation method as indicated by Eringen and Samuels [12] has been applied to find the frequency equation. The present analysis has studied the effect of porosity on the dispersion equation due to the irregularity of the common boundary. 2. MATHEMATICAL FORMULATION SOLUTION OF THE PROBLEM
AND
Consider a fluid-saturated porous layer of finite thickness (average) H overlying a semiinfinite medium. Both the layer and the medium are considered as isotropic and homogeneous. The upper surface of the porous layer is assumed to be free and horizontal. The irregularity in the common boundary is taken in the form of a rectangle of length 26 and depth h. The z-axis is taken vertically downward in the lower medium and passing through the source of disturbance tPresent address: Laboratoire de Mtcanique Thtorique Associt au Centre National de la Recherche Scientifique (CNRS), Universite Pierre-et-Marie-Curie, Tour 66, 4 Place Jussieu 75230Paris Cedex 05 France.
1296
A. CHATTOPADHYAY
POROUS
AND R. K. DE
T
LAYER
H
Fig. I. Model of the figure
S which is at a depth H + d from the upper surface of the porous layer such that d is greater than h. The x-axis is chosen parallel to the layer in the direction of wave propagation, the origin being taken at a depth H below the upper surface of the porous layer (Fig. 1). The equation of the common boundary between the upper layer and lower semi-infinite medium is assumed to be of the form z = ef(x) where E is small. The equations of motion without body forces, neglecting the viscosity of fluid, are [S]
aT
a2
etc. where a,,, r,,, . . . are the components of stress-tensor with the fluid pressure p by the relation - T = cup where cwis the porosity of the layer, u,, uy, u, are the components of the diplacement vectors of the solid and U,, UY,Uz are that of the fluid. The coefficients p,,, p,?, p2?are the mass codfficients related to the densities p, ps,pf of the layer, solid, fluid espectively given by, PI1 + PI2 = (1 -(YIP,, PI2 + P22 = LyPf
so that the mass density of the aggregate is p = PI1 + 2Pl2 f p22 = ps + 4Pf
- Pd.
Also these mass coefficients follow the following inequalities ~~~>0,~22>0,~12<0
and
P~~P~~-P?~>O.
The stress-strain relations for the fluid-saturated porous layer (isotropic) are given by a,, = 2Ne,, + Ae + QE, uYY= 2Ne,, + Ae + QE, a,, =
2Ne,, + Ae + QE,
rxy = Nexy,
= Neyl, r,, =
and T = Qe + RE, e = div u, E = div U
(2)
Love
type waves
in a porous
layer with irregular
interface
1297
where A and N correspond to the familiar Lame coefficients in the theory of elasticity, R is the measure of pressure required on the fluid to force a certain volume of the fluid into the aggregate while the total volume remains constant, and Q is of the nature of a coupling between the volume change of the solid and that of the fluid. Inserting (2) in (1) and using the conventional Love-wave conditions, viz., 4 =
0=
M,, II, =
v, (x, z, t) ,
u, = 0 = u:, u, =
(x,
2,
2,
t)
the only equation of motion for the y-component in absence of body forces is
where +, = Y(N/b) in which b=o,,-P:2
_ Pzr
If pz and pz are the rigidity and density of the lower semi-infinite medium which is isotropic and homogeneous, the only equation of motion without body forces is
where 02
=
v%dP2).
For a wave propagating along x increasing direction, it is assumed v = V(x,z)e’“‘,
(6)
By virtue of (6) the eqn (3) becomes (7) Also by (6), the eqn (5) takes the form
a%, a2v2 i2 ax’+~fjp2=0.
(8)
In order to find the solutions of the eqns (7) and (8), the following Fourier Transforms are used: V(x, z) eisx dx
so that
(9)
Using these transforms, the eqns (7) and (8) reduce to 2-
y&alv, =o
(10)
1298
A.CHAITOPADHYAYANDR.K.DE
and d2ii -&+-q2V2=o,
(11)
where 2 a2=[2-+
(12)
CN
and q2 = p-$
(13)
The appropriate solutions of the eqns (10) and (11) are
v,=A e-“* + B e“‘,
-H 6 z < d(x)
(14)
and
V2= C em”, z 2 ~j(x).
(15)
Hence, according to (9), the displacements V, and V2by virtue of (14) and (15) will be the form r
Vlk z) =
$ J_02[A eeax+ B e”‘] emi*’d[
(16)
and
V2k
2)
C
= k
e-4z
+ 2
e4z
e-itx
e-d
4
d[
I
(17)
where the second term in the integrand of V2has been introduced due to the contribution of the source. Since E is small, expanding A, B, C etc. in the ascending powers of E and retaining the term up to the first order of E the following approxiations are obtained,
(18) The boundary conditions suitable for the problem are
av,_ az -0 on
z=-H,
V, = V2 on z = cj(x),
av,
ah
av2
(19)
av2
1Nax+nNaz=1CL2ax+nCL2azonz=Ej(x), where I, 0, n are the direction cosines of the normal to boundary z = l j(x). The first two boundary conditions in (19) together with (18) yield A0 eaH - B0 emaH+ E(A, e OH-B,e-“H)=O
(20)
Love type waves in a porous layer with irregular interface
1299
and
=E
I-mm[A,,a - Boa - COq+ 2 edqdlf(x) em’*’d&
(21)
Introducing the Fourier Integral of f(x) defined by,
f(X)= & SO
1-1 f(q) eei9” dn,
that
(22) where
f(s) = f-1 f(x) eiv dx , the eqn (21) becomes, [Aoa - Boa - COq+ 2 ePd]f(n) e-ics+V’XdEdr,
Substituting .$+ q = 5 so that de = dc and replacing 5 by 5 on the r.h.s., the above equation reduces to (A,+B,,-Co)+e(A,+BI-CI)-~e~qd=e~,
(23)
where .?, =
& [_= [Aoa - Boa - Coq +2 e-qd]S’c-rl f(q) dv. x
(24)
For rectangular irregularity of the boundary z = l f(x), the direction cosines of the normal are
where
From the third boundary condition in (19), using (18) and proceeding as before, the similar result is deduced, as aN(Ao - B,) - C,,p2q + 2~~ eeqd+ l[aN(A, - B,) - qp2C1] = E$
(25)
1300
A.CHATTOPADHYAY ANDR.K.DE
where -
s2=%
lx
1-_x[{N(AotSd-p2
(C”+~e~qd)}&) 5=5-q
I
+ {a’N(A,, + B,) - ~~(q%, + 2q emqd)}
f(s) dn.
(26)
From (20), (23), (25), equating the absolute terms and the coefficients of E, the following relations are obtained, = 0, AleaH - B,emnH= 0, AoeaH- BOemaH A,,+B,,-Co=temqd, A1+B1-C1=SI,
aN(Ao - BO)- Co p2q = - 2~~ eeqd, aN(Ar - Br) - ct2q = 32. Solving the above equations, Ao, BO,A,, Br, C, are determined, viz.,
&=me
41*2 -aH e-qd,
A,
= (~26
-
s2)
e-aH
w?
4p7 B”=Doe
uH
-qd,
B,
&Z(I%
’
-
$2)
Wf)
e
euH ’
(27)
ceaH + ,-QH) _ i e-qd,
1 Cr = 0(5) aH(e -a” _ e”“)S, _ (ea” + e-““)S2 where D(t) = p2q(e”” + ePH) - aN(eeaH - eaH).
(28)
With the help of the relations in (27) the eqn (24) becomes,
ae G
((p2 - N) (eeaH - euH)}]c=i-q
f(q) dn.
(29)
In a similar way, the eqn (26) takes the form (eaH + e-O”) {(N - p2)571+ a2N - p2q2}]*=‘-’ f(n) dn .
(30)
Considering the effect on an abrupt increase in the thickness of the upper layer (porous) on the displacements in Love wave motion, the boundary between the upper layer and the lower half-space will have its equation in the form 2 = Ef(X) where Ef(X) = h, R - 6 < x < R +
= 0, (x + RI > 8,
s,
Love type waves in a porous layer with irregular interface
1301
R being the distance of the middle point of the rectangular irregularity from the origin. With h l=j$<
p2q&$2=2cL? x 7r
I K
UC& - N)(emaH - eaH)
-z
-(eaH
+ emaH){(N - p&q
+ a2N -
p2qz}) &]‘=‘-’f(q) dn (32)
where e-qd
8(5)=
-D(5)
[aq(p2- N)(emaH- eaH) - (eaH t emoH){(N - p&$q -t a"N - p2q2}]ei”*.
Applying the Willis [13] asymptotic formula to the integral in (32) and neglecting the terms containing the powers of l/S higher than the first for large values of S, [see Tranter 121 the integral on the r.h.s. of (32) becomes
I
‘[g(i-‘f)+g(i+‘))l~d$l~Zg(i)=?rg(i).
0
Consequently,
[aq(p2 - N)(eeaH - eaH)- (eaH+ e-“H)(a2N -
=y
f$.
p2q2)]
(33)
Thus, the displacement in the porous layer is obtained from (16) with the help of (18) and (33), as T 4&qd D(l) 1
=
=212I+
(1 + hg(&)eqd}[e-a(H+zf+en(H+r)]
4p,2e-qd
a(Hlz)+ ea(H+z)~e-i@
D(i){1 - hg(~)e4~ ]e-
e-ii”
df:
dc
(34)
Hence the value of the integral depends entirely on the contribution of the poles of the integrand, the dispersion equation for Love-type waves is given by D(5) - ~g(~)D(~) eqd = 0. Substituting the value of D(J) from (28) and g(c) from (33), the above dispersion equation takes BES Vol. 21 No. II--B
A. CHATTOPADHYAYAND R. K. DE
1302
the form tanh (oHI _ _ /+q + k(Nq’- p2q2) Nu + ~~q(~* - N) *
(35)
Replacing o by ck and 5 by k where c is the common wave velocity, the dispersion eqn (35) becomes, ?J(-$)-hk[g(l-$)+($l)] tan[kH ,/(-$-I)]= ~(~-~)+~k(~-*)~((~-~)(~-l))’
(36’
Putting 01= 0, ps = pl, N = pi, the eqn (36) reduces to ~~(l-~)-~k[~(~-~)~~-~]
tan[kH ,/($-I)]= d($-l)+hk(;-1),&&f+))
(37)
which is the frequency equation obtained by Ma1 [ 101. Substituting h = 0, in (37), the famous Love wave equation can be obtained. 3. DISCUSSION
AND CONCLUSION
The frequency eqn (36) expresses the fact that the phase velocity of Love waves depends upon the frequency, thickness, porosity of the layer and also on the perturbation parameter
1.725
1.665
1.655 (
kHFig. 2. Phase velocity curves vs kH.
Love type waves in a porous layer with irregular interface
1303
roots of the frequency eqn (36) are real when c > cN and c < pz. Thus the propagation of Love waves in a porous layer underlain by an isotropic medium having a rectanbular irregularity at the interface is possible provided
where cN is giVen by (4). For the purpose of graphical representation of phase velocity (c/cN). The following parametric values as suggested by Biot have been considered, T = O&y
= 0.0, f = 1.05,$ = 4.0 and H = 37.5 km.
when h/H = 0.0, 0.15 and 0.2, the values of kH for different values of C/CN have been calculated. The graphs in Fig. 2 show that the phase velocity (c/cN) decreases rapidly with small increase of kH for h/H = 0.15 and 0.2. But when h/H = 0.0, the rate of decreasing of corresponding phase velocity is less as compared with the previous one. Thus, the above analysis indicates that the rate of decrease of phase velocity (c/cN) is prominent for low frequency range when h/~ = 0.15, 0.2, whereas the corresponding rate of decrease is noted for high frequency range when huh = 0.0. Acknowledgemenf-One of the authors (R. K. De) is grateful to the authorities of the Indian School of Mines, Dhanbad for financial support. (Receiued 15 December 1982)
REFERENCES [I] Y. SATO, Brtfl. Ear&q. Res. Inst. 30. 101-120(1962). [2] J. DE NOYER, Bull. Seism. Sot. Am. Q(2), 227-235(1961). I31 A. K. MAL, Geof. Pure Appl 52; 59-68 (1%2). [4] J. BHATTACHARYA, Gerl. Beitr. Geophys. 71(6), 324-333(1962). [S] A. CHATTOPADHYAY, Bull. Cal. Mafh. Sot. 70, 303-316(1978). I61 A. CHATTOPADHYAY and B. K. KAR, Geoph. Res. Bull. 16, (1) 13-23 (1978). [7] A. CHATTOPADHYAY and R. K. DE, Rev. Roum. Sot. Techn. Met. Appl. Tome 26, (3), 449-460 (1981). 181M. A. BIOT, J. Acousf. Sot. Am. 28(a) 168-178,(b) 179-191(1956). [9] H. DERESIEWICZ, Butl. S&m. Sot. Am. 50(4), 599-607(1960). lOI S. K. BOSE, Geof. Purer e AppI 52-27 (1962). ,111G. H. F. GARDNER, .I. Acoust. Sot. Am., 34, (1962). 121 A. C. ERINGEN and C. J. SAMUEL& .I. Appl. Me&. 26491-498 (1959). 131C. J. TRANTER, Integral Transforms in Math. Phys. 63-67 (1966). l4] H. F. WILLIS, Phil. Msg. 39, 455-459 (1948).