Problem 72 - Calculus Early Transcendentals

Question

A tsunami is an ocean wave often caused by earthquakes on the ocean floor; these waves typically have long wavelengths, ranging between 150 to \(1000 \mathrm{km}\). 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 \(d=4000 \mathrm{m}) ?\)

Step-by-Step Solution

Answer

Answer: The shallow-water velocity equation can be applied to tsunamis because they have long wavelengths (150 to 1000 km), which are much larger than the water depth. The wave behaves as if it is in shallow water when its wavelength is significantly larger than the water depth. As a result, we can apply the shallow-water velocity equation for tsunamis despite the large water depth.

Step 1: Justification for applying the shallow-water velocity equation to tsunamis

Tsunamis are characterized by their long wavelengths, which range between 150 to 1000 km. When the wavelength of a wave is much larger than the water depth, the wave behaves as if it is in shallow water. In the case of the Pacific Ocean with an average depth of 4000 m, the wavelengths of tsunamis are significantly larger compared to the water depth. Therefore, we can apply the shallow-water velocity equation for tsunamis despite the large water depth.

Step 2: Calculate the speed of the tsunami using the shallow-water velocity equation and given depth

Now, let's use the shallow-water velocity equation to determine the speed of a tsunami in the Pacific Ocean: \(v = \sqrt{gd}\) Here, \(g\) (acceleration due to gravity) is approximately \(9.81\,\mathrm{m/s^2}\), and the given depth \(d\) is 4000 meters. \(v = \sqrt{(9.81\,\mathrm{m/s^2})(4000\,\mathrm{m})}\) \(v \approx 198\,\mathrm{m/s}\) According to the shallow-water velocity equation, the speed of a tsunami in the Pacific Ocean is approximately 198 meters per second.

Key Concepts

Tsunami

A tsunami is a powerful and large ocean wave typically triggered by underwater disturbances such as earthquakes, volcanic eruptions, or landslides. Unlike regular waves that are generated by wind, tsunamis travel at high speeds across vast ocean distances. What sets tsunamis apart is:

• Long wavelengths, ranging from 150 to 1000 km, much longer than typical ocean waves.
• Immense energy, capable of traveling across entire ocean basins without losing power.
• Ability to cause significant destruction upon reaching shallow coastal areas.

These characteristics enable tsunamis to move as fast as jet planes, making early detection and warning systems essential for reducing the risks they pose.

Wavelength

The wavelength of a tsunami, the distance between two successive wave crests, is key to understanding its behavior. Tsunamis have exceptionally long wavelengths compared to regular ocean waves. This factor influences:

• Their speed and energy, allowing them to travel long distances efficiently.
• Their ability to behave as shallow-water waves despite deep ocean environments.

In essence, a tsunami's wavelength determines how it propagates across the ocean, behaving as if the ocean is shallow, because the wavelength is significantly greater than the ocean's depth.

Ocean Depth

While the Pacific Ocean is the deepest in the world, with an average depth of roughly 4000 meters, tsunamis can still be treated as shallow-water waves. This is because:

• Their long wavelength makes the depth seem small in comparison.
• Shallow-water equations become applicable, simplifying calculations of speed.

Understanding the relationship between ocean depth and wavelength is essential to predict a tsunami's behavior and impact effectively.

Velocity Equation

To calculate the speed of a tsunami, the shallow-water velocity equation is used: \[ v = \sqrt{gd} \] where \( g \) is the acceleration due to gravity (approximately \(9.81\,\mathrm{m/s^2}\)), and \( d \) is the ocean depth. This equation provides a simple yet accurate way to estimate the tsunami's speed. For instance, in the Pacific Ocean with a depth of 4000 meters, the calculated speed is approximately 198 meters per second.

• This equation is crucial for predicting how quickly a tsunami can reach coastlines.
• It helps in formulating warnings and evacuation plans.

By understanding this equation, one can grasp the basics of how tsunamis move through the ocean much like shallow waves.