Transient and impulse responses of a one-dimensional linearly attenuating medium — I. Analytical results

Summary. The transient and impulse responses (Green's function) for onedimensional wave propagation in a standard linear solid are calculated using a Laplace Transform method. The spectrum of relaxation times is chosen so as to model a constant Q medium within an absorption band covering a broad frequency range which may be chosen so as to include the seismic frequencies. The inverse transform may be evaluated asymptotically in the limit of very long propagation times using the saddle point method. For shorter propagation times the method of steepest descent may be modified so as to yield an accurate first motion approximation. The character of the small amplitude precursor to the large amplitude Visible' signal is investigated analytically. It is shown that the signal velocity is intermediate between the high-frequency ('unrelaxed') and the low-frequency ('relaxed') limits of the phase velocity.