The development of shallow water equations are seen to be nonlinear at the first step of approximation with the requirement of the parameter epsilon = (n^2)/(m^2), to be sufficiently small, where m = horizontal length and n = vertical length. This restriction of the smallness of epsilon is perhaps not a very severe constraint on the applicability of shallow water equations. And if it is so then the shallow water equations/approximation should be applicable to extremely large water depths as long as the wave is sufficiently long. I wish to know is it a correct argument?
Most certainly - yes. For example - tsunamis are usually analysed using shallow water approximation, although they travel the deep ocean
Mathematically, "shallow" water approximation refers only to the ratio of the transversal and longitudinal scales of patterns under consideration. As long as the longitudinal scale is large compared to the transversal one (which cannot be larger that depth), "shallow" water approximation is accurate.
Absolutely, it is commonly used for tsunami waves (as has been pointed out) as well as for tidally driven waves.
@Bahareh, @Legena Henry, @ Denis, The linearized swallow water approximation is useful if the following two length ratios are small: /mu=kh
Thess equations apply for open water flow, what effects do they have in internal flow conditions like in a pipeline?
Aminu - They are utterly unsuitable for pipe flow. There are standard relations to deal with that which are well developed.
For the shallow water approximation to hold reasonable the aspect ratio of water depth to wavelength should be sufficiently small. D/L
Wavelength-to-depth ratio is the only parameter in the small parameter approximation for the problem of plane waves moving through water with no other boundary conditions. Therefore, under these conditions, yes!
Did u read james lighthill s water waves? Your answer is in there.
Re your second question, yes you can do so using approximations. Have you read svendsens book on near shore hydrodynamics? I think you will enjoy reading it.
Of course, once the wave is extremely large with respect to the depth, the Shallow water equation applies. It does not matter if the depth is also large. What matters is the largeness of the wavelength with respect to the depth. The depth may be large but if the ratio of wavelength to depth is very small, then the Shallow water equations should apply.
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