Asymptotic solution of the Helmholtz equation in a three-dimensional layer of variable thickness with a localized right-hand side

The asymptotics of the solution to the Helmholtz equation in a three-dimensional layer of variable thickness with a localized right-hand side in the absence of “trap” states and under the asymptotic radiation conditions at infinity is constructed. The wave part of the solution has a finite number of modes. The resulting formula makes sufficiently clear the influence of the shape of the source on the wave part of the solution.