On linear and non-linear wave equations for the acoustics of high-speed potential flows

Abstract

The wave equation is derived for the acoustic potential in the following cases: (a) linear/non-linear sound waves, respectively of small/large amplitude; (b) medium at rest, or steady potential flow, the latter either of low Mach number, or high speed; (c) three-dimensional propagation in free space, or quasi-one-dimensional acoustics of ducts of varying cross-section. Thus there are 2×3×2=12 cases, for which 34 distinct forms of the wave equations are derived (Table 1); of these, 16 forms of the wave equation in 10 cases, appear explicitly in the references given. The wave equations are derived from a variational principle for linear sound (Part 1), and checked by elimination among the equations of potential, unsteady flow for non-linear acoustics (Part II). It is shown that the non-linear wave equations (partly new), can be written in a form similar to the linear acoustics ones (mostly known), by taking into account the self-convection of sound by sound; this implies replacing the linear local and material derivatives, byself-convected and non-linear extensions of these (Table 2), which also affect the sound speed.