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*To*: AUDITORY@xxxxxxxxxxxxxxx*Subject*: Tsunamis.*From*: Reinhart Frosch <reinifrosch@xxxxxxxxxx>*Date*: Sat, 1 Jan 2005 16:57:00 +0000*Delivery-date*: Sat Jan 1 12:26:12 2005*Reply-to*: reinifrosch@xxxxxxxxxx*Sender*: AUDITORY Research in Auditory Perception <AUDITORY@xxxxxxxxxxxxxxx>

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

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