The acoustic analogy and the prediction of the noise of rotating blades

The acoustic analogy was introduced into acoustics by Lighthill in 1952 to understand and predict the noise generated by the jet of an aircraft turbojet engine. The idea behind the acoustic analogy is simple but powerful. The entire noise generation process is mathematically reduced to the study of wave propagation in a quiescent medium with the effect of flow replaced by quadrupole sources. In jet noise theory, Lighthill was able to obtain significant and useful qualitative results from the acoustic analogy. The acoustic analogy has influenced the theoretical and experimental research on jet noise since the early 1950s. This paper, however, focuses on another area in which the acoustic analogy has had a significant impact, namely the prediction of the noise of rotating machinery. The governing equation for this problem was derived by Ffowcs Williams and Hawkings in 1969. This equation is a wave equation for perturbation density with three source terms, which have become known as thickness, loading, and the quadrupole source terms, respectively. The Ffowcs Williams-Hawkings (FW-H) equation has been used for the successful prediction of the noise of helicopter rotors, propellers, and fans. Several reasons account for the success and popularity of the acoustic analogy. First, the problems of acoustics and aerodynamics are separated. Second, because the FW-H equation is linear, powerful analytical methods from linear operator theory can be used to obtain closed-form solutions. Third, advances in digital computers and computational fluid dynamics algorithms have resulted in high-resolution near-field aerodynamic calculations that are suitable for noise prediction. We present some of the mathematical results for noise prediction based on the FW-H equation, including examples for helicopter rotors. In particular, we discuss the prediction of blade-vortex interaction noise and high-speed impulsive noise of helicopter rotors. For high-speed propellers, we briefly discuss the derivation of a singularity-free solution of the FW-H equation for a supersonic panel on a blade.