Part IV - 2D & 3D Wave Equations

The 2D wave equation
We can use 2D affine functions to model 2D waves. We use the affine function as the parameter to the cosine function so that each point in the xy-plane is assigned a corresponding height. The resulting 2D wave function is:

a cos(Ax + By),
where a is the amplitude. It gives a graph like the following:
Figure 4.1

In 2 dimensions, the wavelength λ becomes a vector, as we see in Figure 4.1. It expresses the distance and direction from the crest of one wave to the crest of the next. Note that λ is perpendicular to the level lines of Ax + By.

Calculation of λ
Let's find out what λ is. Since it is perpendicular to the level lines of Ax + By, we know that it will be a multiple of [A, B].

We know from Part III that the signed distance from level line Ax + By = 0 to level line Ax + By = 2Π is

2Π / √(A2 + B2).
This is the length of λ, since it is the distance between crests. To find λ itself, we simply normalize the vector [A, B] to get a unit vector
[A, B] / √(A2 + B2).
This vector has length one and points in the direction of λ, so we simply scale it by the length of λ to get λ:
λ = 2Π [A, B] / (A2 + B2).

Moving with time
To make the graph shift with time we can add a term ωt:

a cos(Ax + By - ωt).

Extending to 3D
Extending the above ideas to 3D is straightforward (although not as easily visualized) and follows the same methods that were used to derive the 2D equations. We get

a cos(Ax + By + Cz),
λ = 2Π [A, B, C] / (A2 + B2 + C2), and
a cos(Ax + By + Cz - ωt).