Question

A tsunami that originated off the Indonesian coast has propagated towards the east-coast of India. It enters the continental shelf at 150 km away from the coast of Chennai. If the average water depth is 80 m from the coast to the continental shelf and 20 minutes is the tsunami period, the time taken by the tsunami to reach the coast of Chennai on entering the continental shelf is .................... hours (round off to two decimal places)

Hint:

For wave problems, the first step is to determine if it's a deep water wave (\(h>\lambda/2\)) or a shallow water wave (\(h<\lambda/20\)). Tsunamis, with their extremely long wavelengths, are always treated as shallow water waves, making their speed calculation simple: \(c=\sqrt{gh}\).

Answer 1

Step 1: Understanding the Concept:
A tsunami is a long-wavelength wave. When the wavelength is much larger than the water depth (\(\lambda >> h\)), it behaves as a "shallow water wave". The speed of a shallow water wave depends only on the depth of the water and the acceleration due to gravity. The period of the wave is irrelevant for calculating its speed.
Step 2: Key Formula or Approach:
1. Calculate the wave propagation speed (celerity), \(c\), using the shallow water wave formula: \(c = \sqrt{gh}\). 2. Calculate the time taken to travel the given distance: \(t = \frac{\text{Distance}}{\text{Speed}}\). 3. Convert the time from seconds to hours.
Step 3: Detailed Explanation or Calculation:
Given values:
Distance, D = 150 km = 150,000 m
Average water depth, h = 80 m
Acceleration due to gravity, g \(\approx\) 9.81 m/s²
1. Calculate the tsunami speed (c): \[ c = \sqrt{gh} = \sqrt{9.81 \text{ m/s}^2 \times 80 \text{ m}} = \sqrt{784.8} \approx 28.014 \text{ m/s} \] 2. Calculate the travel time (t) in seconds: \[ t = \frac{D}{c} = \frac{150000 \text{ m}}{28.014 \text{ m/s}} \approx 5354.4 \text{ s} \] 3. Convert the time to hours: There are 3600 seconds in an hour. \[ t_{\text{hours}} = \frac{5354.4}{3600} \approx 1.4873 \text{ hours} \] Step 4: Final Answer:
Rounding to two decimal places, the time taken by the tsunami is 1.49 hours.
Step 5: Why This is Correct:
The solution correctly identifies a tsunami as a shallow water wave and uses the appropriate formula for its speed. The subsequent calculation of travel time is straightforward. The given wave period is extraneous information not needed for the solution. The result of 1.49 hours falls within the provided answer range.