How Does Tsunami Wavelength Affect Particle Displacement and Wave Equation?

[snickersnee]

Homework Statement

Wavelength of a tsunami:

Propagates toward shore as a sinusoidal plane wave.
- determine the peak value of the horizontal particle displacement. Is your answer consistent with the assumption that the water momentum is mainly horizontal? Explain by considering the magnitude of the ratio uz/ux (vertical/horizontal components of particle velocity)

- rederive the wave equation in terms of zeta (amplitude) allowing for nonconstant equilibrium depth h_0

Homework Equations

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Wave equation

The Attempt at a Solution

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I thought the horizontal displacement would just be the wavespeed times the period but that would just give the wavelength so that couldn't possibly be right. As for deriving the wave equation, I don't even know where to start on that one. Any help would be appreciated

 

[Greg Bernhardt]

Have you derived wave equation before?