Solitary Wave Calculator

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A solitary wave is a shallow water wave that consists of a single displacement of water above the mean water level. John Scott Russell (1808-1882) was the first to report on the wave, caused by the stopping of a barge in a canal. It has been used by Munk as a model for waves in the surf zone. It also is a reasonable description for the leading wave of a tsunami.
The wave form of a solitary wave is given as a function of distance x and time t by

where H is the maximum wave height (which occurs at x=0 when t=0), C is the wave celerity (speed) and sech ( ) is the hyperbolic secant.
The parameter
k

is defined by . To use the applet, you need to enter the maximum wave height H and the water depth, h.

The length of a solitary wave is theoretically infinite. However, for practical purposes, the water surface elevation (in the first equation) decays to zero reasonably fast with x. So, in much the same way as for other types of waves, we can (arbitrarily) define a wave length, L, as

At a distance of x = L/2 away from the crest, the water surface displacement is reduced to 0.74% of its maximum value. This is the definition used in the applet.

Alternatively, the wave length can be defined such that, at x =L/2, h/H = 0.01, say. This gives

. This wavelength is 95.3% (from ) as long as the previous definition.

The speed of a solitary wave is . Thus an apparent wave period could be defined as T=L/C. The applet also gives the surface particle velocity at the wave crest.

Question: How does the height of the wave affect its length?