Linear 1-d Gravity Waves

The linear one dimensional gravity wave equations are:

(66)\[\begin{split} \frac{\partial \eta}{\partial t} &+ H\frac{\partial u}{\partial x} = 0\\ \frac{\partial u}{\partial t} &+ g\frac{\partial \eta}{\partial x} = 0 \end{split}\]

where \(t\) is time, \(x\) is the horizontal coordinate, \(\eta\) is the free–surface displacement, \(u\) is the horizontal velocity, \(H\) is the undisturbed fluid depth and \(g\) is the acceleration of gravity.

The equations describe the time evolution of the shape of the free-surface and of the horizontal fluid velocity under the action of gravity, in an one-dimensional domain (unit thickness in the horizontal coordinate \(y\), normal to \(x\)).

The system of equations can be transformed in to a single equation with respect to \(\eta\) by taking the time and space derivatives of the first and second equations, respectively. We thus obtain the following second-order equation for the free–surface elevation:

(67)\[ \frac{\partial^2 \eta}{\partial t^2} = gH\frac{\partial^2 \eta}{\partial x^2} \]

The wave equation (67) has a general solution of the form \(\eta(x,t)=E_1(x-ct)+E_2(x+ct)\), which represents two waves propagating in opposite directions with the same phase speed \(c=\sqrt{gH}\).