A modal approximation for the mutual radiation impedance for spherical sources and acoustic wave scattering using an improved ATILA Finite Element code

Abstract

A modal approximation for the self and mutual radiation impedances has been derived for arrays of spherical transducers with small ka values, where ka is the acoustic wave number multiplied by the radius of the sphere. I term this the "Modal Pritchard" approximation, as it is related to the so-called Pritchard approximation, often employed to calculate mutual radiation impedances. I investigated the utility of the approximate mutual radiation impedance expression for three two-body problems (monopoles, aligned dipoles, and aligned-linear quadrupoles). For these cases, approximate values were found to be in good agreement with those obtained using a full Spherical Addition Theorem calculation, and are an improvement over the simple Pritchard approximation. Additionally, I investigated the mutual radiation impedance expression in one particular three-body problem. Because of the Modal Pritchard approximation's inability to correctly handle scattering, we recommend using the full Spherical Addition Theorem calculation when scattering is important. Finally, I investigated the use of a new finite element mesh to calculate the T- matrix for a given transducer. The T-matrix relates the incident and scattered waves for a single transducer, in an orthogonal (spherical harmonic) basis set. The monopole element showed an increase in error, while we saw some improvements in the higher-order diagonal elements. Off-diagonal elements, which should be zero for a spherical scatter, were satisfactorily small in most cases. Although the results were less than favorable, I was able to streamline the T-matrix calculation while providing a new method of examining the off-diagonal elements.