Question

A small tsunami is moving at a speed of 36 kilometers per hour
in the positive x-direction. The height, from the crest to the trough, of each wave is 4m
(h = 4m). The distance between the first and third crest is 10m (d=10m).
a) Find the wave equation assuming that at time zero the oscillation is at the crest as
depicted in the diagram. Also, assuming that the vertical axis is the y-axis and the
horizontal axis is the x-axis. (10 points)
b) In the diagram above, there are four completes oscillations. Assuming that all four
oscillation strike the coast, transferring all of its energy, how much energy will be
transferred to the coast? The linear mass density of the wave, u, is 1 000 000 kg per
meter (although you can't see it on the diagram the width of the wave is vast)

Answer 1

a) The wave equation for the following diagram is

$$y(x)=\frac{h}{2}\text{ sin}\bigg(\frac{2\pi}{\lambda }x\bigg)=\frac{4}{2}\text{ sin}\bigg(\frac{2\pi}{10/2}x\bigg)=\\ \space\\=2\text{ sin}\bigg(\frac{2\pi}{5}x\bigg).$$

b) Find the angular frequency $\omega$ of the wave:

$$\omega=2\pi f=\frac{2\pi}{T}=\frac{2\pi v}{\lambda}.$$

The total energy of a single wavelength is

$$E_\lambda=\frac{1}{2}\mu A^2\omega^2\lambda=\frac{2\mu}{\lambda}(\pi vA)^2.$$

The energy of four wavelengths:

$$E=4E_\lambda= \frac{8\cdot10^6}{5}\bigg(\pi\cdot\frac{36\cdot10^3}{3600}\cdot2\bigg)^2=6.3\cdot10^9\text{ J}.$$