The dynamic surface boundary condition \(\left( \frac{\partial \phi}{\partial t} + \frac{p}{\rho} + gz \right)_{z = \eta} \cong \left( \frac{\partial \phi}{\partial t} \right)_{z=0} + g \eta \cong 0 \text{ or } \left( \frac{\partial \phi}{\partial t} \right)_{z=0} \cong -g \eta\) enforces \( p = 0 \) on the liquid surface
and substitution of \(\eta(x, t) = a \cos[kx - \omega t]\) and \(\phi = \frac{\omega a}{k} \frac{\cosh(k(z + H))}{\sinh(kH)} \sin(kx - \omega t)\) into \(\left( \frac{\partial \phi}{\partial t} + \frac{p}{\rho} + gz \right)_{z = \eta} \cong \left( \frac{\partial \phi}{\partial t} \right)_{z=0} + g \eta \cong 0 \text{ or } \left( \frac{\partial \phi}{\partial t} \right)_{z=0} \cong -g \eta\) produces \[ \left( \frac{\partial \phi}{\partial t} \right)_{z=0} = -\frac{\omega^2}{k} \frac{\cosh(kH)}{\sinh(kH)} \cos(kx - \omega t) \cong -g\eta = -ag \cos(kx - \omega t) \] which simplifies to a relation between \( \omega \) and \( k \), or equivalently, between the wave period \( T \) and the wavelength \( \lambda \): \[ \omega = \sqrt{gk \tanh(kH)} \quad \text{or} \quad T = \sqrt{\frac{2\pi \lambda}{g} \coth\left(\frac{2\pi H}{\lambda}\right)} \] The \(\boxed{\omega = \sqrt{gk \tanh(kH)}}\) specifies how temporal and spatial frequencies of the surface waves are related, and it is known as a dispersion relation. The phase speed \( c \) of these surface waves is \[ c = \frac{\omega}{k} = \sqrt{\frac{g}{k} \tanh(kH)} = \sqrt{\frac{g\lambda}{2\pi} \tanh\left(\frac{2\pi H}{\lambda}\right)} \] This result is of fundamental importance for water waves. It shows that surface waves are dispersive because their propagation speed depends on wave number, with lower \( k \) (longer wavelength) waves traveling faster. A concentrated wave packet made up of many different wavelengths (or frequencies) will not maintain a constant waveform or shape. Instead, it will disperse or spread out as it travels. The longer wavelength components will travel faster than the shorter wavelength ones so that an initial impulse evolves into a wide wave train.