Determination of the inhomogeneity laws for the sandwich panel core with minimal sound permeability

The direct and inverse problems of sound transmission through a three-layer sandwich panel with elastic homogeneous outer layers and a core in the form of an elastic continuously inhomogeneous layer thickness are considered. It is assumed that the panel is in an ideal compressible liquid, and the wave incident on it is plane and harmonic.
In the direct problem, with the well-known law of inhomogeneity of the core, wave fields in the sandwich panel and the half-spaces bordering it are determined. The velocity potentials of the sound waves reflected from the sandwich panel and transmission through it, as well as the displacement potentials of elastic waves excited in the outer layers of the panel, are in the form of decompositions according to Cartesian basic solutions of the Helmholtz equation. To determine the displacement field in the core, a boundary value problem for a system of linear ordinary differential equations of the second order is obtained. The influence of the inhomogeneity of the core on the transmission of sound through the sandwich panel is numerically investigated.
In the inverse problem, the laws of inhomogeneity of the core are determined, ensuring minimal sound permeability of the sandwich panel. Based on the solution of the direct problem, a functional is constructed, defined on a class of linear functions describing the mechanical parameters of the core and expressing the average sound permeability coefficient of the sandwich panel in a given frequency range. The minimization of the functionality is carried out using an algorithm based on a combination of random search and coordinate descent methods.