Tsunamis.

[Reinhart Frosch <reinifrosch@xxxxxxxxxx>]

Many List members will have noticed that tsunamis are similar
to Békésy waves (i.e., to cochlear waves without viable outer hair cells).

Tsunamis are "long-wavelength" or "shallow-water" waves
(wavelength much greater than water depth).

See, e.g., www.geophys.washington.edu/tsunami.

In the "Lagrange approach" to small-displacement liquid-surface waves,
the coordinates x (direction of wave motion) and y (vertical) denote
the no-wave position of the considered water droplet;
y = 0 at ocean floor; y = h at surface.

The correct small-displacement formulae can be found, e.g., in
Equation (6.2.23) of "Physics of Waves" by Elmore and Heald (Dover, 1969):

xi = eta_m [cosh(ky) / sinh(kh)] cos(kx - omega t);
eta = eta_m [sinh(ky) / sinh(kh)] sin(kx - omega t).

Here, xi is the horizontal displacement of the water droplet, and eta is
its
vertical displacement. The water droplets run on elliptical closed trajectories.


For a typical tsunami, on the open 3 km deep ocean, the wavelength lambda

is 100 km and the wave height is 2 eta_m, where
eta_m = vertical amplitude at surface = typically 25 cm.

The phase velocity (velocity of wave crests with respect to ocean floor)
is,
for shallow-water waves, equal to the group velocity (velocity of wavy zone

with respect to ocean floor), namly square-root of (gh);
acceleration g = 9.8 meters per square second;
h = water depth = 3000 m; thus velocity c = 171 m/s = 617 km/h.

(Tsunami travel time from Nicobar Islands to Sri Lanka was about 2.5 hours).

Wave period T = 2 pi / omega = lambda / c = 583 s = 9.7 minutes.
Wave number  k = 2 pi / lambda = 0.0628 per km.

The horizontal half axis of the water-droplet trajectories at the surface
is eta_m / tanh(kh) = 134 cm.

The maximal water-droplet velocity at the surface is only about 1.4 cm/s.

On the open, 3 km deep ocean, one does not notice the tsunami.

The typical width of the wavy zone is about 2 wavelengths,
i.e., in the above example, about 200 km.

At the ocean floor, the water droplets move back and forth;
period = T = 583 s. Amplitude = eta_m / sinh(kh) = 132 cm.
Maximal water-droplet velocity at floor 1.4 cm/s.

Because of the large spatial extension, the total wave energy is
large in spite of the small water-droplet velocity.

On approaching the shore, the phase velocity c [= group velocity
= square root of (gh)] gets small because h gets small.
Similar to Békésy waves, the wave period stays the same
(T = 583 s) but the wavelength and the width of the wavy zone
get smaller. Since the total wave energy is not very strongly
reduced, the half wave height eta_m gets bigger.

Example (approximate numbers, derived from small-displacement
formulae):

Water depth h = 3 m; half wave height eta_m = 2 m;
c = square root of (9.8 times 3) = 5.4 m/s = 20 km/h;
wave length lambda = T times c = 3.2 km;
width of wavy zone = 2 lambda = 6.4 km;
wave number k = 2 pi / lambda = 0.0020 per m;
horizontal half axis of water-droplet ellipse at surface =
eta_m/ tanh (kh) = 340 m;
horizontal amplitude of water droplet trajectory at floor =
eta_m / sinh (kh) = 340 m;
maximal water droplet velocity = 3.6 m/s = 13 km/h.

Reinhart Frosch
(r. Physik-Dept., ETH Zürich.),
CH-5200 Brugg.
reinifrosch@xxxxxxxxxx