Question

Tsunamis A tsunami is an ocean wave often caused by earthquakes on the ocean floor; these waves typically have long wavelengths, ranging from 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 75) 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)?

Accepted Answer

Understand the shallow-water wave velocity equation, which is given by $$v = \sqrt{g d}$$, where $$v is $$the wave speed, $$g is $$the acceleration due to gravity, and $$d is $$the water depth. Recognize that the shallow-water wave approximation applies when the wavelength $$\lambda is $$much greater than the water depth $$d. $$For tsunamis, the wavelength ranges from 150 km to 1000 km, which is much larger than the average ocean depth of 4000 m (or 4 km). Since $$\lambda \gg d$$, the tsunami behaves like a shallow-water wave despite the ocean's large depth, making the shallow-water velocity equation applicable. Use the given depth $$d = 4000 m $$and the known value of $$g \approx 9.8 m/s$^2 to $$set up the tsunami speed calculation using the formula $$v = \sqrt{g d}. $$Express the tsunami speed as $$v = \sqrt{9.8 	imes 4000} m/s$$, which gives the theoretical speed of the tsunami traveling across the Pacific Ocean according to the shallow-water wave model.

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Key Concepts

Shallow-Water Wave Approximation

Shallow-Water Wave Speed Equation

Relationship Between Wavelength, Depth, and Wave Behavior