The fluid velocity components for deep-water waves are \[ u = a\omega e^{kz} \cos(kx - \omega t) \quad \text{and} \quad w = a\omega e^{kz} \sin(kx - \omega t) \] At a fixed spatial location, the velocity vector rotates clockwise (for a wave traveling in the positive \( x \) direction) at frequency \( \omega \), while its magnitude remains constant at \( a\omega e^{kz} \)

For deep-water waves, the perturbation pressure from \(p' = -\rho \frac{\partial \phi}{\partial t} = \rho \frac{a \omega^2}{k} \frac{\cosh(k(z+H))}{\sinh(kH)} \cos(kx - \omega t) = \rho g a \frac{\cosh(k(z+H))}{\cosh(kH)} \cos(kx - \omega t)\) simplifies to \[ p' = \rho g a e^{kz} \cos(kx - \omega t) \] which shows the wave-induced pressure change decays exponentially with depth

A bottom-mounted sensor used to record wave-induced pressure fluctuations will respond as a low-pass filter. Its signal will favor long waves while rejecting short ones

The shallow water limit is obtained by \[ \tanh(x) \approx x \quad \text{as } x \to 0 \] For \( H/\lambda \ll 1 \) \[ \tanh(2\pi H / \lambda) \approx 2\pi H / \lambda \] The phase speed from \(c = \frac{\omega}{k} = \sqrt{\frac{g}{k} \tanh(kH)} = \sqrt{\frac{g \lambda}{2\pi} \tanh\left( \frac{2\pi H}{\lambda} \right)}\) simplifies to \[ c = \sqrt{g H} \] Surface waves are regarded as shallow-water waves only if they are 14 times longer than the water depth, \(c = \sqrt{g H}\) shows that the wave speed increases with water depth, and that it is independent of wavelength, so shallow-water waves are non-dispersive

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