Introduction
Up to this point, we have focused primarily on the height of tsunamis and how far onshore they can travel. We have paid less attention to the speed of the wave. However, the speed of the wave determines whether people have time to evacuate from the coast. The sooner the wave arrives, the less time there is to move people away! And as we talked about in lecture, some people may move more slowly: children, older people, or people with disabilities. Even fit, young people might have trouble moving quickly if they have to navigate buildings or other infrastructure that collapsed in the earthquake. This means that a crucial element of tsunami preparation is determining how fast tsunami waves move.
We have the scientific understanding of tsunami wave propagation in the ocean to estimate the wave speed and therefore the warning time. As we talked about in class, the speed at which a tsunami travels depends on the water depth. The deeper the water, the faster the tsunami travels:
c=√(gL/2π tanh(2πd/L) )
c: the speed of the tsunami wave (we use the letter “c” for “celerity”)
g: the acceleration due to gravity (9.8 m/s2)
L: the wavelength of the tsunami (typically 100s of kilometers)
π: 3.14159265…
d: the water depth (0-10 km)
tanh: the hyperbolic tangent function
This is a complicated equation! But when the tsunami wavelength L is much larger than the water depth d (specifically, when d≤L⁄10), then the equation becomes much simpler:
c=√gd
We will apply this formula to the observed tsunami in Japan to determine how much warning time they had in 2011, and then investigate the warning time before a potential tsunami in Cascadia.
What happens if you cannot evacuate in time? We know the wave is powerful, but how fast does it move onshore? Maybe you could outrun the water! The physics of this problem is more difficult to solve, and scientists are still working on understanding it. But we can measure how fast the tsunami travels using video footage from Japan and determine whether you could outrun the wave.
As usual, explain your reasoning and show your work to get full credit!
- First, we should double check that the tsunami speed equation actually works. In other words, we need to “calibrate” our tsunami speed calculations. We will do this using observations from the 2011 Japan tsunami. Below is a map of water depth along a line from the coast of Japan to a DART buoy. Remember: DART buoys measure tsunami waves in the ocean by measuring the pressure of the water above them. They record the arrival time of tsunami waves, allowing us to evaluate the speed equation.
We will work through this calibration calculation systematically (you will use the same steps in later questions, so pay attention and take notes!):
1A. What is the distance between the tsunami source region and the DART buoy? Provide the answer in meters.
1B. What is the average depth of the ocean along the path from the tsunami to the DART buoy? Provide the answer in meters.
1C. Multiply the acceleration due to gravity (g) (provided in the Introduction) and the average depth you measured in Part 1B. Double check your units!
1D. Take the square root of your answer. You can do this on your calculator or you can use the graphs of the square root function provided below. This is the tsunami speed in meters/second.
Sample Solution