Normal‐mode solutions for radiation boundary conditions with an impedance contrast

Wave propagation problems with radiation boundaries cannot be solved by the ordinary eigenfunction expansion method because not all of the eigenfunctions are mutually orthogonal due to non-Hermitian boundary conditions. We present a method for solving such problems in terms of a superposition of eigenfunctions, using the biorthogonal eigenfunction expansion method outlined by Morse & Feshbach (1953). We develop their method, using a variational equation, so that the calculations other than those of eigenfunctions are unnecessary to construct the solution when there is an impedance contrast at the radiation boundary. We present numerical computations for a 1-D semi-infinite continuum that has an impedance contrast. This method may be applicable to such problems as the vibration of a magma chamber embedded in the crust, the acoustic coupling between the solid Earth and the atmosphere, and wave propagation in a layered half-space.