A tsunami is an ocean wave often caused by earthquakes on the ocean floor; these waves typically have long wavelengths, ranging between 150 to 1000 km. Imagine a tsunami traveling across the Pacific Ocean, which is the deepest ocean in the world, with an average depth of about 4000 m. Explain why the shallow-water velocity equation (Exercise 71) applies to tsunamis even though the actual depth of the water is large. What does the shallow-water equation say about the speed of a tsunami in the Pacific Ocean (use d = 4000 m)?

Data from Exercise 71

a. Confirm that the linear approximation to f(x) = tanh x at a = 0 is L(x) = x.

b. Recall that the velocity of a surface wave on the ocean is In fluid dynamics, shallow water refers to water where the depth-to-wavelength ratio d/λ < 0.05. Use your answer to part (a) to explain why the shallow water velocity equation is v = √gd.

c. Use the shallow-water velocity equation to explain why waves tend to slow down as they approach the shore.

Transcribed Image Text:

gл tanh 2тd У— 2т λ