Lamb's problem for an inhomogeneous medium with constant velocities of propagation

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Abstract
A recently introduced method of separation of the vector wave equation of elasticity for certain types of inhomogeneous, elastic media is employed to study the two-dimensional problem of an inhomogeneous, isotropic, half-space whose elastic parameters and density vary as the square of the depth, leaving the velocities of propagation of P and S waves constant. The Rayleigh wave dispersion curve for this medium is examined. It is found that the medium yields a single Rayleigh mode with Rayleigh wave phase velocity varying between the shear velocity and the Rayleigh wave velocity of a homogeneous medium of the same Poisson's ratio. The disturbance generated by an impulsive surface line load is considered. An exact solution to this problem is obtained and numerical results for the displacements are presented and compared to the solutions for the corresponding homogeneous medium. The displacements vary inversely with depth and the relative effects of the inhomogeneity as compared to the homogeneous medium increase with distance from the source. On the surface the amplitude of the Rayleigh wave varies sinusoidally with distance from the source. The displacements at times before and behind the Rayleigh wave have similar, i.e. approximately sinusoidal, variation with distance from the source.

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