The typical speeds of a tsunami wave in water depths of 10 m, 100 m, and 1000 m, respectively, are ________.
GATE XE - 2023,
Answer: C
The speeds are approximately 10, 31 and 100 m s-1.
A tsunami's wavelength, often hundreds of kilometres, is far greater than the ocean depth, so it travels as a shallow-water wave whose speed depends only on depth:
\[ \begin{aligned} c &= \sqrt{gh} \\ \sqrt{9.81 \times 10} &\approx 9.9\ \text{m/s} \\ \sqrt{9.81 \times 100} &\approx 31.3\ \text{m/s} \\ \sqrt{9.81 \times 1000} &\approx 99\ \text{m/s} \end{aligned} \]
Speed rises with depth, so the options with speeds falling as depth increases, or out of order, are wrong; 50, 75 and 100 m s-1 does not follow the square-root law. This depth dependence is why a tsunami slows and steepens as it reaches shallow coastal water.
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