= Solution
Let $\rho'$ and $p'$ be the density and pressure perturbations and let $\mathbf u$ be the velocity perturbation. The <linear acoustics> equations about a uniform quiescent state are
$$
\rho'_t+\rho_0\nabla\mathbin{\cdot}\mathbf u=0,
\qquad
\rho_0\mathbf u_t=-\nabla p',
\qquad
p'=c_0^2\rho'.
$$
The last relation is the <homentropic pressure perturbation> law.
Taking the scalar product of the momentum equation with $\mathbf u$ gives
$$
\frac{\partial}{\partial t}\left(\frac12\rho_0|\mathbf u|^2\right)
=-\mathbf u\mathbin{\cdot}\nabla p'
=-\nabla\mathbin{\cdot}(p'\mathbf u)+p'\nabla\mathbin{\cdot}\mathbf u.
$$
The continuity and pressure relations imply
$$
p'\nabla\mathbin{\cdot}\mathbf u
=-\frac{p'p'_t}{\rho_0c_0^2}
=-\frac\partial{\partial t}
\left(\frac{p'^2}{2\rho_0c_0^2}\right).
$$
Therefore the <acoustic energy conservation> law is
$$
\boxed{
\frac\partial{\partial t}(K+W)+\nabla\mathbin{\cdot}\mathbf I=0
},
$$
with
$$
\boxed{
K=\frac12\rho_0|\mathbf u|^2,
\qquad
W=\frac{p'^2}{2\rho_0c_0^2},
\qquad
\mathbf I=p'\mathbf u
}.
$$
Back to article page
