Wave conversion in stratified media with hyperbolic-tangent nonuniformities

Abstract

A systematic procedure is developed for finding analytical solutions of certain ordinary linear differential equations of arbitrary order, describing reflection and conversion of waves in multiwave media having a concentrated region of nonuniformity. This procedure involves transformation to equations solvable by generalized hypergeometric functions which have the form of normal modes, exp(−ikz), in uniform regions. It is applicable when the variable coefficients in the original equation contain only linear combinations of one varying parameter, p(z), and in certain cases its derivatives or ratios thereof, if p(z) is proportional to tanhz. Arbitrary gradients and dissipation are allowed. The applicability of this procedure to solve for the reflection and conversion coefficients is indicated for second-, fourth-, and sixth-order problems in magnetostatic, spin, and elastic-wave conversion in magnetic insulators such as yttrium iron garnet, and for electromagnetic-to-electrostatic wave conversion in hot plasmas. Some of these problems involve fourth-order singular turning points formed by the coalescence of resonance and cutoff. Explicit expressions for the reflection and conversion coefficients are given for fourth-order systems in terms of Γ functions of complex arguments. In the absence of dissipation, the absolute magnitude of these coefficients may be expressed in terms of ordinary transcendental functions.