The acoustic analogy in four dimensions

Abstract

The classical acoustic analogy theory is incomplete in the sense that the original research on the subject focused only on the prediction of acoustic pressure. There were no provisions for predicting the three components of acoustic velocity which are needed as input for aeroacoustic scattering applications. This is because the scalar wave equations of Lighthill and Ffowcs Williams and Hawkings were derived from the fluid conservation equations by eliminating three of the four governing differential equations from which the acoustic velocity could have been obtained. Recently developed acoustic analogy methods for predicting the acoustic velocity lead to complex formulations whose numerical evaluation can be problematic when providing input for large-scale scattering problems. Their calculation can overwhelm the numerical scattering process when the incident sound has high-frequency content, such as that produced by the rotating blades of various propulsion devices. To obtain improved acoustic analogy formulations for scattering and other applications, the three discarded differential equations are returned to the analysis in this paper and the historical acoustic analogy equations are derived anew using four-dimensional tensor methods. The 4-D formulation is an application of the electromagnetic analogy (EMA), a concept based on the equivalence of Maxwell’s equations and the fluid conservation equations of mass and momentum. The scalar equations of Lighthill, Ffowcs Williams and Hawkings, Farassat, and Kirchhoff are extended to four equations – one equation for the acoustic pressure (or acoustic density) and three equations for the acoustic velocity. The 4-D tensor representation provides significant theoretical and computational simplification relative to the classical approach. For each of the original acoustic analogy results, a single, concise formulation in 4D is derived that enables the simultaneous prediction of acoustic pressure and the three components of acoustic velocity.