# Speed of sound at phase transition — does percolation theory describe it?

Tl;dr: How does the speed of sound vary across a phrase transition? Although a phase transition might be "instantaneous" in terms of temperature, if we instead control Pressure and Volume then we should see a continuous change in speed of sound. How can we approach the physics here?

Discussion:

I was discussing with a friend whether we expected liquid water or ice to have a higher speed of sound, near freezing; we figured it would be water since it was more dense. It turned out to be ice (by a factor of 3), which I guess is due to the much stronger coupling of molecules. Okay.

The only literature I could find talked about speed of sound as a function of temperature, though. If we control the pressure and volume of water instead, we get a mixture; how does the speed of sound appear here? I mean at the macroscale, so that it is approximately uniform (despite the phase mixing).

I'm wondering if this something that can efficiently be predicted by knowing the solid speed and liquid speeds. As we vary the solid concentration by varying the pressure, I think we could model the structure of the solid component with percolation theory. Then if we get arbitrarily large clusters of solid, they would mediate the sound at that speed. If we don't get large clusters, then the sound travels through the solid for parts and alternately travels through liquid. I would expect this model to not work if the extra "noise" in the structure of the liquid, at the small scale, would cause more reflections back and forth and reduce the overall speed.

I would be really curious to see if there's a way percolation theory could be applied here, but maybe not. Maybe something like just a clever way to 'average' the speeds of sound is all that's needed. It is known though that water with air bubbles gets a speed of sound that is higher than either water or air individually (because the compressibility etc. combine in different ways, and these determine the speed of sound). But I couldn't find anything on mixtures involving solids, which is what I think is really the interesting thing here.

• Exactly what are you asking for an explanation of? And why do you think percolation theory might explain it? – sammy gerbil Jul 26 '16 at 0:16
• I'm asking for a reference on the speed of sound being measured in a mixed-phase state. And I think, if we view mixed solid and liquid states as two states random states, then percolation is relevant because it describes the path of phonons through homogeneous phase. A connected component is all solid, and so can all propagate a phonon at the solid's speed locally. Then it reaches the end of the component and has to pass through liquid again. A simpler model (for instance) would be just expecting that it is a linearly weighted mean of the liquid's and solid's speeds of sound, when mixed. – Alex Meiburg Jul 28 '16 at 9:53
• So the question amounts to (1) is there a reference on the speed of sound, and (2) are there good predictive models for this using the pure-phase speeds as inputs? – Alex Meiburg Jul 28 '16 at 9:54