Abstract
The paper presents a new type of functionally invariant Smirnov–Sobolev solutions for the wave equation, which can be used for solving many homogeneous problems of elastodynamics. The derived solution has a unique property: the double preimage of Laplace and Fourier transforms of this function with a new argument and coefficient coincides with the function itself. Thus, this property allows us to find an inversion formula for double integral transformations. The method for obtaining this formula is demonstrated by solving the Lamb problem for a half-plane under the transition from the image to the original.
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