Scaling and Shifting | Basic properties of Wave Functions | Wave Equations | Affine Functions | Waves in 2-D and 3-D |

Waves in 2-D and 3-D

Now that we have seen some properties of affine functions, we can understand wave motion in 2-D and 3-D with a little more ease. Similar to the wave equation in 1-D, the wave motion is describe by an oscillating function "cosine", however there are a few more variables involve. I will describe the case of 2 dimension in detail and explain the difference in 3 dimension later on.

Since we are in the 2 dimension space, we have to incorporate one more variable "y" to the wave equation.

f(x,y) = Acos(ax + by)

The wave motion can be represented through level lines with each line being the crest of the wave. With the extra dimension, the wavelength is no long a value, it is a vector. So in order to calculate the wavelength, it is similar to calculating the distance between 2 level lines.

First we note that

= 2 / (a^2 + b^2)^(1/2)
and
is a multiple, k, of [a,b]

So to calculate the wavelength or signed distance from a level line of 0 to 2, we do the following,

|||| = 2 / (a^2 + b^2)^(1/2) = k[a,b]
so k = 2 / (a^2 + b^2)
therefore = (2 / (a^2 + b^2))[a,b]

As for 3-D wave motion, just add another variable z and a constant c to the above equations.

So becomes

= (2 / (a^2 + b^2 + c^2)^(1/2))[a,b,c]

Similar to the wave equation in 1-D, when time = t is also a variable, the height of the wave in 2-D becomes a function of

f(x,y,t) = Acos(ax + by - t) where is the radian frequency

As for wave in 3-D the height can be calculated with the following

f(x,y,z,t) = Acos(ax + by + cz - t)

By: Edmund Lai and Leo Cheng