A seismic wave propagating from one region of constant velocity to another, through a smooth transition zone, will differentially reﬂect or transmit across the zone, depending on the relative sizes of the transition zone and the wavelength of the propagating wave. This work presents an exact analytic solution for the case of a linear ramp velocity in the transition zone, and demonstrates that for long wavelengths, the ramp looks essentially like a jump discontinuity in the medium, with the corresponding reﬂection and transmission coefﬁcients. For short wavelengths, the ramp provides essentially 100% transmission and no reﬂection. Energy conservation is veriﬁed for all wavelengths.
A careful consideration is given to the two cases of varying the velocity parameter, one via variations in the density of the propagation medium, the other in varying the modulus of elasticity. The results are different, in particular there is a sign difference in the reﬂection coefﬁcient, and a large amplitude difference in the transmission coefﬁcient.
We also present the numerical result for the transmission and reﬂection of a delta spike through the velocity ramp, and observe the reﬂection is a broadened “boxcar” response, while the transmission results in a spike.
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