Question

A tsunami with a 2 m amplitude (A) and 100 km wavelength (L) is approaching an island coast traveling through 1000 m deep water. An oceanographer quickly applies the equation c=sqrt[gL/(2pi)] to figure out the phase speed (c) is 400 m/s. Observations later show that the tsunami arrived 4 times slower than the oceanographer’s predictions. Did the oceanographer do something fundamentally wrong or was this the best theoretical prediction someone could make quickly? Explain your answer in detail. Also, calculate the period (T) of this tsunami.

Answer 1

The oceanographer's theoretical prediction was fundamentally wrong. The equation that is given by the oceanographer is used only to calculate the velocity of waves travelling in high water depth. Tsunami waves entering shallow water near the coastal change its height and wavelength. This is due to the fact that group velocity which carries wave energy changes with water depth, so decrease in transport velocity is compensated by the increase in energy density leads to increase in wave height in order to maintain constant energy flux. This bunching up causes a reduction in wavelength. So velocity decreases as wavelength decreases (frequency remains constant).

For shallow water more appropriate equation to find the velocity of the wave is,

      

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Since time period( or frequency) is independent on the depth of seawater we can use deep water velocity(given by the oceanographer) and wavelength in deep water to get,

Time period,   

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