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\title{Reflection and Transmission of Elastic Waves at a Loosely Bonded Interface between an Elastic Solid and a Viscoelastic Porous Solid Saturated by Viscous Liquid}
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             \author[1]{Baljeet  Singh}

             \author[2]{Surjeet  Singh}

             \affil[1]{  Post Graduate Government College, Sector 11, Chandigarh,160011, India}

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\date{\small \em Received: 14 December 2013 Accepted: 31 December 2013 Published: 15 January 2014}

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\begin{abstract}
        


In the present paper, a problem on reflection and transmission of elastic waves at a loosely bonded interface between an elastic solid and a viscoelastic porous solid saturated by viscous liquid is studied. The study is carried out with the assumption that the interface behaves like a dislocation which preserves the continuity of stress and allows a finite amount of slip. The appropriate potential functions for reflected and transmitted waves satisfy the required boundary conditions at the interface. The relations between amplitude ratios of different reflected and refracted waves are obtained for incidence of P and SV waves. The amplitude ratios of various reflected and refracted waves are computed for a particular model. The effects of loosely boundary and viscoelasticity are observed on these amplitude ratios.

\end{abstract}


\keywords{reflection and transmission of elastic waves, reflected and refracted waves are computed for a particular model.}

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\let\tabcellsep& 	 	 		 
\section[{Introduction}]{Introduction}\par
he observed attenuation of the seismic wave in the earth helps in getting information regarding the composition and state of deep interior. This attenuation cannot be explained by assuming the earth to be an elastic solid.  {\ref Biot (1956a)} studied the propagation of the plane harmonic seismic waves in liquid saturated porous solids. \hyperref[b2]{Biot (1962)} presented a unified treatment of the mechanics of deformation and acoustic propagation in porous media, where liquidsolid medium is treated as a complex physico-chemical system with resultant relaxation and viscoelastic properties. \hyperref[b4]{Deresiewicz (1960)} and \hyperref[b5]{Deresiewicz and Rice (1962)} studied the reflection at the plane tractionfree surface of non-dissipative and dissipative liquid saturated porous solids respectively. They considered the porous solid as perfectly elastic with no internal energy loss.\par
Viscoelasticity is an important property of many rocks in the crust, which is a major cause of seismic attenuation. In the presence of porosity, a viscoelastic solid permeated by pores and fractures and saturated with viscous fluid becomes a more realistic model for sedimentary or reservoir rocks. \hyperref[b1]{Biot (1956b)} established Author ? ?: Department of Mathematics, Post Graduate Government College, Sector 11, Chandigarh, India. e-mail: bsingh@gc11.ac.in the equations for the deformation of a viscoelastic porous solid containing a viscous fluid under the most general assumptions of anisotropy. \hyperref[b9]{Sharma and Gogna (1991)} studied the seismic wave propagation in a viscoelastic porous solid saturated by viscous liquid.  {\ref Vashishth et al. (1991)} investigated a problem on reflection and transmission of a plane periodic wave incident on the loosely bonded interface between an elastic solid and a liquid-filled porous solid with the assumption that the interface behaves like a dislocation which preserves the continuity of stress allowing a finite amount of slip.  {\ref Vashishth and Gogna (1993)} studied a problem of reflection and refraction of plane seismic waves incident on an interface of two loosely bonded half-spaces, an elastic solid half-space and a liquidsaturated porous solid half-space, which permits a finite amount of slip. \hyperref[b14]{Vashishth and Sharma (2008)} discussed the wave propagation in a medium considered as a viscoelastic, anisotropic and porous solid frame such that its pores of anisotropic permeability are filled with a viscous fluid. Recently, Sharma (2012) studied the propagation of Rayleigh waves on the stress-free surface of a viscoelastic, porous solid saturated with viscous fluid.\par
In the present paper, a problem is considered on reflection and transmission of elastic waves on loosely bonded interface between an elastic solid and a viscoelastic porous solid saturated by viscous liquid. For incidence of P and SV waves, the amplitude ratios of various reflected and refracted waves are computed for a particular model. The effects of loosely boundary and viscoelasticity are shown graphically on these amplitude ratios. 
\section[{II.}]{II.} 
\section[{Basic Assumptions}]{Basic Assumptions}\par
Murty (1976) introduced a real bonding parameter to which numerical values can be assigned corresponding to a given degree of bonding between half-spaces and discussed the particular cases of ideally smooth and fully bonded interfaces corresponding to the values 0 and ? of the bonding parameter. He considered three basic assumptions. The first assumption is that the stresses are continuous T Global Journal of Researches in Engineering ( ) Volume XIV Issue III Version I across the interface. The second assumption is that the microscopic structure of the material at the interface is such that a finite amount of slip can take place at the interface when a periodic wave is propagating. The third assumption is that there exists a linear relation between slip and shear stress at the interface which implies that different degrees of bonding correspond to different values of the constant of proportionality. The principle behind the third assumption is that there must exist some relation between the local shearing stress and the 'slip' at the interface such that when the shearing stress is zero, the 'slip' is infinite implying that the interface behaves like an ideally smooth interface and when the 'slip' vanishes the interface behaves as a fully bonded interface. We may assume that shearing stress = K × slip at the interface of loosely bonded media so that the vanishing of K corresponds to an ideally smooth interface and an infinitely large value of K corresponds to a welded interface. The intermediate values of K represent a loosely bonded interface.\par
We assume a model having a viscous liquid layer between the elastic half-space and liquid-saturated porous viscoelastic solid half-space. Let H be the thickness of the layer and ? be the coefficient of viscosity and ? 0 H implying that the thickness of the layer is infinitely small. It is reasonable to assume that the shearing stress at the interface is given by( ) ( ) ( ) ( ) µ ? µ ? ? ? ? ? ? + + + ? ? + ? ? = + ? ? ? ? ? ? 2 2 2 2\par
.\par
.f u M u M w u w t , (\textbf{4}) ( ) ( ) ( ) ? ? ? ? ? ? ? ? + ? = + + ? ? ? ? ? ? ? 2 2 [ . . ] f w M u M w u mw t t ,\textbf{(5)}\par
Where, the vector ? is pore fluid viscosity, and ? is permeability. ? and M are the elastic coefficients related to the coefficient of fluid content ? , unjacketed compressibility ? and jacketed incompressibility by1 K ? ? = ? , ( ) ? ? ? = + ? 2 1/ M K .\par
The stresses ij ? and liquid pressure f p are given by solid ( )2 2 ij ij ij e M e M ? µ ? ? ? ? ? ? ? = + + + ? ? , ? ? = ? f e p M M ,\textbf{(6)}\par
Where ( ) , ,\textbf{1}2 ij i j j i e u u = +J xz u z ? ? ? ? ? = ? ? ? ? ? ? ,\textbf{(1)}\par
where u ? is the component of velocity parallel to the interface (dot represents time derivative) and the partial derivative is taken normal to the interface. Equation ( \hyperref[formula_5]{1}) can be approximated as( ) xz e u u H ? ? = ? ? ? ,\textbf{(2)}\par
where ( )e u u ? ? ?\par
is the jump in the x-component of velocity across the layer. If we assume the waves to be time harmonic, then equation ( \hyperref[formula_8]{2}) can be written as ( )xz e i u u H ? ? ? ? ? = ? ? ? ? ? ? , (\textbf{3})\par
where ? is the angular frequency and u and e u are the displacement components parallel to the interface at the boundaries of the infinitesimal thin layer of viscous liquid.\par
III. 
\section[{Basic Equations}]{Basic Equations}\par
According to \hyperref[b2]{Biot (1962)}, the differential equations governing the displacement u ? of solid matrix and U ? of interstitial liquid in a homogeneous isotropic porous solid saturated by viscous liquid are\par
Reflection and Transmission of Elastic Waves at a Loosely Bonded Interface between an Elastic Solid and a Viscoelastic Porous Solid Saturated by Viscous Liquid2 3 K ? µ ? ? = + ? ? ? ?\par
and, the stress-strain relations are ( )2 * * * * * * 2 ij ij ij e M e M ? µ ? ? ? ? ? ? ? = + + + ? ? , ? ? = ? ? * * * f p M e M ,\textbf{(8)}\par
With the help of (  {\ref 7}), the equations of motion ( \hyperref[formula_1]{4}) and ( \hyperref[formula_2]{5}) become ( )( ) µ ? µ ? ? ? ? ? ? ? + + + ? + ? = + ? ? ? ? 2 2 * 2 * * * * * * 2 f u M e M u w t ,\textbf{(9)}( ) ( ) ? ? ? ? ? ? ? ? + = + + ? ? ? ? ? 2 * * * 2 f w M e M u mw t t , (\textbf{10}) ( ) ,0, u u w = ? , ( ) ,0, U W ? = ? , ( ) ,0, e e e u u w = ? ,\textbf{(12)}\par
where 12 11 u x x z ? ? ? ? ? ? = + + ? ? ? , 12 11 w z z x ? ? ? ? ? ? = + ? ? ? ? , ( ) ? ? = ? ? 1 , 12 11 1 2 0 U x x z ? ? ? µ µ ? ? ? ? = + + ? ? ? , 12 11 1 2 0 d W z z x ? ? ? µ µ ? ? ? = + ? ? ? ? , ,\textbf{e}? ? ? ? ? ? ? ? = + = ? ? ? ? ? , ( ) e e ? ? = ? ? , (\textbf{)}( ) * * * 2 * 2 / , 1,2 f j j f v j m i ? ? ? ? µ µ ? ? ? ?? ? + + = = ? ? ? + ? ? ? ? , ( ) ? ? ? ?? = ? + 0 . / f m i IV. 
\section[{Reflection and Transmission}]{Reflection and Transmission}\par
For incidence of P or SV wave, there will be reflected P, SV waves in elastic half-space and refracted * * 2 * 2 , 1,2 j j v j ? µ ? + = = * 2 3 * 3 v µ ? = . (\textbf{11})\par
To consider only two-dimensional reflection problem, we shall restrict the plane wave solutions for the displacement potentials to those that have propagation and attenuation vectors in the x-z plane. Following \hyperref[b9]{Sharma and Gogna (1991)}, the components of displacement vectors are taken as\par
The appropriate potentials in elastic half-space are ( ) ( )0 1 i kx d z t i kx d z t e A e A e ? ? ? ? ? ? ? + ? = + , (\textbf{13}) ( ) ( ) * 0 2 i kx d z t i kx d z t e A e A e ? ? ? ? ? ? ? + ? = + , (\textbf{14})\par
where1 2 2 2 . . d p v k ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? , 1 2 2 2 . . d p v k ? ? ? ? ? ? ? ? ? = ? ? ? ? ? ? ? ? ? ? ? .\par
The appropriate potentials in viscoelastic porous solid half-space are( ) ( ) ? ? ? ? ? ? = ? ? ? ? ? 12 12\par
11 11A r i P r t B e e ,\textbf{(15) ( ) ( ) ? ?}? ? ? ? = ? ? ? ? ? 22 22\par
12 21A r i P r t B e e ,\textbf{(16) ( ) ( ) ? ?}? ? ? ? = ? ? ? ? ? 32 32 12 A r i P r t C e e , (\textbf{17})\par
where, the propagation vectors ij P ? and attenuation vectors ij A ? are defined by( ) Re Re ?1 j ij i P k x dv z = + ? ? , ( ) = + ? ? Im Im ?1 , j ij i A k x dv z with 1 2 2 2 2 i i dv p v k v ? ? ? = ? ? ? ? ? ? . where = ? ? 12 1 ( ) y C C , ? 1 C is arbitrary complex vector chosen such that ? ? ? = ? 0, k is an arbitrary complex number such that Re 0 k ? to ensure\par
propagation in the positive x-direction. The subscripts Re and Im denote the real and imaginary parts of the corresponding complex quantities. Following \hyperref[b3]{Borcherdt (1982)}, the displacement potentials given by ( \hyperref[formula_22]{13}) to \hyperref[b16]{(17)}  ? ? ? ? ? ? ? = = = = = = = ? ? ? * * *( ) ? ? = ? = ? * Im 2 sin ,\textbf{( 1,2,3)}j j j k A j (19)\par
Which is the extension of Snell's law. We also obtain the following non-homogeneous system of five equations= = = ? 5 1 , ( 1,2,..,5) ij j i j a Z b i ,\textbf{(20)}\par
where ( )( ) ? ? µ ? µ ? = ? ? + = 2 2 11 12 2 , 2 a k d a kd , ( )( ) ( ) \{ \} 2 2 * * * 2 * * * * * * 13 1 1 1 1 a H M dv k M M M ? ? ? ? µ µ = ? + + ? ? ? ? ( ) ( ) µ ? + ? ? 2 * * * 1 1 , M M dv ( )( ) ( ) 2 2 * * * 2 * * * * * * 14 2 2 2 1 a H M dv k M M M ? ? ? ? µ µ ? ? = ? + + ? ? ? ? ? ( )( ) µ ? + ? 2 * * * 2 2 , M M dv ( ) ? ? = ? ? 2 * * * * 15 3 , a kdv H M ( ) µ ? µ ? µ ? ? = ? = ? ? = ? ? ? 2 2 *µ ? ? µ = ? = ? ? ? ? ? ? ? ? ? = ? = ? ? ? 2 51 52 0 0 1 1 , 1 sin 1 sin a k a kd ? ? ? ? ? ? = ? ? = ? ? ? ? 2 2 53 1 54 2 0 0 1 1 2 , 2 , 1 sin 1 sin a kdv k a kdv k ( ) \{ \} ? ? ? = ? ? ? ? 2 2 55 3 3 0 1 , 1 sin a kdv dv k (a)\par
For incident P wave, A Z A = , 2 2 0 A Z A = , 11 3 0 B Z A = , 21 4 0 B Z A = , 12 5 0 C Z A =\par
, Are amplitude ratios of reflected P, reflected SV, refracted P 12 , refracted P 22 and refracted P 32 waves, respectively. A Z A = , 2 2 * 0 A Z A = , 11 3 * 0 B Z A = , 21 4 * 0 B Z A = , 12 5 * 0 C Z A =\par
, Are amplitude ratios of reflected P, reflected SV, refracted P 12 , refracted P 22 and refracted P 32 waves, respectively. For ? = 1 , the above system of equations (20) reduces for welded interface.  , , , Z Z Z Z and 5 Z , given by (20), are computed for incident P and SV waves. The angle of incidence 0 ? , is considered to be varying from normal incidence ( ) 0 0 ? = ° to grazing incidence ( )0 90 ? = °.\par
We restrict the numerical computations for homogeneous case only. a) Loosely Boundary Effect i. Incident P wave The amplitude ratios of reflected P and SV waves for ? = 0.25, 0.5, 0.75 and 1.0 are plotted against the angle of incidence (0 o < ? 0 < 90 o ) of P wave. These variations are shown in Figures 2 and 3 by black, blue, red and green curves, respectively. In each case, the amplitude ratios of reflected P and SV waves are same at normal and grazing incidence. The comparison of the different curves shows the effect of loose boundary on amplitude ratios of reflected P and SV waves. This effect is observed maximum in the range 45 o < ? 0 < 90 o . The amplitude ratios of refracted P 12 , P 22 and P 32 waves for ? = 0.25, 0.5, 0.75 and 1.0 are plotted against the angle of incidence of P wave. These variations are shown in Figures 4 to 6 by black, blue, red and green curves, respectively. These amplitude ratios are also affected due to loosely boundary at angles other than grazing and normal incidence.\par
ii. Incident SV wave\par
The amplitude ratios of reflected P, SV waves and refracted P 12 , P 22 and P 32 waves for ? = 0.25, 0.5, 0.75 and 1.0 are plotted against the angle of incidence (0 < ? 0 < 50) of SV wave also. These variations are J V. 
\section[{Numerical Results and Discussion}]{Numerical Results and Discussion}\par
For numerical computations of reflection and transmission coefficients, we resolve the operators t c c f f f f < < =\par
. Following Murphy III (1982), we consider water-saturated Massilon-sandstone with the following parameters: Porosity = 23 per cent, Grain density = 2.66 gm/cm 3, Pore diameter = 3 × 10 -3 cm. Following \hyperref[b1]{Biot (1956b)}, in case of water in the pores at 15°C, we find where, for uniform circular pores with axes parallel to the pressure gradient, c would be equal to 1.\par
Following \hyperref[b6]{Fatt (1959)} and Yew and Jogi (1976) relevant elastic parameters for water-saturated sandstone are chosen to be shown in Figures 7 to 11 by black, blue, red and green curves, respectively. The comparison of the different curves shows the effect of loosely boundary on amplitude ratios of reflected and refracted waves. 
\section[{b) Viscoelastic effect}]{b) Viscoelastic effect}\par
To observe the viscoelastic effect on reflected and transmitted coefficients, we consider the incidence of P wave and ? = 0.25. On comparing the solid and dotted curves in Figures 12 to 16, it can be seen that the coefficients of reflected and transmitted waves change due to viscoelastic effect. 
\section[{VI.}]{VI.} 
\section[{Concluding Remarks}]{Concluding Remarks}\par
Relations between reflection and transmission coefficients are obtained for incident of P and SV at a loosely bonded interface between an elastic solid halfspace and a viscoelastic porous solid half-space. Numerical values of these coefficients are computed for a particular model of the interface. It is observed that these coefficients are affected significantly due to the presence of loosely boundary. These coefficients are also affected due to the presence of viscoelasticity in upper half-space.       Figure  {\ref 14} : Viscoelastic effect on the amplitude ratios of transmitted P 12 wave against the angle of incidence of P wave for ? = 0.25\par
Figure  {\ref 15} : Viscoelastic effect on the amplitude ratios of transmitted P 22 wave against the angle of incidence of P wave for ? = 0.25 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-2.png}
\caption{\label{fig_0}}\end{figure}
 \begin{figure}[htbp]
\noindent\textbf{}\includegraphics[]{image-3.png}
\caption{\label{fig_1}}\end{figure}
              			\footnote{© 2014 Global Journals Inc. (US)} 			\footnote{© 2014 Global Journals Inc. (US) J} 		 		\backmatter  			  				\begin{bibitemlist}{1}
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