A multiple scales approach to sound generation by vibrating bodies

• Published in: Journal of sound and vibration Vol. 185; no. 5; pp. 767 - 790
• Main Authors: Geer, J.F., Pope, D.S.
• Format: Journal Article
• Language: English
• Published: London Elsevier Ltd 07.09.1995; Elsevier
• Subjects: Acoustics; Exact sciences and technology; Fundamental areas of phenomenology (including applications); Physics; Structural acoustics and vibration; Vibration; Multiscale method; Sound fields; Numerical methods; Perturbation techniques; Acoustic radiation; Spheres
• ISSN: 0022-460X; 1095-8568
• DOI: 10.1006/jsvi.1995.0417

The problem of determining the acoustic field in an inviscid, isentropic fluid generated by a solid body the surface of which executes prescribed vibrations is formulated and solved as a multiple scales perturbation problem, using the Mach number, M, based on the maximum surface velocity as the perturbation parameter. Following the idea of multiple scales, new “slow” spacial scales are introduced, which are defined as the usual physical spatial scales multiplied by powers of M. The governing non-linear differential equations lead to a sequence of linear problems for the perturbation coefficient functions. However, it is shown that the higher order perturbation functions obtained in this manner will dominate the lower order solutions unless their dependence on the slow spatial scales is chosen in a certain manner. In particular, it is shown that the perturbation functions must satisfy an equation similar to Burgers’ equation, with a slow spatial scale playing the role of the time-like variable. The method is illustrated by a simple one-dimensional example, as well as by three different cases of a vibrating sphere. The results are compared with solutions obtained by purely numerical methods and some insights provided by the perturbation approach are discussed.