I don't have much knowledge on gas dynamics but I read that acoustic wave is discontinuous solution since properties ahead and behind the wave are different but the equation of a sound wave is a sine wave as I remember so why is it discontinuous?
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$\begingroup$ I think you need to say where you have read this, if you want an answer. It sounds more like a statement about shockwaves than about sound waves. $\endgroup$– mike stoneCommented Jan 1, 2021 at 13:49
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$\begingroup$ I came across it in a lecture video and I also read it in a book named fundamentals of gas dynamics., the exact statement was "when points 1 and 2 are located across a wave (say, a sound wave or shock wave),the derivatives of the flow properties will be discontinuous" , where 1 and 2 are 2 points over which the basic 1- d flow equations (for frictionless, adiabatic, steady, one dimensional flow of a calorically perfect gas.) were integrated. $\endgroup$– VivekCommented Jan 2, 2021 at 8:18
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$\begingroup$ That sentence makes little sense, I'm afraid. $\endgroup$– mike stoneCommented Jan 2, 2021 at 12:55
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Acoustic wave may indeed be considered as an infinitesimally weak shock wave (see Fundamentals of Aerodynamics by J. Anderson, for example), in the sense that fluid properties (pressure, for example) change by an infinitesimal amount across the acoustic wave. It obeys the usual wave equation (see derivation) precisely because it is a small amplitude wave.
It is not true that the wave equation allows only sinusoidal solutions. For example, the general solution of a 1-dimensional non-dispersive wave equation is $F(x-ct)+G(x+ct)$, where $F$ and $G$ are arbitrary functions; sine wave is just one among several possibilities. Here $x$ is spatial coordinate along the direction of propagation, $t$ is time and $c$ is phase speed of the wave.
