Explanation and analysis of the solution of flow-induced sound governing equations under solid boundary

Flow-induced sound in the presence of solid boundaries is an important research topic in acoustics, as it involves the influence of wall-boundary interactions on fluid-generated sound. Ffowcs Williams and Hawkings pioneered the FW-H equation, employing Heaviside step functions to model moving surfaces. Later, Goldstein adopted Green’s second identity to derive a solution without recourse to step functions—termed the “unified approach”. While numerous publications introduce and explain these two methods, a systematic, step-by-step exposition of their underlying conceptual framework and full mathematical derivation remains scarce. This article focuses on Goldstein’s unified approach, with three main contributions: (1) unifying symbolic notation across formulations; (2) presenting an enhanced schematic diagram and providing an explicit expression for the surface velocity when an object undergoes small-amplitude elastic vibration; and (3) standardizing the formulation of governing equations—including a rigorous, line-by-line derivation of all key steps. Special attention is given to two subtle but critical points raised by Goldstein: the zero-divergence condition for the auxiliary variable and the vanishing integral of the Green’s function over the entire domain. We clarify that the final result expression is strictly valid only when both the mean flow and the acoustic source move at constant velocity relative to the observer. Furthermore, the integration domain of the resulting expression is explicitly illustrated for the first time. Finally, we summarize the overall derivation strategy, identify and justify the approximations involved, and outline how physical quantities are extracted in practical numerical implementations.