Aeroacoustics, moving boundaries, and bursting balloons: acoustic sources revisited

The use of equivalent acoustic sources to describe scattering in a nonuniform medium dates back to Rayleigh's theory of sound. The idea of equivalent sources in a uniform medium at rest was later developed by Lighthill into his "acoustic analogy," capable of describing the generation of sound by turbulence and other vortical flows. In the present paper Lighthill's acoustic analogy formulation is generalized to encompass initial-value problems; the initial conditions are represented by impulsive sources and dipoles distributed over the domain, and boundary conditions are represented in the usual manner by surface sources and dipoles. David Blackstock's bursting balloon example, discussed in Chapter 3 of Fundamentals of Physical Acoustics, can be solved by this method. However, in situations where the medium is of nonuniform density (for example, a gas with a specified temperature distribution at the initial time), the impulsive source distribution obtained by a direct application of time windowing to the acoustic analogy is nonphysical. The apparent paradox is resolved by introducing the energy conservation equation, and reformulating the acoustic analogy with pressure, rather than density, as the wave variable