Application of full and simplified acoustic analogies to an elementary problem

Abstract

The axisymmetric impulse-heating case used by Tam in a critique of Lighthill's acoustic analogy is revisited. Linear and nonlinear sound levels are considered, as are three versions of the acoustic analogy: full; linearized; and one with the quadrupoles frozen at their initial distribution. Numerical solutions confirm that the full version reproduces the Euler solution and, in our analysis but contrary to Tam's, correctly identifies both the source of sound and nonlinear steepening. The linearized version is somewhat inaccurate at high sound levels, near 175 dB. The frozen version is almost as accurate as the linearized version at these levels, and identical at linear levels. In this admittedly artificial problem, it provides the radiated sound from the initial conditions only, i.e. without using the Euler solution. This is in contrast with the full acoustic analogy, which can be viewed as ‘circular’ in the sense of taking the history of all the Euler variables and returning the density.