PandSwave travel times and spheroidal normal modes of a homogeneous sphere

The physical relation between P and S wave travel-time curves and torsional and spheroidal dispersion curves for a homogeneous sphere has been demonstrated. The dispersion curves represent interference conditions for multiply reflected body waves: SH waves in the case of torsional oscillations and P and Sv waves in the case of spheroidal oscillations. The interference equations have been derived both by considering constructive interference requirements for rays and by substituting asymptotic expressions for Bessel functions in the period equations. For the case of torsional oscillations the interference equation is Ttor = [ts - Δs/c]/[I - 1 + G(c,T)] for the case of spheroidal oscillations, when c > Vp (the P wave velocity) the interference equation is Tsph = [ttps - Δps/c]/[I - 1 ± 1/π · sin-1{A sin (ω/2(tp-s - Δp-s/c) - π/2)} + H(c,T)] where the negative value of the inverse sine is taken when I−1 is odd. When c < VP there is no reflected P wave and the equation becomes Tsph = [ts - Δs/c]/[I - 1 + 0.25 + β/2π + H(c,T)] where β is the phase shift for an Sv wave reflecting at a plane free surface. In these equations Ttor and Tsph are the torsional and spheroidal periods corresponding to a phase velocity c; ts and tp are the S and P wave travel times corresponding to the arc distances Δs; and Δp; for a phase velocity c; tps = tp + ts; Δps = Δp + Δs; tp−s = tp − ts; Δp−s = Δp − Δs; and I is an integer corresponding to the mode number. A is the reflection coefficient for P waves at a free interface. G(c, T) and H(c, T) are parameters relating the torsional and spheroidal dispersion curves to the travel-time curves. In the limit of high frequencies G(c, T) approaches ¼ and H(c, T) approaches 0.