Acoustical Holography in Spherical Coordinates for Noise Source Identification

During a study of reciprocating refrigerator compressor noise, it was desired to develop a new technique for visualizing of the vibrations of the shell surface. Previous investigators have shown the value of acoustical holography for this purpose in other applications. However, the spherical geometry of the compressor of interest required the use holography in spherical coordinates. The present work involved checking the accuracy of acoustical holography in spherical coordinates through numerical simulations, and applying the technique to experimental cases including a shell driven by a shaker and the actual compressor in operation on a test stand. INTRODUCTION Acoustical holography for the purpose of noise source identification has most often been applied in planar geometries, as summarized by Maynard et al [1]. However, many small machines have shapes that make planar holography inconvenient. In these cases, spherical or cylindrical analysis geometries may yield better results. Several investigators have developed holographic algorithms incorporating solutions of the wave equation in cylindrical and spherical coordinates. For spherical geometries the technique has been applied in cases where the sound source may be entirely enclosed by a sphere on which sound pressure is measured. Weinreich and Arnold [2] have described a method in which a boom system was employed to measure the complex sound pressure on two concentric spheres. They then expanded the solution of the wave equation in spherical coordinates in terms of the spherical harmonics whose amplitudes were determined from the measured pressures; their method can be used to characterize the sound field when both incoming and outgoing waves are present. Laville et al. [3] have developed a technique based on the assumption that the sound field comprises outgoing waves only, and which allows the complex pressure to be expressed as a function of sound intensity and the mean square pressure measured on a single spherical surface. The present work was conducted in order both to confirm the accuracy .of spherical holography in simulation, and to extend its application to noise source identification for small machines, specifically a refrigeration compressor. As employed here, the technique involves using sound pressure and referenced phase data on the measurement sphere to determine the spherical harmonic coefficients. Those coefficients are then used to propagate and reconstruct the sound field in either of two directions: outward, to visualize the sound field, or inward for the purpose of noise source identification. The acoustic particle velocity and radial displacement can be calculated so that their distributions over the surface of the source may be visualized. The pressure and velocity results can then be combined to yield the acoustic intensity. THEORY The radiation of sound from vibrating spherical bodies is described in detail by Morse and Ingard [4]. In short, the wave equation is solved in spherical coordinates by separation of variables so that the radial dependence of the sound field is described by spherical Hankel functions, and the polar and azimuthal dependence is described by the Legendre functions, through which the spherical harmonics can be calculated. The general series solution may be expressed as [ 4]: n P (M,co) = L hn(kr) L [AnmYri'm(O,q>) + BnmYiim(O,q>)] (1) n=O m=O