Solution of Ffowcs Williams & Hawkings’ Equation
4.1 Formal Solution
Since the solid surfaces responsible for sound generation are replaced by equivalent sources, the wave equation of the analogy as stated by Ffowcs Williams & Hawkings is solved in an unbounded medium at rest by making use of the free-space Green’s function. The formal solution is described below for the standard application with integration on the solid surfaces. After making use of the general properties of convolution products and performing the change of variable y ^ n where n is the source coordinate vector in the reference frame attached to the surfaces, the acoustic pressure fluctuation at point x and time t is expressed as
with Rn = |x — y(n, t’)|. Note that the surfaces are assumed rigid.
Note that here the summation required for the two possible roots of the retarded-time equation in supersonic motion is not written for simplicity. In eq. (25) P is the net force on the fluid from each surface element, Vn is the normal velocity field on the surfaces for a normal unit vector pointing inwards, M is the Mach number of the sources, corresponding to the velocity in the stationary frame of reference, 1 — Mr with Mr = M • R/R is the Doppler factor related to the projected motion on the line from the source to the observer, and the squared brackets mean that the embedded quantity is to be evaluated at the retarded time. Equation (25) clearly separates the contributions of source motion with respect to the stationary axes, and source physics described in the moving axes, leading to generality the simple arguments introduced with translating point sources in sections 3.2 and 3.4.
Though exact the result involves a surface distribution of monopoles whereas the total mass is constant for rigid surfaces. There is only fluid displacement induced by the passage of the surfaces. In fact the instantaneous balance of monopoles is exactly zero and sound is radiated only because of retarded-time differences between the monopoles. For this reason an alternative and mathematically equivalent form of the result has been proposed by Ffowcs Williams & Hawkings (1969) and Goldstein (1976) as
introducing volume integrals of equivalent dipoles and quadrupoles over the inner volume Vi of the surfaces. Note that the volume and surface boundaries do not depend on time anymore. Notations (V(0), Г) stand for the absolute velocity and acceleration fields defining the solid-body motion of the surfaces. All quantities are now defined in the sense of ordinary functions. Finally the noise radiated by flows and surfaces in arbitrary motion can be thought of as produced by dipoles and quadrupoles only. Moreover, when the inner volume of the surfaces tends to zero, as in the case of thin airfoils or blades of compressors, the noise from the third and fourth source terms in eq. (26) is expected negligibly small. This justifies the terminology of thickness noise. Both formulations are singular at the sonic radiation angle for which the denominator vanishes. The physical meaning of the singularity has been discussed in section 3.2. As shown by Ffowcs Williams &
Hawkings (1969) and Farassat (1983), it is removable at the price of other sophisticated solving procedures, not detailed here. The present expressions hold away from the singular condition, for either subsonic or supersonic regimes. For simplicity the subsonic regime is retained later on.