We derive an asymptotic expansion as |x|→∞, uniform with respect to the direction x/|x|, varying over double-struckS2, of solutions to the reduced wave equation (the Helmholtz equation) in locally perturbed two‐layered media in double-struckR3. The main difficulties arise from the conical singularities of the generalized eigenfunctions, caused by the transmitted (refracted) waves at the interface. They are overcome by the classical method of stationary phase.

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Exponential decaying states for the wave equations are considered and some examples are given. Moreover, the spectral structure of the operator with coulomb type dissipation is investigated.