Tsunami

Classical Mechanics Level 3

A tsunami is a giant wave generated by an earthquake. Consider a tsunami wave far away from its source that has $\lambda=2km$ wavelength in water that is $d=1km$ deep. The amplitude of the wave is $A=1.5m$ . This wave is barely noticable, since the largest slope (the angle relative to the horizontal) is $s= 2 \pi A/\lambda\approx 0.27^{\circ}$ . A ship could easily sail over the wave. As the tsunami approaches the shore, the amplitude increases. Estimate the amplitude of the wave in 10m deep water.

Hint: The speed of the wave is related to the depth of the water by $v=\sqrt{gd}$ , where $g$ is the acceleration of gravity (this is true if the wavelength is longer than the water depth). Similar to other harmonic oscillations, the energy the wave is proportional to the amplitude-squared. The energy density, defined as the energy in a unit area of water, integrated over the depth, is $\epsilon= C \rho g A^2$ , where $\rho$ the density of water and $C$ is a numerical constant of order of 1. The energy flux carried by the water is $\Phi= v \epsilon$ . We may assume that the energy flux is constant, whereas the velocity and the amplitude changes.

Bonus: How long will it take for the wave to pass under the ship in $1km$ deep water?

2m 10m 5m 20m

1 solution

Laszlo Mihaly
Nov 15, 2017

Since the quantity $v \epsilon$ is constant, we can write $\sqrt {gd} A^2 = const$ and therefore $A_1^4d_1=A_2^4d_2$ . This is known as the "Green's law for shallow water waves". The amplitude will be

$A_2=\left(\frac {d_1}{d_2}\right)^{1/4} A_1= 4.74m$

At this height the amplitude is comparable to the water depth, and the wave is very close to crushing. That can be very dangerous to ships. That is why in a tsunami situation ships try to go out to deep water (although they do not have much time to do that, since a wave moves really fast).

The speed of the wave out in the open ocean is $v=\sqrt{gd}= 140 m/s \approx 500km/h \approx 300mph$ and it takes for the full 2km wave about 14 seconds to pass under the ship. Closer to shore, in water depth of 10m, the velocity is 10m/s (36 km/h, about 20 miles per hour). Note that the tsunami wave rarely gets extremely high (like 10-15 m), because of the 1/4 power law in the equation. It can still be very destructive, especially if the conditions (the depth profile of the ocean) causes the wave to focus to a certain area. There is also a huge amount of displaced water that gets pushed to shore - that phenomenon is well beyond the simple wave motion we discussed here.

Tsunami

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