The problem of flexible vibrations of a rectangular orthotorque plate clamped along the contour is considered. The general solution of the problem, which satisfies the vibartion equation identically, is constructed on the basis of the superposition method in the form of two Fourier series. Clamped boundary conditions lead to a homogeneous infinite system of linear algebraic equations with respect to unknown coefficients in the general solution. The uniqueness of a bounded non-trivial solution of an infinite system for the natural frequency is proved, the asymptotics of the unknowns are found, and an effective solution algorithm is constructed. Examples of the numerical implementation of the developed algorithm for calculating the natural frequencies and natural modes of the plate vibrations are given.
Keywords: plate, vibrations, natural frequencies, planar forces, superposition method, infinite system of linear equations, asymptotics
An analytical representation of the velocity potential of a point source of sound is constructed for a model of a marine acoustic waveguide with a rigid stepped bottom, where a sound velocity profile varies with depth. Inhomogeneity of the bottom in the form of a cylindrical protrusion is modeled on the basis of the method of partial regions. The sound field is represented in the form of the sum of normal modes to construct the sound velocity potential in each partial region. The continuity of solutions at the boundary of partial regions leads to an infinite system of linear equations with respect to the coefficients under normal modes. In this work, formulas are obtained that describe the energy characteristics of the propagation of each of the normal modes along the waveguide. Examples of numerical modeling are given. An analysis of the excitation coefficients of normal modes is carried out for waveguide parameters are true to type of the Black Sea region.
Keywords: waveguide, normal modes, bottom inhomogeneity, excitation coefficient, partial regions, infinite system of linear equations, asymptotics