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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.3.76
Chapter 7, Problem 7.3.76

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)?

Verified step by step guidance
1
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 \(v = \sqrt{9.8 \times 4000}\) 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

Here are the essential concepts you must grasp in order to answer the question correctly.

Shallow-Water Wave Approximation

The shallow-water wave approximation applies when the wavelength of a wave is much longer than the water depth. In this case, the wave's speed depends primarily on the depth, not the wavelength. Tsunamis have extremely long wavelengths (hundreds of kilometers), making the ocean depth effectively 'shallow' relative to the wave, so the shallow-water equations are valid.
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Shallow-Water Wave Speed Equation

The shallow-water wave speed is given by the formula v = √(g * d), where g is gravitational acceleration and d is water depth. This equation shows that wave speed increases with depth. For tsunamis in the Pacific Ocean with an average depth of 4000 m, this formula predicts their high propagation speed across the ocean.
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Relationship Between Wavelength, Depth, and Wave Behavior

Wave behavior changes depending on the ratio of wavelength to water depth. When wavelength is much greater than depth, waves behave as shallow-water waves, with speed dependent on depth. Tsunamis’ long wavelengths compared to ocean depth mean they travel as shallow-water waves, despite the ocean being deep.
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