ИСТИНА

We derive an asymptotic solution of the Helmholtz equation in a three-dimensional layer of variable thickness with a localized right-hand side. Using method of adiabatic dimension reduction [1] and recently developed approach [2], assuming the absence of ”trap” states and the fulfillment of radiation conditions at infinity (such as the Sommerfeld condition) we construct the asymptotic solution of the formulated problem. The asymptotic solution can be represented as an decomposition into a finite number of modes, each mode is connected with pair of Lagrangian manifolds. One of the corresponding manifolds defines a localized (”singular”) part of the solution in the neighborhood of the point (x = ξ1, y = ξ2). The second manifold defines the oscillating (“ wave ”) part of the solution over the entire layer (taking into account the possible appearance of caustics and focal points). In the limit F(x, y)g(z) → δ(x)δ(y)δ(z) the obtained formulae describe the asymptotics of the Green’s function for the considered Helmholtz equation, but unlike such asymptotics, the obtained formula allows to describe influence of the source shape on the wave part of the solution quite explicitly. [1] Belov, V. V., Dobrokhotov, S. Y., Tudorovskii, T. Y. (2004). Asymptotic solutions of nonrel- ativistic equations of quantum mechanics in curved nanotubes: I. Reduction to spatially one- dimensional equations. Theoretical and mathematical physics, 141(2), 1562-1592. [2] Anikin, A. Yu, et al. ”The Maslov canonical operator on a pair of Lagrangian manifolds and asymptotic solutions of stationary equations with localized right-hand sides.” Doklady Mathe- matics. Vol. 96. No. 1. Pleiades Publishing, 2017.