P-SV wave diffusion in fluid-saturated medium

Propagating P-SV waves in the fluid-saturated mediums are categorized to fall into two distinct groups: insoluble and soluble mediums. These waves are known as surface Rayleigh waves. By introducing these waves with slowness in accordance to Snell Law, the diffusive and rotating waves are obtained. The results bear out that the propagating P-SV waves in soluble medium share similar diffusion characteristic as of insoluble medium while the discussions on fluid density in the mediums show that high density fluid promotes diffusive characteristic while low density fluid endorses nondiffusive P-SV waves. There is a compressed zone during the propagation of P-SV waves in medium saturated with high density fluids. | Diffusion| Attenuation | Diffraction | Shock waves | ® 2010 Ibnu Sina Institute. All rights reserved. http://dx.doi.org/10.11113/mjfas.v6n1.178


INTRODUCTION
P-waves refer to the primary waves when an earthquake is discussed.Engineers estimated the wet-rock P-wave velocity from the dry-rock P-wave velocity [1].In geophysics, S waves are categorized to SV and SH waves while the combination of P-SV waves are known as surface Rayleigh waves in [10].The dependence of P-and S-wave attenuation on strain amplitude and frequency had been studied experimentally in dry and water-saturated sandstone samples under a confining pressure [2].The protection of underground structures against dynamic loadings has long been a topic of great interest in defense engineering.Hence, the P-wave propagation and attenuation in rock shelter layer with an inclusion or filled medium had been studied [3].The problem of diffraction of waves due to plane harmonic P-wave incident normally on a line crack situated in an infinite micro-polar elastic medium was studied as well in [4].Nevertheless, an analytical solution is obtained for twodimensional scattering and diffraction of plane P waves by circular-arc alluvial valleys with shallow saturated soil deposits [12].In Cardiology, P-waves had been used for monitoring in risk assessment of the patient after surgery [13].
The dependence of P-and S-wave attenuation to strain amplitude and frequency in saturated medium is studied in [2].Investigation on saturation on horizontal and vertical of porous soils was carried out in [14].
As the wave's amplitude increases, the frequency increases too.Effects of saturation on the velocities of P-and Swaves in several types of soil are studied and the outcomes show that the velocity is higher in dense sand.Similar outcome is found in [1].The thickness of the porous medium is related to the frequency of the waves.The frequency decreases when the thickness of the medium increases [7] Hence, this study aims to probe the connection of P-SV wave's refraction velocity and the attenuation.The derivation of P-SV waves will be shown by the analytic discussion.Some figures will be plotted to give better viewing pleasure.

EXPERIMENTAL
The governing equations: Boundary conditions for 0 z = : Initial conditions for 0 t = : In linear elasticity for isotropic medium, λ and µ denote the Lame parameters for the stress σ z , τ xz , τ yz and the displacements x u , y u and z u are continuous everywhere.
F is the body force in the direction of x, y, z respectively and ρ is the density.
Provided that ω is the angular velocity or frequency, k is the wave number, c is the wave velocity along with the dispersion relation ck ω = , the group velocity of a wave is the velocity with the overall shape of the wave's amplitudes which is also known as envelope of the waves.
In other words, the phase velocity is the average velocity of the components, given by / The group velocity is velocity of the envelope, given by / When this is applied to the problem of P-SV wave's propagation in fluid saturated medium, the group velocity reflects the apparent velocity of the surface displacement or the overall shape of the P-SV wave's amplitude at the boundary of the medium.The envelope is formed by the phase velocity of P-SV waves.In this research, the apparent velocity or surface displacement velocity is measured along the boundary of the similar density medium in accordance to Snell law: where e is the incident angle and f is the refraction angle made by the P-SV waves.However, this research aims to study P-SV waves in horizontal direction only.For the case of fluid-saturated medium, there exists variation in density within medium [7,8].There exists slowness for fluidsaturated medium [5,6] i.e. the apparent velocity of the displacement at the boundary is slower than phase velocity of the wave in the medium that gives .
In this study, the elastic wave equation will be solved for extracting P-SV wave's displacements in the different types of medium that satisfied ( 5) and (6).By using the divergence operator eq. ( 1) is reduced to where α and β are the velocities for the P-wave and S- wave and Eq. ( 8) is the elastic wave equation [10].Hence, this modelling is only valid for elastic medium and it is necessary to reduce the right hand side (RHS) of ( 8) by letting By inserting second order derivative of (10) into RHS of (8), the equation is given by ( ) Since elastic wave consists of irrotational P-wave and solenoidal by S-wave [10], the displacement shall be written as .
For irrotational P-waves, the vorticity 0 P u ∇ × = .Thus, the relation ( 12) becomes 0 By inserting ( 12) and ( 13) into (11), the equation yields 0, , The Helmholtz equations [11] for P-and S-waves are given by 2 2 0 Here, the Helmholtz equations are solved by utilizing Hansen vector [10] that gives ( )exp where x a , y a and z a are the unit vectors while l and n are the vector components.Eqs. ( 17) and (18) will be considered in this study since the aim is to study the P-SV waves only.P-SV wave's incidence is illustrated in Fig. 1.The amended eqs.( 17) and (18) in which the amplitudes reduce with depth are where α η and β η are always positive.ω η η = is in complex [9].The diffusive waves are associated with attenuation of the amplitudes with the time due to certain dissipation mechanisms present in the system.Eq. ( 21) should be polarized to give the real quantities (frequency) equation that reads [10].We note that the relations ( 23) and ( 25) are meant for the insoluble medium such that the variation of velocities c and α or β is significant.When the velocity c is similar to α or β , we presume this will explain the soluble medium such that the fluid mixes well with the medium to give similar velocity.Yang and Tadanobu [14] and Kahraman [1] show that the high density medium promotes high wave's velocity.Hence, the relation (23) will only be present when the low density fluid is saturated in the insoluble medium; the low density fluid will reshuffle the ray velocity or reduce the P-SV wave velocity.Eventually, the relation ( 25) is meant for the high density fluid saturated in the insoluble medium.
The detailed explanations about the tie between medium's solubility and the fluid's density will be discussed next for showing the vital roles play by the relations (23-25) especially the attenuation's characteristic.When condition (25) is applied to the Eq. ( 21), a complex solution will be obtained for α η and β η that reads , This indicates a modification is required for Eq. ( 22) to give (27).Hence, the P-SV wave's displacements for 3 types of mediums are

DISCUSSIONS
For soluble medium, Eq. (28) gives The amplitude is not reduced immediately for which the displacement remains until the higher density fluid or soluble medium exists.

CONCLUSION
The studies show that the P-SV waves in soluble medium are non-diffusive.However, the discussion on fluid density has linked the P-SV waves to the diffusion.High density fluid in insoluble medium promotes diffusive P-SV waves while low density fluid in insoluble medium promotes non-diffusive P-SV waves.The non-diffusive waves have induced shock wave in the insoluble medium after diffraction.Parts of the rotating P-SV waves are compressed during the propagation of shock waves in the medium saturated with higher density.However, there are less compressed waves for which the shock waves are assumed less disastrous to the medium.

Figure 1 :
Figure 1: The P-SV wave's incidence and the fluid-saturated medium.( ) ( )( ) Eq. (27-29) show that the P-SV wave's propagation in xdirection whiles the diffusion is in z-direction.The Eq. (27-29) are arranged such that the density of the saturated fluid reduces from ,

Fig. 2 (
a) illustrates the rotating behaviour of P-SV waves.There is a compression zone by P-SV wave's rotation which is known as shock waves[15].Insoluble medium with low density fluid (29) is nondiffusive given that the term

Fig. 3
Fig. 3 is plotted for the non-diffusive P-SV waves with c α > , c β > ， 5 5 x − ≤ ≤ , implies that the P-SV wave in soluble medium is non-diffusive plane wave and there is no diffraction.Insoluble medium with displacement (27) is diffusive given that .Apparently, the shock wave is illustrated in medium saturated with both high and low density fluid in accordance to Figs.2 and 3.The compressed zone of waves in figure2(a) shows that there is possibility of generating significant displacement during an earthquake.The existence of lower density fluid is disastrous since the waves are not diffusive.