Asymptotic Formulas for Active and Reactive Mutual Power of Free Vibrations of a Clamped Annular Plate

Normalized mutual power of radiation, active and reactive, has been analyzed in this paper. The sound source is an annular plate, clamped to a planar, rigid and infinite baffle. The plate’s vibrations are axisymmetric and sinusoidal with respect to time. A Kirchhoff-Love’s model of a perfectly elastic plate has been used. First, the integral formulas for the mutual sound power have been presented and then the asymptotic formulas for fast numerical computations of the mutual sound power have been computed, which can be used for acoustically fast waves. The contour integral computation and stationary phase methods have been used. The analysis presented can be used as a basis for analysis of the total power of excited and damped vibrations. 1 INTRODUCTION A modal analysis has been made to get the normalized mutual sound power, active and reactive, of a clamped annular plate. Free, axisymmetric vibrations sinusoidally varying with respect to time with no friction and damping forces have been considered. A Kirchhof-Love model of a perfectly elastic plate has been used. Results of this analysis can further be used to compute the total sound power of a clamped annular plate. The frequency equation and its solution have been presented in several works (e.g. [2], [5, 6]). The contour integral computation and stationary phase methods have been used to transform some integral formulas to asymptotic formulas with known approximation error. And those methods are presented in several papers for computation of mutual sound power of a circular plate (e.g. [3, 4]) or sound power of an annular plate [5]. A number of papers treat of several theoretical as well as experimental methods for obtaining the sound power of circular plates (e.g. [1]). 2 INTEGRAL REPRESENTATION The mutual sound power lost via mode (0,m) and used to overcome resistance produced by the same source via mode (0,n) of free vibrations of a clamped annular plate is