声波在薄流体中的传播分析
Analysis of Acoustic Wave Propagation in a Thin Moving Fluid
摘要：
我们研究了声波在包含在一个薄的二维管中的流体中的传播，并且它以仅取决于管的横坐标的速度分布移动。控制方程是Galbrun方程，或者等价地，线性化的Euler方程。我们分析了最近由Bonnet-Bendhia，Duruflé和Joly导出的近似模型，该模型描述了当管的宽度为零时，声波在极限中的传播。我们研究这个模型的严格单调稳定的速度剖面。我们证明了Bonnet-Bendhia，Duruflé和Joly模型的方程是适定的，即存在唯一的全局解，并且该解连续依赖于初始数据。此外，我们证明，对于光滑的配置文件的解决方案的增长最多为$t^3$作为$t \to \infty$，和分段线性配置文件，它的增长最多为$t^4$。这在弱意义上建立了模型的稳定性。这些结果是通过构造解的一个准显式表示得到的。我们的准显式表示给出了声波在流体中传播的物理解释，它提供了一种有效的方法来计算数值的解决方案。
英文摘要：
We study the propagation of acoustic waves in a fluid that is contained in a thin two-dimensional tube, and that it is moving with a velocity profile that only depends on the transversal coordinate of the tube. The governing equations are the Galbrun equations, or, equivalently, the linearized Euler equations. We analyze the approximate model that was recently derived by Bonnet-Bendhia, Duruflé and Joly to describe the propagation of the acoustic waves in the limit when the width of the tube goes to zero. We study this model for strictly monotonic stable velocity profiles. We prove that the equations of the model of Bonnet-Bendhia, Duruflé and Joly are well posed, i.e., that there is a unique global solution, and that the solution depends continuously on the initial data. Moreover, we prove that for smooth profiles the solution grows at most as $t^3$ as $t \to \infty$, and that for piecewise linear profiles it grows at most as $t^4$. This establishes the stability of the model in a weak sense. These results are obtained constructing a quasi-explicit representation of the solution. Our quasi-explicit representation gives a physical interpretation of the propagation of acoustic waves in the fluid and it provides an efficient way to compute numerically the solution.
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