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.

中文翻译：

---

四个维度的声学类比

经典的声学类比理论是不完整的，因为该主题的原始研究仅关注声压的预测。没有规定用于预测作为气动声学散射应用的输入所需的声速的三个分量。这是因为 Lighthill 和 Ffowcs Williams 和 Hawkings 的标量波动方程是从流体守恒方程中推导出来的，通过消除四个控制微分方程中的三个可以获得声速。最近开发的用于预测声速的声学类比方法导致复杂的公式，当为大规模散射问题提供输入时，其数值评估可能会出现问题。当入射声音具有高频成分时，例如由各种推进装置的旋转叶片产生的高频成分，他们的计算可以压倒数值散射过程。为了获得用于散射和其他应用的改进的声学类比公式，本文将丢弃的三个微分方程返回到分析中，并使用四维张量方法重新推导出历史声学类比方程。4-D 公式是电磁类比 (EMA) 的应用，该概念基于麦克斯韦方程组的等效性以及质量和动量的流体守恒方程。Lighthill、Ffowcs Williams 和 Hawkings、Farassat 的标量方程，Kirchhoff 和 Kirchhoff 扩展到四个方程 - 一个用于声压（或声密度）的方程和三个用于声速的方程。相对于经典方法，4-D 张量表示提供了重要的理论和计算简化。对于每个原始的声学类比结果，都导出了一个单一、简洁的 4D 公式，可以同时预测声压和声速的三个分量。