A Multiple Scales Approach to Sound Generation by Vibrating Bodies

Abstract

The problem of determining the acoustic field in an inviscid, isentropic fluid generated by a solid body whose surface executes prescribed vibrations is formulated and solved as a multiple scales perturbation problem, using the Mach number M based on the 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 spacial scale multiplied by various powers of M. The governing nonlinear 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 spacial 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 spacial scale playing the role of the time-like variable. The method is illustrated by a simple one-dimensional example, as well as by the pulsating sphere. The results are compared with solutions obtained by purely numerical methods and some insights provided by the perturbation approach are discussed.