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.

An Investigation into the Four-Dimensional Acoustic Analogy Model for Homogeneous Media and the Prediction of Flow-Induced Noise in Spatio-Temporal Fields

“…Sound intensity, expressed by acoustic pressure and acoustic velocity, is a more representative measure to reveal the radiation and propagation paths of noise energy. Therefore, the classical acoustic analogy method is incomplete [4]. Meanwhile, the study of acoustic velocities contributes to accurate calculations of acoustic scattering from large structures, such as the fuselage or wing [5,6].…”

“…Kambe [20] was the first to introduce the flow field and electromagnetic field analogy to study the vortex sound theory, and the presented formulation of wave equations with entropy and acoustic velocity as integral variables is innovative. Based on Kambe's work, Dunn [4] developed a 4D Lighthill acoustic analogy for stationary media by considering the acoustic pressure and acoustic velocity in three directions as four acoustic variables. This approach resulted in a simple integral solution, which was further improved by incorporating a permeable boundary.…”

Predicting Flow-Induced Noise Based on an Improved Four-Dimensional Acoustic Analogy Model and Multi-Domain Feature Analysis

“…Using sound intensity visualization technology, Lee et al [8] revealed the impact of scattering surfaces on sound energy redistribution, and Crocker and Jacobsen [9] mainly discussed the theoretical and practical application of sound-intensity measurement, especially in engineering problems. It can be seen from the above that the classical acoustic analogy method only focuses on the prediction of sound pressure and does not stipulate the extrapolation of the acoustic velocity vector, making it incomplete [10].…”

“…Scholars have proposed that some combinations of flow field variables and their derivatives can be defined as "fluid electric fields" and "fluid magnetic fields," thereby converting fluid dynamics equations into "fluid Maxwell's equations" [21,22]. Dunn [10], using the concept of electromagnetic analogy, reorganized the mass and conservation equations of fluid motion, establishing a fourdimensional acoustic analogy formulation that includes both sound pressure scalars and acoustic velocity vectors (three components). The four-dimensional acoustic analogy formula is easy to implement and can predict the characteristic distributions of both acoustic pressure and velocity.…”

“…The four-dimensional acoustic analogy formula is easy to implement and can predict the characteristic distributions of both acoustic pressure and velocity. However, Dunn's method [10] is only reliable when the object surface is chosen as the integration boundary. When using a permeable surface as the integration boundary, the surface source terms related to acoustic velocity in Dunn's four-dimensional acoustic analogy integration formula are not fully extracted, leading to inaccurate prediction results for acoustic velocity.…”