AI 中文总结
本章从经典连续介质力学原始拉格朗日量系统推导声学控制方程，回顾拉格朗日框架下可压缩无耗散流体的作用量构造，通过摄动展开从变分原理恢复线性声学方程，得到弱非线性近似，理想气体结果为特例。
AI 中文摘要
文献中已出现许多声学变分原理，通常以特设方式引入，通过在拉格朗日密度中选取项以得到期望的控制偏微分方程。在本章中，我们表明这种猜测是不必要的，因为声学的控制方程可从经典连续介质力学的原始拉格朗日量系统推导得出。此外，我们注意到使用欧拉坐标并非推导声学变分原理的唯一方式。为此，我们回顾了可压缩无耗散流体在拉格朗日框架下作用量的构造，探究了物质重标记对称性及其相关赝动量平衡的结果，以及其蕴含的边界和跳跃条件。通过摄动展开，我们展示了如何从每个变分原理中恢复线性声学方程，物质框架还额外给出二阶声能平衡。我们展示了具有变分结构的弱非线性近似如何在两种框架中出现，适用于一般正压流体，其中非线性由流体的参数$B/A$携带，理想气体结果作为特例被恢复。
英文摘要
Many variational principles for acoustics have appeared in the literature, often introduced in an \textit{ad hoc} manner by picking and choosing terms in a Lagrangian density to yield a desired governing partial differential equation. In this chapter, we show that such guesswork is unnecessary, as the governing equations of acoustics can be derived systematically from the primitive Lagrangians of classical continuum mechanics. Furthermore, we note that using Eulerian coordinates is not the only way to derive variational principles in acoustics. To this end, we review the construction of an action in the Lagrangian frame for compressible nondissipative fluids. We explore the consequences of material relabeling symmetry and the associated pseudomomentum balance, together with the boundary and jump conditions it implies. Via perturbation expansions, we show how to recover the equations of linear acoustics from each variational principle, the material frame yielding in addition the second-order acoustic energy balance. We show how weakly nonlinear approximations with variational structure emerge in each frame, in both cases for a general barotropic fluid, with the nonlinearity carried by the fluid's parameter $B/A$ and the perfect-gas results recovered as a special case.
Comments33 pages, 1 figure; submitted for consideration in an edited volume of the "Advances in Continuum Mechanics" series
