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Chapter 15

Q. 15.5

Most of the waves we observe are generated by the action of the wind over the surface of the water. Tsunamis are waves generated by seismic activity such as earthquakes and volcanoes, or by catastrophic events such as asteroids impacting in the ocean. A tsunami would seem quite harmless if observed on the open ocean, where its amplitude might be as small as 10 cm. However these waves transmit a tremendous amount of energy. What is the wave speed of a tsunami traveling across the Pacific, where a typical depth is 4000 m? Compare this with the speed of a wave of the same amplitude in 1 m of water.

Step-by-Step

Verified Solution

The wave is sketched in Figure 15.21A. Clearly the small amplitude approximation, Eq. 15.20, is appropriate for the tsunami in the open ocean. Inserting the data, we have

c=\sqrt{gy}=\sqrt{(9.81\ m/s^2)(4000\ m)}=198\ m/s

which is over 400 mph! When this wave approaches a coastline, it will slow down because the water is shallower, but its amplitude will grow because the energy flux, which is a function of the speed and amplitude, is constant. Tsunamis are very destructive because their amplitude can easily exceed several meters. Figure 15.21B shows the effect of a tsunami that struck Hawaii in 1960.

For a 10 cm amplitude wave in 1 m deep water, we can calculate the wave speed from

c=\sqrt{gy\left[1+\frac{\Delta y}{2 y} \right]\left[1+\frac{\Delta y}{y} \right] } =\sqrt{(9.81\ m/s^2)(1\ m)\left[1+\frac{0.1\ m}{2(1\ m)} \right]\left[1+\frac{0.1\ m}{1\ m} \right]}

= 3.37 m/s

Using the small amplitude approximation c=\sqrt{gy}=\sqrt{(9.81\ m/s^2)(1\ m)}  for this wave yields a wave speed of 3.13 m/s, which is in error by only about 7%.

15.21