Tsunami - Mathematical classification

Question

I'm interested in knowing which type of wave a tsunami is, in a strictly mathematical sense.

I'm gonna cite from Wikipedia for this:

As the tsunami approaches the coast and the waters become shallow, wave shoaling compresses the wave and its speed decreases below 80 kilometres per hour (50 mph). Its wavelength diminishes to less than 20 kilometres (12 mi) and its amplitude grows enormously – in accord with Green's law.

and also

Except for the very largest tsunamis, the approaching wave does not break, but rather appears like a fast-moving tidal bore.

While the first property strongly hints at a tsunami being a shallow water wave, the latter fact, namely that they usually don't break would rather suggest for the tsunami being a soliton-solution of a Korteveeg-de-Vries-type equation.
I thought the latter property would not be inherent to shallow water waves although Wikipedia, again seems to argue differently, however only with a picture.

So maybe someone can clarify:

• Which equations govern the surface profile of a tsunami? Has this model been tested against satellite measurments?

Answer 1

Short answer: Tsunami models use the shallow-wave mathematical approach, because their wavelength is usually much larger than the relevant water depth determining their propagation.

Long answer: Tsunamis are often called 'tidal waves' to highlight the idea that their characteristic time response is closer to tydes than to the standard wind waves we are used to see in a beach. The full spectrum of water wave periods is shown in this figure (tsunamis being classified as long-period waves, together with waves caused by meteorological events): Classification of the spectrum of ocean waves according to wave period. Redrawn from Figure 1 in: Walter H. Munk (1950) "Origin and generation of waves". Proceedings 1st International Conference on Coastal Engineering, Long Beach, California. ASCE, pp. 1–4.

This image shows that the wave frequency depends on the cause behind the perturbation of the water surface.

Oceanographers generally treat ocean waves as free surface waves in an ideal fluid [2].

Mathematically, tsunamis are surface gravity waves approached as shallow-water waves (see chapter 4.1.5 in here), meaning that the wavelength λ > 20H (H being the water depth). The involved shallow-water-wave mathematical approximations oppose to the deep-water waves (λ < H, see section 4.1.4 in the same [ref 3]).

Whereas for shallow waves the wave frequency and the wave speed are (ref. 3):

$ω = \sqrt {g k 2H}$ with $k = 2\pi / \lambda$

$c = \sqrt {g H}$

for deep-water waves, instead:

$ω = \sqrt {g k}$

$c = \sqrt {\frac{g \lambda}{2\pi}}$

As the wave approaches the shore and H decreases, the wave velocity decreases and energy is preserved by increasing the wave amplitude.

Detailed information here: 3.