Ffowcs Williams–Hawkings equation

Definition

Consider a compressible fluid in $\R^3$ with density $\rho$, velocity $u$, pressure $p$ and viscous stress tensor $\tau_{ij}$. Let $\rho_0$, $p_0$, $c_0$ be the constant density, pressure and speed of sound of the fluid at rest. Let a closed surface move in the fluid, given as the zero set of a smooth function $f(x, t)$. Assume $f > 0$ outside the surface and $\abs{\nabla f} = 1$ on it. Let $n = \nabla f$ be the outward unit normal, $v_n$ the normal velocity of the surface and $u_n = u \cdot n$ the normal velocity of the fluid. Let $H$ be the Heaviside function and $\delta$ the Dirac delta. Set $\rho' = \rho - \rho_0$, $P_{ij} = (p - p_0) \delta_{ij} - \tau_{ij}$, and let $T_{ij}$ be the Lighthill stress tensor of Lighthill's acoustic analogy. The Ffowcs Williams–Hawkings equation is the equation, in the sense of distributions, $$\left( \frac{1}{c_0^2} \frac{\partial^2}{\partial t^2} - \Delta \right) \big( c_0^2 \rho' H(f) \big) = \frac{\partial}{\partial t} \big[ Q \, \delta(f) \big] - \frac{\partial}{\partial x_i} \big[ L_i \, \delta(f) \big] + \frac{\partial^2}{\partial x_i \partial x_j} \big[ T_{ij} H(f) \big],$$ where $Q = \rho_0 v_n + \rho (u_n - v_n)$ and $L_i = P_{ij} n_j + \rho u_i (u_n - v_n)$, with summation over repeated indices.