On reducing the order of Rayleigh system. Protect II. Cases of reduction

A. N. Kuznetsov and V. M. Markushevich


We study two sets of differential equations in unknown parameters of a flat layered medium obtained in the first part of the work. The first set represents conditions for the system of equations for P-SV vibration transforms to reduce to two Sturm-Liouville's equations. The second set expresses conditions for the same system of equations to be factored. Solutions to each of these sets depend on an arbitrary function. We integrated both sets by quadratures and found the first integral of the first set. We proved the first set to be equivalent to the system found by Alverson, Gair, and Hook and later by Young. The second set, being more general than the first one, strengthens their result. We obtained the system of simplified equations for P-SV vibrations in the case where a medium obeys the first set. We considered a class of media where parameters are power functions of depth and obey the first set. This class is capacious; is contains the examples given by Gupta and Young and some media indicated by Hook. We show that media considered by Pekeris and Zvolinskii obey the second set but not the first. The same is true for the elasticity operator, which factores into two Dirac's operators and thereby demonstrates some parallelism between the theory of elasticity and the quantum theory.

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Computational Seismology, Vol. 5.