# Does this wave propagation problem make physical sense?

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1. Nov 17, 2015

### hoomanya

Hi,

I'm trying to make sense of a wave propagation problem. It's a 1D problem, modelling propagation of density perturbations which travel like waves in a fluid. The problem is governed by the mass and momentum equations and density is related to pressure using the bulk modulus of the fluid. The variables are density ($\rho$), pressure (p) and velocity (v). Temperature (T) is constant. So the waves are essentially due to an interplay between p and $\rho$. The fluid is water.

I wanted to know whether the isothermal condition makes physical sense. I read somewhere that the isothermal conditions could be explained by the waves having enough time to exchange heat and maintain a constant T. I am guessing the problem is correct but struggling to understand it physically.

2. Nov 17, 2015

It really depends on the strength of the wave. If it is weak enough, then assuming the medium to be isothermal is likely a decent approximation. If the wave is strong, especially if it is strong enough to form a shock, then it is absolutely not likely to be reasonable to assume the fluid is isothermal. How weak is "weak enough" is based on your own needs, as ultimately even small changes in $\rho$ and $p$ will result, most likely, in small changes in $T$.

3. Nov 17, 2015

### hoomanya

Thanks very much. By strong or weak, do you mean in terms of speed of propagation of the wave?
Also does this statement make sense "I read somewhere that the isothermal conditions could be explained by the waves having enough time to exchange heat and maintain a constant T"? Thanks again.

4. Nov 17, 2015

No I mean amplitude. Speed and amplitude may well have a relationship, but amplitude is the one that would be important for the effect on other parameters like temperature.

And yes the idea that heat can be exchanged to maintain a constant temperature makes some sense as far as I can reason. It would require a sufficiently small amplitude, though, such that any temperature disturbance introduced requires only negligible time to diffuse back into the nearby medium.

5. Nov 18, 2015

### nasu

In gases the compressions in sound wave are adiabatic, not isothermal.
Newton assumed isothermal conditions and his calculated speeds of sound were significantly off.

6. Nov 18, 2015