Is it possible to constrain the height, volume flow, or distance of a waterfall from the quantitative analysis of a high-quality recording of its sound?
As an aside, the simulated sounds of fluid splashing or pouring water have been synthesized by computer. The group that did this research was at Cornell University. This seems like an example of a solution to the forward problem. If you listen to the simulation with closed eyes, can you distinguish the waterfall from the running faucet?
Raindrop sizes have been identified by different physical mechanisms associated with the drop splashes and used as a basis for acoustic rain-gauges.
My question is about how to find a (partial) solution for the inverse problem. From sound measurements, how could one differentiate very high falls from modest height ones – say compare Snoqualmie Falls (82m) and Angel Falls (979m.)
The spectrum of waterfall sound is qualitatively described as ‘broadband’ or ‘brown noise’ and I have not yet found a good example of an actual acoustic spectrum for a waterfall.
I think that the total sound energy radiated into the air (and into the ground) would equal the gravitational energy released by the falling water – less a small amount of energy that is turned into heat and warms the water.
It seems to me that the spectra of a 100m versus a 1000m waterfall might have predictable different ratios of low-frequency and high-frequency power. Are there fluid mechanical or acoustic mechanisms that come into play for the higher falls which do not operate at lower heights? Could there be diagnostic sounds related to cavitation?
Would there be too much sound reflection and scattering in a real waterfall basin to make measurements?
My Conclusions
OK, I think I have a satisfactory answer to my question now and I have awarded the bounty to zhermes. I believe his response correctly describes the underlying physics of the problem, and once I got that I was able to find a lot more relevant information and perform some pretty approximate preliminary calculations.
In a nutshell, the important physical process may be the resonant scattering of ambient sound inside the turbulent waterfall by ‘bubble clouds.’ This is referred to as Minnaert resonance and has been extended to describe ‘bubble clouds’ as well as individual air-bubbles. The approach has been used productively to analyze the noise of propeller blades and the sound of ocean waves. I found it a useful way to begin thinking about how the sound of a waterfall may be affected by the height of the falls.
The resonant frequency of a spherical ‘bubble cloud’ is inversely proportional to the radius of the cloud. I also found examples of analyses that showed ~1/f dependency for the power spectrum of this type of noise (as suggested by zhermes too.)
So we might anticipate that the low-frequency cut-off in the power-spectrum of waterfall noise may be determined by the maximum size of the ‘bubble clouds.’
One (obvious) insight is that it seems that it has to be sound being generated in the plunge-pool at the base of the waterfall that contains information about the full height. Sounds originating in the flow 10 meters from the top of a 1000m falls should be no different than for a 10 meter falls. Once the water falls the additional 990m, it has also gained more kinetic energy which would be available to generate a ‘bubble cloud’ having a size dependent upon the waterfall height.
Equating the gravitational energy of water at the top of the falls, the kinetic energy of water when it hits the plunge-pool, and the work to push a jet into the pool, I calculated the maximum depth the jet could penetrate into the pool and took that as the maximum size of the ‘bubble cloud.’ This estimate for the ‘bubble cloud’ size is proportional to the inverse square-root of the waterfall height.
Substituting in standard conditions for water and pressure, this analysis yielded a low-frequency cut-off frequency (which was also the frequency of maximum power) that was far into the infrasound range (<20 Hz) and below the frequency range of human hearing.
The conclusion to draw from these calculations may be that the pitch of high and low waterfalls is not very different in the range of human hearing. Differences might be perceptible for falls having large volumetric flows that are capable of generating lots of power at the limits of hearing. Differences might also be perceived (felt rather than heard) as infrasound.
Perhaps this explains why MP3 recordings of waterfalls, cascades, and streams sound so similar? We may be missing the information coming to us as infrasound.
Best Answer
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.
Basic Picture
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).
The Answer
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 ;)
Additional consideration?
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.
Endnotes
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)?