acousticsvelocitywater
Conductivity, temperature, depth sensors are used by oceanography to calculate the velocity of sound at varying depths.
These measurment values are used to comprise sound velocity profiles like the one below to increase the presition of deep water sonars.
My question is, how is the actual calculation from conductivity, temperature and depth to sound velocity conducted?
I'm answering my own question.
Apparently this is one of those rare cases when the physicist must doubt what he observed -- or what he thought he observed -- and believe the numbers his theory yielded instead.
From further experiments I've noticed that the ice tends to form thin plates inside the supercooled water once the crystallization process starts -- this form of ice is apparently called dendritic ice. When the starting temperature of the water was about $-10^\circ$C, the resulting ice-water mixture still contained a lot of water by the time the process finished, and most of it was trapped between those thin ice plates. The latter fact would make it hard to measure the mass percentage of water exactly.
I've found some scholarly articles studying this process -- mostly in the context of formation of ice plugs in pipes. In [1] they measured the temperature at a number of points inside a capsule full of supercooled water during ice formation. From the time-dependent temperature profiles in the article it is obvious that my model above (that energy released by the freezing ice heats up all of the water and ice) is completely wrong. The process happens so fast (at a rate of a few cm/s, depending, among other factors, on the temperature), that the heat transfer between the already frozen (thus heated to $0^\circ$C) and still supercooled regions is practically negligible.
However, based on the observation that ice and water appears well mixed in the already frozen region, we can put forth a new model: the released latent heat of fusion is used up locally and quickly in the boundary layer of the expanding frozen region. As a particular region at the boundary freezes, it heats up rapidly to $0^\circ$C (or close to it), and heats up the water surrounding it. Since the ice plates thus formed are relatively close to each other, the resulting region containing ice-water mixture will mostly be free of temperature inequalities, and those inequalities that do exist will be damped quickly. Therefore the thermal profile of a volume of supercooled liquid undergoing freeze-out will consist of two flat regions, with a relatively sharp boundary between.
It would be quite interesting to look at the process with a thermal infrared camera. Such an observation could confirm or reject the model above. To my knowledge, no one published such an observation -- if such a publication exists, I'd be very interested in seeing it. A video made by such a camera would be especially enlightening.
With some simplifying assumptions (spherical container full of supercooled liquid with uniform temperature, and a single nucleation source at the center), the simple model above could be made quantitative, but I haven't done that yet.
1 Juan Jose Milon Guzman, Sergio Leal Braga: Dendritic Ice Growth in Supercooled Water Inside Cylindrical Capsule, 2004
Please note that the following is all conjectural. I only volunteer it due to the lack of other responses after numerous days, the coolness of the question, and the probably lack of people/references who are explicitly experienced with this specific topic.
As a general relation, I'm sure one can correlate the sound-volume with the total energy being dissipated --- but the noise produced is going to be a (virtually) negligible fraction of that total energy (in general, sound caries very little energy1).
To zeroth order, I think it's safe to assume the waterfall produces white-noise, but obviously that needs to be modified to be more accurate (i.e. probably pink/brown to first order). Also, by considering the transition from a small/gradual slope, to an actual waterfall, I can convince myself that there is definitely dependence on the height of the fall in addition to the water-volume2.
How would height effect the spectrum?
Generally power-spectra exhibit high and low energy power-law (like) cutoffs, and I would expect the same thing in this case. In the low-frequency regime, if you start with a smooth flow before the waterfall, there isn't anything to source perturbations larger than the physical-size scale of the waterfall itself. So, I'd expect a low-energy cutoff at a wavelength comparable to the waterfall height. In other words, the taller the waterfall, the lower the rumble.
There also has to be a high energy cutoff, if for no other reason, to avoid an ultraviolet catastrophe/divergence. But physically, what would cause it? Presumably the smallest scale (highest frequency) perturbations come from flow turbulence3, and thus would be determined primarily by the viscosity and dissipation of the fluid4. Generally such a spectrum falls off like the wavenumber (frequency) to the -5/3 power. But note that this high-frequency cutoff wouldn't seem to change from waterfall to waterfall.
Overall, I'm suggesting (read: conjecturing) the following:
E.g.: The following power spectrum (power vs. frequency - both in arbitrary units).
I'm sure information can be obtained from the sound. In particular, estimates of its height/size, flow-rate, and distance5. I'm also sure this would be quite difficult in practice and, for most purposes, just listening and guessing would probably be as accurate as any quantitative analysis ;)
I suppose its possible waterdrop(let)s could source additional sound at scales comparable to their own size. That would be pretty cool, but I have no idea how to estimate/guess if that's important or not. Probably they would only contribute to sound at wavelengths comparable to their size (and thus constrained by the max/min water-drop sizes6...).
Water, especially in a mist/spray, can be very effective at damping sound (which they used to use for the space-shuttle). I'd assume that this would have a significant effect on the resulting sound for heights/flow-volumes at which a mist/spray is produced.
The acoustic properties of the landscape might also be important, i.e. whether the landscape is open (with the waterfall drop-off being like a step-function) or closed (like the drop-off being at the end of a u-shaped valley, etc).
Finally, the additoinally surfaces involved might be important to consider: e.g. rocks, the surface of the waterfall drop-off, sand near the waterfall base, etc etc.
1: Consider how much sound a 60 Watt amp produces, and assume maybe a 10% efficiency (probably optimistic). That's loud, and carrying a small amount of power compared to what a comparable-loudness waterfall is carrying. The vast-majority of waterfall energy will end up as heat, turbulence, and bulk-motion.
2: I'd also guess that height/volume blend after some saturation point (i.e. 1000 m3/min at 20m height is about the same as 500 m3/min at 40m height)... but lets ignore that for now.
3: Turbulence tends to transfer energy from large-scales to small-scales.
See: http://en.wikipedia.org/wiki/Turbulence
4: Figuring out the actual relation for the smallest size-scale of turbulence is both over my head and, I think, outside the scale of this 'answer'. But it involves things like the Kolmogorov spectrum, and associated length scale.
5: Distance could be estimates based on a combination of the spectrum and volume level - to disentangle the degeneracy between sound-volume and distance.
6: Perhaps the minimum droplet size is determined by it behaving ballistically (instead of forming a mist)?
Best Answer
[Posted as answer instead of comment as suggested by David Zaslavsky]
While not able to give all details, I understand it like this: salinity=salinity(conductivity, temperature), density=density(temperature,salinity), stiffness=stiffness(temperature,pressure), and finally $c=\sqrt{\mbox{stiffness}/\mbox{density}}$). In the end, $c$ is a function of conductivity, temperature and pressure (depth).
There is an eq with references at wikipedia.