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We study the propagation of acoustic waves in a fluid that is contained in athin two-dimensional tube, and that it is moving with a velocity profile thatonly depends on the transversal coordinate of the tube. The governing equationsare the Galbrun equations, or, equivalently, the linearized Euler equations. Weanalyze the approximate model that was recently derived by Bonnet-Bendhia,Durufl\'e and Joly to describe the propagation of the acoustic waves in thelimit when the width of the tube goes to zero. We study this model for strictlymonotonic stable velocity profiles. We prove that the equations of the model ofBonnet-Bendhia, Durufl\'e and Joly are well posed, i.e., that there is a uniqueglobal solution, and that the solution depends continuously on the initialdata. 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 atmost as $t^4$. This establishes the stability of the model in a weak sense.These results are obtained constructing a quasi-explicit representation of thesolution. Our quasi-explicit representation gives a physical interpretation ofthe propagation of acoustic waves in the fluid and it provides an efficient wayto compute numerically the solution.

Analysis of Acoustic Wave Propagation in a Thin Moving Fluid