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On a calm smooth lake, or even a large rain puddle, I've seen transient rough patches on the surface suddenly appear and disappear, and sometimes move across the water some distance before disappearing. The amplitude and wavelengths of the wavelets are centimeter scale, and the motion of the water at the surface appears jumbled and random, almost like boiling, but only at the surface. The disturbance does not seem to propagate far outside the rough patch. I am pretty certain these are being driven by small turbulent gusts of wind (which I can sometimes feel pass around me) and that there are no moving submerged objects.

Can wind blowing on smooth water create speckle interference patterns? Is it necessary that the wavelets all be of the same wavelength for a speckle pattern to develop?

ANSWER SUMMARY

My conclusion is that the phenomenon I observed is not caused by 'speckle interference' although some interference of the ripples is clearly taking place as well.

Chris White rightly pointed out that what I described is commonly referred to as capillary waves, so I marked his answer as the correct one. My search for 'capillary waves' on the web turned up this video which is a vigorous example of what I was seeing on my lake.

There is a minimum in the phase-velocity of capillary-gravity waves on a air-water interface at a wavelength of 1.7 cm which probably provides a better physical explanation for the unique qualitative appearance than 'speckle interference.'

Capillary waves are also thought to have a critical role in the initiation of ocean waves.

The complete mathematical theory of capillary-gravity waves does not appear simple. However, I also found this paper describing Capillary Waves Understood by an Elementary Method for anyone interested in a physical explanation.

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Yes, at a wavelength near the "disturbance" size... –  daaxix Jan 30 '13 at 20:57
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I know the phenomenon you are talking about. My guess is that when the wind velocity is high enough, capillary waves develop on the surface. These are short-wavelength phenomena whose restoring force is surface tension (as opposed to gravity). A gust hitting the water will excite such waves in a patch. As long as there is some non-uniformity in the gust, the surface will consist of an interference pattern from waves propagating in different directions, resulting in a specular effect.

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Thanks Chris, 'capillary waves' looked very interesting and pertinent to my question. That's what I was thinking, but could not recall the name. I'll read the analysis you point to and then post a summary of my conclusions in a day or two. –  Mark Rovetta Jan 31 '13 at 17:40
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Yes, at a wavelength near the "disturbance" size...do you mean speckle from reflections off the water or speckle from the turbulent air? Also, when viewing a "rough" (here rough means feature sizes on the order of the wavelength) surface, polychromatic light will result in interference fringe washout.

This can be formalized by the Wiener-Khinchin theorem : $$ \Gamma(\tau) = 4\int_0^\infty \bar{S}(\nu)e^{-2\pi i\tau\nu}d\nu $$ where $\bar{S}(\nu)$ is normalized power spectral density (representing the polychromatic content of the light illuminating the surface), and $\Gamma(\tau)$ is the visibility function, denoted the temporal coherence function. $\Gamma(\tau)$ characterizes how visible the fringes are from interference.

As you can see, the above relationship is a Fourier transform relationship, so as the illumination light becomes more polychromatic (as $\bar{S}(\nu)$ broadens in frequency), then $\Gamma(\tau)$ becomes narrower and narrower. Thus, you will not generally see very good speckle when illuminating with polychromatic light (like a normal filament bulb light or the sun).

If you want to see speckle, you need to illuminate with a laser, or at least an LED. Then, you will see speckle, provided that the roughness is fine enough...


Edit

I just re-read the question and I think you are talking about wave speckle on the water itself! I apologize.

In general speckle is generated by interference of waves (or wavelets). Different wavelengths can interfere, to an extent. They do have to be fairly close in wavelength to interfere, but can have some bandwidth. This is the coherence length, as defined for light above. The coherence length is the point in time (or space for water waves) where two waves of different wavelengths will no longer apparently interfere. In sound, this is what "beat" waves are, interfering waves of slightly different frequencies. The coherence length would be proportional to how the beat waves fall off as different waves are added together with a time difference between them.

So yes, for a visible speckle pattern to develop, the wavelets should be around the same wavelength...

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