Suppose we release an object and make it fall on the surface of water, then the scatter of water is recorded,

Taking some observations, can we calculate the depth of the water by analyzing the scatter of water pattern?

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    $\begingroup$ Almost certainly not, unless you're willing to drop a sequence of ever-larger objects. For example, the splash a pebble makes is exactly the same whether the lake is 10 feet (sorry, 3 m :0) ) deep or 5 km deep. The range of measureable pressure wave is just not that great. $\endgroup$ – Carl Witthoft Sep 14 '14 at 12:24

Looks like it is possible to some extent: http://iusti.univ-provence.fr/Local/iusti/dir/user-4263/documents_CPI/pdf/2003_CR.pdf (C. R. Mecanique 331 (2003) 61–67 ):

"In 1908,Worthington [1] observed with a high-speed photography technique an odd but frequent phenomenon: milk drop impacts on shallow milk surfaces. His observations show complex dynamics of the crater’s formation due to the drop impingement and droplet emissions all around a thin corona. Different types of drop impacts are described in the literature depending on the type of liquid surface: shallow (less than 10 drop diameters) or deep (more than 50 drop diameters) and other parameters such as the impact drop velocity."

You may wish look at the references in that article.

EDIT (09/14/2014): You may wish to consider another approach as well. An object falling into liquid creates acoustic wave in the liquid. When the echo from the bottom returns to the surface, it should create a pattern on the surface. This pattern can be observed, and the travel time can be measured (say, using the same camera - you can calculate the frames). I would think depth of relatively deep tanks can be measured in this manner. Let me note that ultrasonic level sensors are quite popular (http://en.wikipedia.org/wiki/Level_sensor#Ultrasonic )

  • $\begingroup$ Agreed - there's a correlation so long as the depth is measurable in a small amount of drop-diameters:-) $\endgroup$ – Carl Witthoft Sep 14 '14 at 16:16
  • $\begingroup$ @Carl Witthoft: Please see the edit to my answer. $\endgroup$ – akhmeteli Sep 14 '14 at 17:16
  • $\begingroup$ Good point, but stretches the OP's definition of "measuring water scatter pattern" :-) $\endgroup$ – Carl Witthoft Sep 14 '14 at 18:39
  • $\begingroup$ @Carl Witthoft: Difficult to say - the echo can affect the scatter pattern as well, as scattering takes some time. $\endgroup$ – akhmeteli Sep 15 '14 at 1:04

In my answer, I assume that "scatter" can be generalized to wave motion.

In general, we recognize different kinds of surface waves: gravity waves (main driving force is gravity), and capillary waves (surface tension dominates).

Now gravity waves have a velocity that is a function of both wavelength and depth:

$$v=\sqrt{\frac{g\lambda}{2\pi}\tanh{\frac{2\pi d}{\lambda}}}$$ where
$\lambda$ = wavelength
$d$ = depth
$g$ = gravitational acceleration

So if you can set up an experiment in which "dropping an object" causes waves of different wavelength, you should be able to observe the propagation speed and deduce the depth. This only works if you are able to set up a wave with wavelength similar to the depth, and if this wavelength is long compared to the size of capillary waves (which dominate at wavelengths < 2 cm).


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