= Ffowcs Williams-Hawkings equation
{c}
= FW-H equation
{c}
{synonym}
The <Lighthill acoustic analogy> for a moving body has a volume <acoustic quadrupole>, a surface <acoustic monopole> from <acoustic thickness noise>, and a surface <acoustic dipole> from <acoustic loading noise>. For an impermeable moving surface, with outward normal $n$, normal surface speed $v_n$, and fluid and body <normal velocities> equal, the surface <mass> coefficient is $\rho_0v_n$ and the surface loading is the <force> exerted on the fluid. Schematically, the density-source equation is
$$
(\partial_t^2-c_0^2\nabla^2)\rho'=\partial_i\partial_j(T_{ij}H(f))+\partial_t(Q\delta_s)-\partial_i(L_i\delta_s),
\quad \delta_s=\delta(f)|\nabla f|.
$$
The density perturbation is understood with the chosen interior extension. Permeable-surface versions have additional <mass> and momentum flux terms. Source approximations must distinguish local thickness radiation from cancellation of its compact net-volume contribution.
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