As the wave-length of the seismic wave is very large compared with the roughness of corrugated surface for most cases, it is generally negligible. But if the period of incident seismic wave is very short, we cannot overlook the effect of the roughness.
Lord Rayleigh had studied the theory of reflection and refraction of sound in his ‘The Theory of Sound’, where plane waves were supposed to be incident perpendicularly upon a regularly corrugated boundary, whose form was limited to simple trigonometrical functions. After that he extended the above theory and discussed it more in detail in ‘Pro. Roy. Soc., A, Vol. LXXIX. pp. 399-416, 1907’.
We investigated in this paper, after Rayleigh, the theory of reflection of elastic waves on corrugated free surface, supposing the
depth of the corrugations is small in comparison with the length of the waves.
The refleected waves are composed of regularly reflected waves and various spectra. If the wave-length of corrugation is small compared with that of distortional wave, the waves reflected as bodily waves are only regularly reflected waves; besides, the waves, whose amplitudes diminish as
z increases, are propagated in both directions of
x with the velocity depending upon the wave-length of corrugated surface and the angle of incidence. When the wave-length of the corrugations becomes large, the spectra of reflected bodily waves gain in number but so long as the wave-length of the corrugations is finite, the above waves are propagated along the surface.
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