The formulation of sound transmission problems through the boundaries of three-dimensional media and through plates

Introduction. Modern physical models for calculating the propagation of plane longitudinal waves through media boundaries, based on the conditions of continuity of sound pressure and vibrational velocity, are implemented only with normal wave incidence. In all directions of wave propagation other than normal, the continuity conditions are not observed, which does not allow to obtain the correct formulas for the coefficients of reflection and transmission of waves.

Materials and methods. In the proposed formulation of the problem, the physical model of the wave propagation medium consists of pieces of wave rays, tightly, without breaks and mutual penetrations, occupying its entire space (as it clearly happens with waves on the surface of water). A method for determining the volumes of these pieces is given. Their masses are approximated by material points having effective values of vibrational wave velocities. The passage of a plane harmonic wave through a plane boundary of media is described by the equations of conservation of kinetic energy and conservation of the amount of motion. The solution of these equations gives the correct formulas for the coefficients of reflection and refraction of waves in terms of vibrational velocity at any angles of their propagation.

Results. The proposed formulation of the problem of wave propagation through the boundary of continuous semi-infinite media is suitable for solving the problems of sound propagation through layers and, in particular, through plates. The problem of sound propagation through a plate separating the air medium is fundamental in the sections of architectural and technical acoustics, since applied theories of sound insulation of walls and ceilings of buildings and vehicles enclosing premises are based on it.

Conclusions. The conservation equations, up to the boundary frequency of the wave coincidence, will include the effective value of the vibrational velocity in the incident wave, the angle of wave propagation, the values of the mass of a discrete body represented by the surface density of the plate, the reduced masses of the pieces of the medium and unknown coefficients of reflection and passage of the vibrational velocity. At frequencies above the boundary, the mass of the discrete body changes to the reduced mass of the plate. The solution of the system of conservation equations gives the correct formulas for the coefficients of sound transmission and reflection and the correct formulas for sound insulation, in accordance with changes in the physical models of wave propagation in different frequency ranges.

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