This paper considers the problem of calculating the reflection coefficient of a plane wave incident on an inhomogeneous elastic solid layer of finite thickness, overlying a semi-infinite, homogeneous solid substrate. The physical properties of the inhomogeneous layer, namely the density, compressional (sound) speed, shear speed, and attenuation, are all assumed to vary with depth. It is shown that, provided terms involving the gradient of the shear speed and of the shear modulus are ignored, and volume coupling between compressional and shear waves is neglected, analytical solutions of the resulting equations can be obtained. The assumptions made are justified at most frequencies of practical interest in underwater acoustics. In the case of a solid whose density and shear speed are constant, the results obtained are exact solutions of the full equations of motion, and may usefully be compared with numerical solutions in which the variation in sound speed through the layer is represented by a number of homogeneous sublayers. It is concluded that, with realistic sediment and substrate properties, a surprisingly large number of sublayers can be needed to give accurate results.
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