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Propagation of Love waves in a transversely isotropic poroelastic layer bounded between two compressible viscous liquids is presented. The equations of motion in a transversely isotropic poroelastic solid are formulated in the framework of Biot’s theory. A closed-form solution for the propagation of Love waves is obtained in a transversely isotropic poroelastic layer. The complex frequency equation for phase velocity and attenuation of Love waves is derived for a transversely isotropic poroelastic layer when it is bounded between two viscous liquids and the results are compared with that of the poroelastic layer. The effect of viscous liquids on the propagation of Love waves is discussed. It is observed that the presence of viscous liquids decreases phase velocity in both transversely isotropic poroelastic layer and poroelastic layer. Results related to the case without viscous liquids have been compared with some of the earlier results and comparison shows good agreement.

Deresiewicz [

In the present analysis, propagation of Love waves in a transversely isotropic poroelastic layer bounded between two viscous liquids is discussed in the frame of work―Biot’s theory. The governing equations of motion in a transversely isotropic poroelastic layer are derived. The frequency equation for Love waves in a transversely isotropic poroelastic layer bounded between two viscous liquids is obtained and the results are compared with that of poroelastic layer. Phase velocity has been computed and analyzed against non-dimensional wavenumber.The effect of presence of viscous liquids is studied and it is observed that presence of viscous liquids decreases phase velocity of Love waves in transversely isotropic poroelastic layer and in poroelastic layer. Also, it is observed that phase velocity for transversely isotropic poroelastic layer is more compared to poroelastic layer. In general, phase velocity decreases as wave number increases.

Consider a rectangular co-ordinate system (x, y, z) with x and y axes taken as horizontal and z-axis as positive downwards normal to the plane. Propagation of waves is taken as two-dimensional (i.e. propagation in xz-plane) along the x direction. A transversely isotropic poroelastic layer of thickness ‘h’ bounded between two compressible viscous liquids is considered. The boundaries of the poroelastic layer are taken as z = 0 and z = h . At z = 0 the poroelastic layer is interacting with upper viscous liquid, whereas at z = h it is interacting with lower viscous liquid. The physical parameters of two viscous liquids are denoted by a superscript j (1, 2) enclosed in parentheses. The parameters with superscript (1) & (2) refer to upper viscous liquid and lower viscous liquid, respectively. The parameters of poroelastic layer are without any superscript. The geometry of the problem is shown in

The equations of motion of a homogeneous, transversely isotropic poroelastic solid in the presence of dissipation b are

[ ( A + 2 N ) ∂ 2 ∂ x 2 + N ∂ 2 ∂ y 2 + L ∂ 2 ∂ z 2 ] u x + ( A + 2 N ) ∂ 2 u y ∂ x ∂ y + ( F + L ) ∂ 2 u z ∂ x ∂ y + M ∂ ∈ ∂ x = ∂ 2 ∂ t 2 ( ρ 11 u x + ρ 12 U x ) + b ∂ ∂ t ( u x − U x ) , M ∂ 2 u x ∂ x 2 + M ∂ 2 u y ∂ x ∂ y + Q ∂ 2 u z ∂ x ∂ z + R ∂ ∈ ∂ x = ∂ 2 ∂ t 2 ( ρ 12 u x + ρ 22 U x ) − b ∂ ∂ t ( u x − U x ) ,

( A + 2 N ) ∂ 2 u x ∂ x ∂ y + [ N ∂ 2 ∂ x 2 + ( A + 2 N ) ∂ 2 ∂ y 2 + L ∂ 2 ∂ z 2 ] u y + ( F + L ) ∂ 2 u z ∂ y ∂ z + M ∂ ∈ ∂ y = ∂ 2 ∂ t 2 ( ρ 11 u y + ρ 12 U y ) + b ∂ ∂ t ( u y − U y ) , M ∂ 2 u x ∂ x ∂ y + M ∂ 2 u y ∂ y 2 + Q ∂ 2 u z ∂ y ∂ z + R ∂ ∈ ∂ y = ∂ 2 ∂ t 2 ( ρ 12 u y + ρ 22 U y ) − b ∂ ∂ t ( u y − U y ) ,

( L + F ) ∂ 2 u x ∂ x ∂ y + ( L + F ) ∂ 2 u y ∂ y ∂ z + [ L ∂ 2 ∂ x 2 + L ∂ 2 ∂ y 2 + C ∂ 2 ∂ z 2 ] u z + Q ∂ ∈ ∂ z = ∂ 2 ∂ t 2 ( ρ 11 u z + ρ 12 U z ) + b ∂ ∂ t ( u z − U z ) , M ∂ 2 u x ∂ x ∂ z + M ∂ 2 u y ∂ y ∂ z + Q ∂ 2 u z ∂ z 2 + R ∂ ∈ ∂ y = ∂ 2 ∂ t 2 ( ρ 12 u z + ρ 22 U z ) − b ∂ ∂ t ( u z − U z ) , (1)

where ( u x , u y , u z ) and ( U x , U y , U z ) are displacements of the solid and liquid media, respectively, while e and Î are dilatations of the solid and liquid respectively; A,N,Q,R,F,L,M and C are all poroelastic constants and ρ 11 , ρ 12 , ρ 22 are the mass coefficients following Biot [_{ij} and the liquid pressure s of the transversely isotropic poroelastic solid given by Biot [

σ x x = P e x x + A e y y + F e z z + M ∈ , σ y y = A e x x + P e y y + F e z z + M ∈ , σ z z = F e x x + F e y y + C e z z + Q ∈ ,

σ x y = N e x y , σ y z = L e y z , σ z x = L e z x , s = M e x x + M e y y + Q e z z + R ∈ , (2)

The physical meaning of other parameters A, N, Q, R, F, L, M and C are given by Biot [

P = E ( 1 − E E ′ ν ′ 2 ) ( 1 + ν ) ( 1 − ν − 2 E E ′ ν ′ 2 ) , F = E ν ′ ( 1 − ν − 2 E E ′ ν ′ 2 ) , C = E ′ ( 1 − ν ) ( 1 − ν − 2 E E ′ ν ′ 2 ) , L = G ′ , N = E 2 ( 1 + ν ) and M = R f 2 .

where P is a poroelastic constant given by P = A + 2 N , the two constants A , N correspond to familiar Lame constants in purely elastic solid, which are positive in sign. The coefficient N represents the shear modulus of the solid. The coefficient R is a measure of the pressure required on the liquid to force a certain amount of the liquid into the aggregate while total volume remains constant. The coefficient Q represents the coupling between the volume change of the solid to that of liquid. E are E ′ Young’s moduli in the plane of transverse isotropy and in a direction normal to it, respectively. ν and ν ′ are Poisson’s ratios characterizing the lateral strain response in the plane of transverse isotropy to a stress acting parallel or normal to it, respectively. G ′ is the shear modulus in planes normal to the plane of transverse isotropy.

For Love waves, the displacement is only along y direction thus the non-zero displacement component of the solid and liquid media are ( 0 , v , 0 ) and ( 0 , V , 0 ) respectively. These displacements are functions of x, z and time t. Then the equations of motion of transversely isotropic poroelastic solid by Biot [

N ∂ 2 v ∂ x 2 + L ∂ 2 v ∂ z 2 = ∂ 2 ∂ t 2 ( ρ 11 v + ρ 12 V ) + b ∂ ∂ t ( v − V ) , 0 = ∂ 2 ∂ t 2 ( ρ 12 v + ρ 22 V ) − b ∂ ∂ t ( v − V ) . (3)

We assume the propagation mode shapes of solid and liquid u y and U y are

v = ϕ ( z ) e i ( k x + ω t ) , V = φ ( z ) e i ( k x + ω t ) , (4)

where t is time, ω is circular frequency, k is wave number and i is the complex unity.

Substitution of Equation (4) into Equation (3) yields

L ϕ ″ − k 2 N ϕ = − ω 2 ( K 11 ϕ + K 12 φ ) , 0 = − ω 2 ( K 12 ϕ + K 22 φ ) , (5)

where

K 11 = ρ 11 − i b ω , K 12 = ρ 12 + i b ω , K 22 = ρ 22 − i b ω . (6)

From the second equation of (5), we get

φ = − K 12 K 22 ϕ . (7)

Substitution of Equation (7) into the first equation of (5) we obtain,

ϕ ″ + γ 2 ϕ = 0 , (8)

where

γ 2 = ( − k 2 N L + ω 2 N L V 3 2 ) and V 3 2 = ( N K 22 K 11 M 22 − K 12 2 ) . (9)

In Equation (9), V_{3} is shear wave velocity as in Biot [

On simplification, Equation (8) gives

ϕ ( z ) = C 1 e i γ z + C 2 e − i γ z , (10)

where C_{1} and C_{2} are constants.

From Equation (7), φ ( z ) can be obtained as

g ( z ) = − M 12 M 22 ( C 1 e i γ z + C 2 e − i γ z ) .

Substituting f (z) from Equation (10) into the first equation of (4), the displacement u_{y} is

v = ( C 1 e i γ z + C 2 e − i γ z ) e i ( k x + ω t ) . (11)

Following Equations (2) and (11), the only non-zero stress can be obtained as

σ y z = ( C 1 ( i γ L 2 ) e i γ z + C 2 ( i γ L 2 ) e − i γ z ) e i ( k x + ω t ) . (12)

From Equation (4), it can be shown that the normal strains of solid and liquid are zero, hence the dilatations of solid and liquid media vanish. Since the dilatations of solid and liquid are zero, the liquid pressure s developed in the solid-liquid aggregate following Equation (2) is zero. Thus, no distinction is made between a pervious and an impervious surface of the solid in case of Love waves.

In the absence of body forces, the equations of motion [

ρ l ( ∂ V ∂ t ) = − ∇ p + η l 3 ∇ ( ∇ ⋅ V ) + η l ∇ 2 V , (13)

where V ( u l , v l , w l ) is the velocity vector, ρ l is density of liquid, η l is coefficient of viscosity, p is over pressure.

For Love waves, V = ( 0 , v l , 0 ) and ∇ ⋅ V = 0 . Hence, Equation (13) reduces to

( ∂ v l ∂ t ) = α l ∇ 2 v l , (14)

where α l = η l / ρ l is the kinematic viscosity.

Solution of Equation (14) can be written as

v l = D e β z e i ( k x + ω t ) (15)

where β 2 = k 2 − i ω α l and D is a constant.

Following (15), stresses in compressible viscous liquid layer can be shown as

τ y z = D η l β e β z e i ( k x + ω t ) . (16)

For contact between the poroelastic layer and the viscous liquids, we assume that the stresses and displacement components are continuous at the interfaces z = 0 and z = h.

Thus, the boundary conditions are given by

at z = 0 ; τ y z ( 1 ) = σ y z , v l ( 1 ) = i ω v

at z = h ; σ y z = τ y z ( 2 ) , i ω v = v l ( 2 ) . (17)

Equation (17) results in a system of four homogeneous algebraic equations in four constants.

For a nontrivial solution, the determinant of the coefficients must vanish. By eliminating these constants the frequency equation of wave propagation in an isotropic poroelastic layer bounded by viscous liquid layers is

| η f ( 1 ) β ( 1 ) i γ L 2 − i γ L 2 0 1 i ω i ω 0 0 i γ L 2 e i γ h − i γ L 2 e − i γ h − η f ( 2 ) β ( 2 ) e − β 2 h 0 i ω e i γ h i ω e − i γ h e − β ( 2 ) h | = 0. (18)

On simplification Equation (18) reduces to

[ γ 2 L 2 + 4 ω 2 η f ( 1 ) β ( 1 ) η f ( 2 ) β ( 2 ) ] tan ( γ h ) = 2 i ω γ L [ η f ( 1 ) β ( 1 ) + η f ( 2 ) β ( 2 ) ] . (19)

Equation (19) represents frequency equation of Love waves in a transversely isotropic poroelastic layer bounded between two viscous liquids.

If the upper liquid is inviscid i.e. η l ( 1 ) = 0 , then the above frequency Equation (19) reduces to

γ L tan ( γ h ) − 2 i ω η f ( 2 ) β ( 2 ) = 0. (20)

Equation (20) represents frequency equation of Love waves in a transversely isotropic poroelastic layer in contact with a viscous liquid.

If both upper and lower liquids are inviscid then the frequency Equation (19) reduces to

sin ( γ h ) = 0. (21)

On simplification of Equation (21), we obtain the frequency ω as

ω = V 3 l 2 π 2 h 2 L N + k 2 , where l = 1 , 2 , 3 , ⋯

Here ω represents frequency of Love waves in a transversely isotropic poroelastic layer when it is free from two viscous liquids.

Frequency Equations (19)-(21) are investigated numerically by considering two distinct poroelastic materials with parameters N = 0.234, L = 0.8. The physical parameters of two viscous liquids are taken as ρ 1 = 0.1 , η 1 = 0.5 and ρ 2 = 1 , η 2 = 2.5 . Poroelastic medium is dissipative in nature and thus the wave number k is complex. The waves generated obey diffusion type process and therefore get attenuated. Let k = k_{r} + ik_{i}, where k_{r} is real and k_{i} is the imaginary part of the wave number k. The real and imaginary part of the wave number corresponds to propagation and attenuation of waves. Hence, the phase velocity C_{p} and attenuation coefficient δ are, respectively

C p = ω k r and δ = k i k r .

The effect of presence of viscous liquids on phase velocity against wave number (k_{r}h) in transversely isotropic poroelastic isotropic layer is depicted in _{r}h increases from 0.2 to 0.4 then onwards it decreases gradually. From

poroelastic layer is more compared to poroelastic layer. Phase velocity decreases as wave number increases and a sudden decrease observed in phase velocity when k_{r}h increases from 0.2 to 0.4 then onwards it decreases gradually. Phase velocity at the interface of solid layer and lower viscous liquid is plotted in _{r}h increases from 0.2 to 0.4 and then from 0.6 to 1 and phase velocity increases when k_{r}h increases from 0.4 to 0.6. Phase velocity against wave number in the absence of liquids is presented in _{r}h increases from 0.2 to 0.4 then onwards it decreases gradually. Attenuation as a function of non-dimensional wave number is depicted in

that attenuation increases for higher values of phase velocity. A similar phenomenon is noticed in the case of poroelastic layer bounded by viscous liquids from

A study of propagation of Love waves in an infinite poroelastic layer bounded by viscous liquids leads to the following conclusions:

1) Presence of upper and lower viscous liquids decreases phase velocity of Love waves for both transversely isotropic poroelastic layer and poroelastic layer.

2) Phase velocity of Love waves is more in transversely isotropic poroelastic layer than in poroelastic layer when they are bounded between viscous liquids as well as when they are free from viscous liquids. In both cases phase velocity decreases gradually.

3) Phase velocity of Love waves in transversely isotropic poroelastic layer is

very much higher than that of poroelastic layer when they are in contact with lower viscous liquid.

4) Attenuation in transversely isotropic poroelastic layer when it is bounded by viscous liquids is more when compared to the case of poroelastic layer bounded by viscous liquids. In both the cases, attenuation increases as phase velocity and wave number increase.

The authors declare no conflicts of interest regarding the publication of this paper.

Chinta, N., Syed, A.S., Modem, R. and Mangipudi, V.R. (2019) A Study on Propagation of Waves in a Transversely Isotropic Poroelastic Layer Bounded between Two Viscous Liquids. Open Journal of Acoustics, 9, 1-12. https://doi.org/10.4236/oja.2019.91001