Question

The typical speeds of a tsunami wave in water depths of 10 m, 100 m, and 1000 m, respectively, are ................

• A. approximately 100 m s$^{-1}$, 31 m s$^{-1}$ and 10 m s$^{-1}$
• B. approximately 50 m s$^{-1}$, 75 m s$^{-1}$ and 100 m s$^{-1}$
• C. approximately 10 m s$^{-1}$, 31 m s$^{-1}$ and 100 m s$^{-1}$
• D. approximately 100 m s$^{-1}$, 10 m s$^{-1}$ and 31 m s$^{-1}$

Hint:

Tsunami speed depends only on water depth ($c = \sqrt{gh}$). Shallow water = slow, deep ocean = very fast.

Answer 1

The Correct Option is C

Step 1: Recall tsunami wave velocity formula.
\[ c = \sqrt{g h} \] where $c =$ wave speed (m/s), $g = 9.81 \, \text{m/s}^2$, $h =$ water depth (m).
Step 2: Calculate for each depth.
- For $h = 10$ m: \[ c = \sqrt{9.81 \times 10} = \sqrt{98.1} \approx 9.9 \, \text{m/s} \approx 10 \, \text{m/s} \] - For $h = 100$ m: \[ c = \sqrt{9.81 \times 100} = \sqrt{981} \approx 31.3 \, \text{m/s} \approx 31 \, \text{m/s} \] - For $h = 1000$ m: \[ c = \sqrt{9.81 \times 1000} = \sqrt{9810} \approx 99.0 \, \text{m/s} \approx 100 \, \text{m/s} \]
Step 3: Match with options.
This corresponds to option (C): 10, 31, and 100 m/s. Final Answer: \[ \boxed{\text{(C) 10 m/s, 31 m/s, and 100 m/s}} \]