EFFECT OF GRAVITY ON VISCOELASTIC SURFACE WAVES IN SOLIDS INVOLVING TIME RATE OF STRAIN AND STRESS OF HIGHER ORDER

A study is made of the surface waves in a higher order visco-elastic solid involving time rate of change of strain and stress under the influence of gravity. A fairly general equation for the wave velocity is derived. This equation is used to examine various kinds of surface waves including Rayleigh waves, Love waves and Stoneley waves. It is shown that the corresponding classical results follow from this analysis in the absence of gravity and viscosity. KEYWORDSAND PIIRASES: Surface waves, effects of gravity and viscosity.


INTRODUCTION
Considerable literature including Bullen [1], Flugge [2] and Stoneley [3] is available on the theory of surface waves in an isotropic homogeneous elastic solid medium.However, the effects of gravity, viscosity and curvature, although important, are not included in the classical problems.Biot [4] has first investigated the effect of gravity on Rayleigh waves on the surface of an elastic solid based on the assumption that gravity produces a type of initial stress of hydrostatic in nature.Subsequently, Biot's   theory has been used by several authors including De and Sengupta [5,6] to study problems of waves and vibrations in solids under the initial stress in various configurations.Further, Sengupta and his associates [7][8][9] have made an attempt to study the problems of surface waves in solids involving time rate of strain and viscosity.In spite of these studies, relatively less attention has been given to surface wave problems in a higher order visco-elastic solid involving time rate of strain and stress under the influence of gravity.The main purpose of this paper is to study such problems.A fairly general equation for the wave velocity is derived.This equation is utilized to examine various kinds of surface waves including Rayleigh waves, Love waves, and Stoneley waves.It is shown that the corresponding classical results follow from this analysis in the absence of viscosity and gravity.

FORMULATION OF THE PROBLEM AND BOUNDARY CONDITIONS
Let Mt and M2 be two homogeneous general visco-elastic solid media involving time rate of strain and stress of higher order in welded contact under the influence of gravity at their common surface of separation.We suppose that the media are separated by a plane horizontal boundary extending to infinitely great distance from the origin, M being above M.We introduce a set of orthogonal Cartesian coordinate axes Oxx3 in the semi-infinite isotropic visco-elastic media, with the origin at the common bounda surface and the x-axis is normal to M. We consider the possibility of a type of wave travelling in the direction of 0x in such a manner that the disturbance is largely confined to the neighborhod of the bounda and at any instant all panicles on any line parallel to 0x2 have equal displacements.Hence the wave is a surface wave and all partial derivatives with respect to the coordinate x2 are zero.Then the components of displacement u and us at any point may be expressed in the form u3=+- Ox Ox where 9 and are the functions ofx,x and and Out Ouz Ou Ou Ou Ox Thus the introduction of the functions 9 and enables us to separate out the purely dilational and rotational disturbances associated with the components u and ua.e component u, of coupe, is associated with purely distoional movement.us 9, and uz are associated respectively with P-waves, SV-waves and SH-waves, as used by  where 0, and are the elastic constants and , k and (k 1,2 n) are the effects of viscosity, e,i is the strain tensor and 6i is the onecker symbol.e displacement equations of motion in the higher order general visco-elastic medium, under the influence of avity, are OA Ou (O x + D,) + D,V=u, + pgD n pD, where p,q,,, ,(k -0,1,2 ,n) denote the properties e medium M and those with dashes the properties of the medium M2.Substituting (2. lab) in equations (2.5)-(2.7),we obtain the wave equations in M satisfied by , and u2, as = ,v' + g V and similar relations in M with p, q,, k,, t, replaced by p', q',, ',, U', and so on (where k O, 1,2,..., n).
The boundary conditions are (i) e components of displacement at the boundary surface between the media Mt and M must be continuous at all times and distances.
(ii) , 0% ] (2.15) D,o DVZ, + 2D u + OxOx ) and similar expressions for Mz, across the boundary surface between M and Mz must be continuous al all times and distances.
Introducing (3.1) in (2.8)-(2.10),we have for the medium M,: Similar relations for M2 can be obtained by replacing ,, fi:,, rh, Vr, V,s, rl, V'r, V,s, , ,, 9 by the same symbols with dashes.Here p', rl',, .',,', (k 0,1, 2 n are the physical properties of the medium g. We assume that , p and u3 represent exponentially decaying solutions in the medium M as x3 oo so that they can be expressed in the form: u.Ce -*A/C- and similar solutions inM can be obtained replacing , ap, u2,A,A2, B,B2, C, rl, V;s, ,., the same symbols with dashes in solutions (3.6)-(3.8).ltere and t' (j 1,2) are respectively the roots of the equations and where (3.9) % igrll(to:' " ;V,s,) 4, ie, n/(C ' i V,s'/'q,') (j 1,2) In evaluating quantities like qq=and V'I Zto2,tl/, the root with positive real part must be taken in each ease.Using boundary conditions (i) and (ii), we obtain 11   It follows from equations (3.11b) and (3.11e) that C =C'-0.Thus there is no propagation of dis- placement u2.Hence there are no SH-waves in this case.
(ii) Love Waves In this case we consider a layered semi-infinite medium in which M is bounded by two horizontal plane surfaces at a finite distance H-apart, while M remains infinite as it was.In this case, we consider the displacement component u only.
For the medium M:, we write down the full solution, since the displacement in M:, may no longer diminish with the increasing distance from common boundary x3--0 and for the medium M the solutions are the same as it was in the general case.