Abstract
The paper explores the physical meaning underlying the surface complex acoustic power of a vibrating body, and its relationship to radiation efficiency under mono-frequency oscillations. The vibrating can be the entire wetted surface, or only a part of the surface with the remaining surface being held rigid. The surface complex acoustic power can be computed by the surface integral of pressure multiplying the complex conjugate of normal velocity. Based on the Gaussian Divergence theorem, it is shown that the real part of the complex power is the power radiated into a far field, while that the imaginary part pertains to the volume integral of the difference between the acoustic kinetic energy density with the potential energy density over the volume between the vibrating surface and the far field. The dynamical behavior of the acoustic field can be viewed as an infinite degree of freedom mass/spring/dashpot system, where the mass and spring are the inertia effects and acoustic compression effects of the acoustic particles and the dashpot is due to the plane wave relationship of the pressure waves at the far field that the acoustic energy propagates away from the acoustic field. By the model of the mass /spring/dashpot system, the phase angle of the complex acoustic power is identified as an indication of the ability of the vibrating surface to radiate acoustic power. The phase angle of the complex power depends on the distribution of the surface normal velocity. In order to study the normal velocity profile in relation to the ability to radiate acoustic energy, the previously established radiation mode (Chen and Ginsberg, 1995) is introduced and extended to situations in which a part of the surface is held rigid. An orthogonal condition for the velocity radiation modes is also established such that arbitrary velocity profiles can be decomposed into radiation modes. The acoustic modal radiation efficiency, defined as the radiated modal acoustic power divided by the surface integral of mean square normal velocity, is investigated in terms of the acoustic eigenvalue of that mode. Several different geometries of vibrating bodies are used to demonstrate the correlation of radiation efficiencies to eigenvalues of radiation modes, which include a rectangular baffled vibrating membrane, a box with only one of the six surfaces vibrating, a slender spheroidal body, and a spherical body. This correlation of acoustic radiation characteristics for different geometries is also demonstrated for a spheroidal body vibrating at some areas with other areas being held rigid.
Extension of Kelvin’s minimum energy theorem to incompressible fluid domains with open regions
