ON PROPERTIES OF SURFACE WAVES IN AN IMHOMOGENEOUS ELASTIC MEDIUM

Abstract

Assuming that the soil ground is composed of a homogeneous isotropic elastic medium, theauthors have previously discussed on the dynamic properties of a half-space and a stratum over arigid medium, and they have also examined the influences of such media on the dynamic respon-ses of structures. It seems almost impossible, however, to investigate the case of a stratifiedground having several layers on the basis of the three-dimensional elastic wave propagationtheory that have been used in the previous studies. It may then be an interesting approachto this problem that the discontinuous properties of a stratified ground are replaced by somecontinuous vertical variations of the constitutive parameters, that is, the soil ground is thuspresumed as an imhomogeneous elastic medium.The general solutions may easily be obtained for homogeneous media having horizontalboundary surfaces, because the vector wave equation can always be separated into the longitu-dinal and transverse components when the two kinds of potentials are introduced. For imbomo-geneous media, on the other hand, such method based on the Helmholtz theorem is not suc-cessfully applied in general. J.F. Hook have recently presented an generalization of thisHelmholtz theorem for some imhomogeneous media whose properties are functions of the singleCartesian coordinate z with constant Poisson's ratio σ. If certain conditions given by a simul-taneous nonlinear ordinary differential equation for the constitutive parameters are satisfied, use of this generalization leads to the separation of the vector wave equation.Making use of Hook's generalization, this paper attempts to investigate the properties of theRayleigh and Love waves sn an imhomogeneous, isotropic, semi-infinite, elastic medium. Theconstitutive parameters are given by Lames constants λ/λ0=μ/μ0= (1+z/z1)r(r-2) and the den-sity ρ/ρ0= (l+z/z1)2(r-2) in which r=(λ+2μ)/μ=2(1-σ)/(1-2σ), z1=an arbitrary con-stant, and λ0, μ0, and ρ0=the corresponding quantities at z=0. The propagation velocitiesof generalized bodily waves are thus proportional to the square root of the depth, and the presentmedium is reduced to a homogeneous one when z5 tends to infinity. It is the only differencesn the general solutions that they contain Whittaker's functions with respect to the Cartesiancoordinate z instead of the exponential functions for a homogeneous medium.Numerical results for the frequency equations are summarized as follows:The dispersion curves, I.e. the relations between the phase velocity and the wave numbershow an extraordinary similarity between the two kinds of surface waves. They have innumer-able modes of propagation all of which are dispersive. All of these modes are separated into thetwo groups. The phase velocities in one group are monotonically decreasing functions of theproduct of the wave number and the arbitrary constant z5, and as the product increases, they ap-proach to the transverse or Rayleigh wave velocity in a limiting homogeneous medium. Themodes in the other group generally have the two velocities for an identical wave number, andthey vanish with increasing wave number or z5. The former group is also observed in a strati-fied homogeneous medium and consequently, it may be concluded that the latter group is in-herent in the imbomogeneous medium.