The wave equation

2.6. The wave equation#

2.6.1. The classical wave equation#

The classical wave equation in 1D can be written:

The wave equation

(2.18)#\[\frac{\partial^2 \eta}{\partial t^2}=c^2\frac{\partial^2 \eta}{\partial x^2}\]

, where \(\eta\) represents the vertical displacement of the wave surface. In the ocean this would mean the deviation from the mean sea surface heigh.

The general solution to the wave equation (2.18) can be expressed as:

(2.19)#\[\eta(x,t)=f(x+ct)+g(x-ct)\]

The solution represents one wave moving eastward (\(f(x+ct)\)) and one wave moving westward (g(x-ct)). If a surface is initially disturbed, such as from depressing the surface, waves will spread in all directions. If you throw a pebble in a puddle, you will see waves spreading like rings from the center of where you threw the pebble. For the 1D case, you can think of it as a transect cutting across the 2D case, or alternatively as throwing a pebble in a very narrow and long puddle. Here, the waves spread in only 2 directions, as is indicated in the solution to the wave equation (2.19).

2.6.2. Inertia-Gravity waves#

Inertia-gravity waves describes the relation between the surface elevation and velocity. We have a set of coupled equations where one equation describes the change in surface elevtion with time caused by a horizontal velocity shear. Additionally, we have one equation (or 2, if northward velocity \(v\ne0\)) describing the acceleration of water parcels associated with horizontal gradients in surface elevation and coriolis.

Inertial Gravity waves

(2.20)#\[\begin{split}\begin{aligned} \frac{\partial \eta}{\partial t}+H\frac{\partial u}{\partial x}&=0\\ \frac{\partial u}{\partial t}-fv&=-g\frac{\partial \eta}{\partial x}\\ \frac{\partial v}{\partial t}+fu&=0 \end{aligned}\end{split}\]

You may see the resemblance between the two latter equations in (2.20)and the inertial oscillations from (2.8). The only term separating them is the right hand side of (2.20), which is nonzero, and contains a term linking the oscillatory motion to gravity and horizontal pressure differences caused by the surface elevation \(\eta\). Here, the surface elevation is only varying in one dimension, representing a wave moving in the east/west dimension (i.e., \(\frac{\partial \eta}{\partial y}=0\)). To derive the right hand side term from the pressure to the surface elevation term, see section (GFD:Shallow Water).

(2.20) are a set of coupled equations. The velocity and surface elevation terms are connected, and you have to solve the equations jointly.

2.6.3. Shallow-water gravity waves#

If we omit the effect of Earth rotation, the Inertia-gravity waves becomes the shallow-water gravity waves. Here, the advection terms are included for completeness, and the surface elevation \(\eta\) is undulating in both the x and y-direction.

Inertial Gravity waves

(2.21)#\[\begin{split}\begin{aligned} \frac{\partial \eta}{\partial t}+H\frac{\partial u}{\partial x}&=0\\ \frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}&=-g\frac{\partial \eta}{\partial x}\\ \frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v\frac{\partial v}{\partial y}&=g\frac{\partial \eta}{\partial y} \end{aligned}\end{split}\]
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