Kinematics in iterated correlations of a passive acoustic experiment

Wave equation and Green's function

We study the wave equation in an acoustic, linear, isotropic, time-invariant, sourceless, constant-density medium. The familiar wave equation for pressure $P = P(\mathbf{x},t)$ is

$$\partial_i^2 P - c^{-2}\partial_t^2 P = 0,$$ (62)

where Einstein's summation convention is applied to lower-case subscripts; for 2D they are summed over 1 and 2. Temporal and spatial derivatives are denoted $\partial_t$ and $\partial_i$ respectively, where the subscripts denote time and spatial directions respectively. Under the constant-density assumption, the characteristic wave velocity $c = c(\mathbf{x})$ fully determines the medium.