A contribution to the theory of acoustic radiation

The field of radiation produced by a harmonically oscillating membrane with arbitrary amplitude distribution in a closely fitting aperture of an infinite rigid plane is studied. The analysis does not take into account the viscosity of the medium nor any other deviation from its "ideal" behavior.Numerical calculations are not given. Attention is paid mainly to the mathematical formalism. In section 1 the problem is mathematically stated by means of the velocity potential in the form of a boundary value problem in connection with the wave equation. A new argument is given as to why the time factor exp (-iwt) is preferred in theoretical considerations. Section 2 contains a brief sketch of the derivation of the now already classical Rayleigh formula, together with some critical remarks. Section 3 is devoted to energy considerations. The definitions of acoustic impedance and radiation characteristic are given. In addition formulae are derived by means of which they can be computed in the case of any prescribed source distribution. In section 4 the general formulae are applied to a circular membrane, oscillating with azimuthal and radial nodal lines. In section 5 King's theory of the circular disc oscillating with uniform amplitude is very much extended. The theory can be developed in terms of cylindrical wave functions. It is possible to express the velocity potential as a definite integral involving Bessel functions. It is shown that the reactive part of the acoustic impedance of any harmonically vibrating membrane of whatever amplitude distribution is always negative. This part can be calculated by integrating the radiation pattem over complex space directions. Section 6 shows expansions in spherical wave functions, applied to the radiation field of a membrane oscillating with nodal lines. King's statement that the theory of Backhaus regarding certain expansions in spherical coordinates does not hold is shown to be erroneous. For the sake of completeness the theory of section 6 is worked out in detail in the case of the Rayleigh plate. This is to be found in section 7. To preserve the reader from many tedious operations the majority of the necessary mathematical calculations are given in a separate mathematical appendix.