# Sound Generation by Turbulence and Surfaces in Arbitrary Motion

J. E. Ffowcs Williams, D. L. Hawkings

## Abstract

The Lighthill-Curle theory of aerodynamic sound is extended to include arbitrary convective motion. The Kirchhoff description of a homogeneous wave field in terms of surface boundary conditions is also generalized to surfaces in arbitrary motion. The extension is at variance with the two previously published accounts of this problem which are erroneous. When both the bounding surfaces and the turbulence are compact relative to the radiated length scales, the turbulence is acoustically equivalent to a volume distribution of moving quadrupoles and the surfaces to dipole and monopole distributions. At low convective speed, their field increases as powers of the Doppler factor $|$1 - M$_{r}|^{-1}$. Convective acceleration generally gives rise to new source terms at this condition. At the Mach wave condition when the Doppler factor is singular, both the turbulence and surfaces are non-compact and are acoustically equivalent to monopole distributions. Convective acceleration then tends to limit the radiation. At this condition the surface sources are quite unrelated to the low-speed sources, being second order in the field variable contrasting with the linear low-speed terms. At high supersonic convective speeds, the field is dominated by an intensive beaming along the directions of Mach wave emission that lie normal to the surface. The magnitude of the field then varies inversely as the Gaussian surface curvature. If the surface has only single curvature the field is proportional to r$^{-\frac{1}{2}}$ and if it is locally plane at this condition, the field no longer decays with distance travelled. There are indications that the surface-induced intensity increases as the square of surface speed at high supersonic speeds.