An eigenvalue problem relating to reactive acoustic power and acoustic radiation

Abstract

The acoustic power radiated for mono-frequency vibrations from submerged bodies can be calculated by evaluating one half of surface integral of surface pressure multiplying the complex conjugate of normal velocity. The real part of the integral is the radiated power from bodies, whereas the imaginary part, or the reactive power, is seldom used in the study of surface acoustics. The present study is to explore the reactive power in the connection of dynamic characteristics for surface pressure and normal velocity. This problem is considered by investigating the quadratic expressions of complex surface acoustic power as discretizations are introduced on the surface. A reciprocal principle for acoustic medium is employed to assert the symmetry of the matrices associated with the quadratic forms. Correspondingly, the investigation of surface acoustics turns into the examination of symmetrical eigenvalue problems associated with the matrices. Eigenvalues are identified as the indication of phase differences between surface pressure and normal velocity in the corresponding eigenvectors. These eigenvalues are then used to classify the modes for strong or poor radiations. The triple quantities, which are eigenvalues, modal modes for surface pressure, and normal velocity, provide a full description of dynamic characteristics for submerged bodies. [Work supported by Chinese National Science Foundation, NSC 82-0209-E-019-063 and United States Office of Naval Research, Code 1132-SM.]