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When reaching the speed of sound in air a shock-wave starts building up in front of the object moving at sonic speeds where the air is compressed.

But what happens, if the medium is not (or almost not) compressible?

In my naive understanding, the pressure would go towards "infinite" and independently of how much power I put into driving the object I'd just increase the force at the front of the object but can't get any faster than the speed of sound.

Can there be anything moving faster than the speed of sound for example in water? Is there an example of something moving faster than the speed of sound in water?

Qmechanic
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kruemi
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  • The speed of sound is proportional to the square root of the "bulk modulus". The Bulk modulus is infinite in an incompressible liquid. https://en.wikipedia.org/wiki/Speed_of_sound#Speed_of_sound_in_liquids and https://en.wikipedia.org/wiki/Bulk_modulus – Quillo Nov 09 '22 at 14:25
  • Related: https://physics.stackexchange.com/q/503017/247642 I am not sure that question is even meaningful when asking about truly incompressible media, whereas if asking about water, it is answered in the link privided – Roger V. Nov 09 '22 at 15:21
  • So in a truly incompressible medium the speed of sound is meaningless and in a compressible medium the problem does not really exist (it's just really hard)? Yeah, so my question does really not make a lot of sense :( – kruemi Nov 10 '22 at 05:52

2 Answers2

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The incompressibility constraint $\nabla \cdot \mathbf{u} = 0$ is an algebraic constraint in incompressible Navier-Stokes equations. This makes the speed of perturbations (the speed of sound) in the medium to be infinite, since the medium must immediately adapt itself in the whole domain even to a local perturbation, in order to comply with the incompressibility constraint.

That said, pefectly incompressible media do not exist. That is a just a mathematical model for media that behaves (in certain conditions) as a nearly incompressible medium. But in that model, you can find a mathematical justification about the infinite velocity of perturbations.

If a medium is not perfectly incompressible, then it's a compressible medium, and thus you can treat it with the governing equation for compressible media. In order to find the speed of sound, i.e. to study the propagation of small-amplitude (usually) high frequency perturbations, you usually want to:

  • find the base-flow;
  • linearize the equations around the base-flow: this allows to find the (local) speed of sound, and determine the (local) sonic limit, between subsonic and supersonic conditions.
basics
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There was some information on a test of a supercavitating torpedo moving under water faster than sound.

akhmeteli
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  • Would you care to provide at least a brief explanation of the material you linked? A bare link provided as an answer is not the best practice. – ZaellixA Nov 12 '22 at 11:29
  • @ZaellixA : I believe I gave the most relevant information: faster-than-sound motion under water was demonstrated, and this was achieved using supercavitation. You would prefer to have more details, I prefer not to dilute the message, sorry. – akhmeteli Nov 14 '22 at 10:36
  • I understand that the linked material may indeed provide a good answer. What I object is that, in my opinion, it would be better if you were to provide, at least, a brief description of the material in that link so that someone looking for an answer won't have to rely on data based on other sites that may (or may not, of course) be removed. This way, your answer would very well constitute a place of reference for similar future questions here on Physics SE. – ZaellixA Nov 14 '22 at 18:11
  • @ZaellixA : I fully agree that my answer could be improved, but it is still an answer, and I don't have an obligation to improve it. Furthermore, upon reading your comment, I had another look at the linked material, and I don't feel I want to add anything. Of course, you are free to provide your own, more complete answer. – akhmeteli Nov 14 '22 at 18:23
  • Of course you don't have an obligation, people don't take part in SE to increase their liabilities. I just thought I should express my opinion on an answer that could very well be improved. – ZaellixA Nov 14 '22 at 23:28