# What's the difference between shock waves and acoustic waves?

What's the difference between shock waves and acoustic waves? I tried and searched around this subject, but I could not find any relative article about it. Please help me find a proper answer.

What's the difference between shock waves and acoustic waves?

In principle, an acoustic wave can be of the same mode as a shock wave if the speed of communication in the medium in question is the speed of sound. As I stated in https://physics.stackexchange.com/a/139436/59023, a shock forms from a nonlinearly steepening wave. In principle, all longitudinal, compressional modes will steepen which can be though of as an amplitude dependence in the phase speed. That is, higher amplitude parts of the wave will propagate faster.

In the fluid equations of motion, steepening comes from the $$\mathbf{u} \cdot \nabla \mathbf{u}$$ term. Further, the speed of sound is defined as $$C_{s}^{2} = \partial P/\partial \rho$$, where $$P$$ is the scalar pressure and $$\rho$$ is the mass density. In a sound wave, the mode is longitudinal and so the peaks(troughs) are compression(rarefaction) regions. In the absence of energy dissipation, a sound wave will steepen and eventually undergo wave breaking. If there is energy dissipation, but it does not become a strong parameter until the steepening causes the gradient scale length becomes comparable to the mean free path for particle-particle collisions, then it is possible for a steepening sound wave to reach a stable discontinuity, called a shock wave.

So the short answer is a shock wave is the nonlinear extrema of an acoustic wave in a collisional medium where the speed of sound is the relevant communication speed.

I should note that the speed of sound is not the relevant speed in all mediums, e.g., in plasmas it is the fast mode speed (i.e., see https://physics.stackexchange.com/a/179057/59023).