Diffraction of low-frequency waves on elastic thin-walled shells of rotation

Abstract

Asymptotic and functional relations connecting the characteristics of scattered near and far fields with elastic and spectral characteristics of thin-walled elongated elastic shells described by the Love theory were found. The study was carried out by the method of two-scale expansions. For the near scattered field, recurrent systems of boundary value problems for Laplace and Poisson equations were obtained, the solutions of which were found explicitly. The radiation patterns of the scattered field were obtained using the theory of wave potentials for the Helmholtz equation. Asymptotic formulas for the potential densities of simple and double layers were found. This made it possible to present the asymptotics of the scattered field directivity diagram in the form of parametric integrals that depend on the angles of incidence and observation, frequency, surface shape, and material characteristics of the shell. The asymptotic method was effective for strongly elongated shells when the ratio of the maximum longitudinal diameter to the maximum diameter of rotation is more than ten. For such highly elongated bodies, the use of various difference and iterative schemes is problematic due to the difficulties of triangulating the shell surface. Numerical implementations of calculations of directional diagrams of a spheroidal steel shell in water at different angles of incidence of plane waves in a wide frequency range are given. The numerical calculations performed in this work are not tied to a specific frequency, since the geometric dimensions are given in wavelengths. Calculations have shown that the radiation pattern for elongated bodies begins to differ from the spherically symmetrical one at values kl > 4. When the wave size of the shell increases, the lobes of the directional diagram appear. The lobes direction depends on the above parameters. The number of lobes, their direction and power can be changed by using special distributions of the shell surface impedances.