# Forward and Inverse Methods using spherical measurement arrays

**At LMSSC, Cnam, Paris, April 25th 2013, 14 p.m.**

Earl. G. Williams

Senior Scientist for Structural Acoustics & Sound Field Reconstruction,

Acoustics Division at the Naval Research Laboratory (NRL), Washington D.C., USA

An array of microphones populating a spherical boundary provides an exciting new tool to study a broad range of acoustic problems from concert halls to the interiors of aircraft and automobiles. These studies have used open arrays (transparent to the incident field) and closed arrays (flush mounted on a rigid surface). In either case and when viewed from a mathematical prospective the open and the closed spheres are perhaps the only practical geometries in which the measured field can be expanded into orthogonal eigenfunctions (called spherical harmonics) and extrapolated without error off the surface (using spherical Hankel/Bessel functions) to predict the pressure in the space outside or inside the sphere. Outside the sphere these extrapolations can be reformulated as plane waves and can predict an equivalent three-dimensional distribution of plane waves that exist outside the array, extremely useful in studying the reflectivity of the bounding surfaces in auditoriums. Smaller confined spaces such as the interior of automobiles are better treated using inverse methods to extrapolate the field throughout the interior again using open or closed array measurements. These methods are a form of near-field acoustical holography (NAH) and are very powerful in their ability to image the acoustic intensity fields – the flow of energy in the interior - providing a powerful visual metric to identify structural sources of noise. Popular source imaging methods often promoted by instrument companies use delay-sum beamforming with open arrays and are attractive because of their mathematical simplicity.

This lecture provides a broad discussion of these approaches discussing the basic theories in a unified way and comparing results to illuminate capabilities. Although most of the literature deals with the frequency domain, we derive a new time-domain formulation using generalized function analysis that we hope will prove useful to the acoustics/ambisonics community. This research is supported by ONR.