Mathematics 309 - the elementary geometry of wave motion

Part III - waves in 2D & 3D

Now that we know the equation describing waves in 1D, what about waves in several dimensions? To Answer this question lets first take a look at an example in 2D.

http://otrc93.ce.utexas.edu/~waves/wowwv3d.html

Similar to waves in 1D, the general shape is determined by the cosine function. However, to come up with an equation, we also need to determine the level lines/planes. (lines/planes where all points on the line/plane have exactly the same height)

wave height = Acos( ax + by - ct) wave function in 2D (eq 1)

wave height = Acos( ax + by + cz - dt)wave function in 3D(eq 2)

Equation 1 and 2 describe waves in 2D and 3D, respectively. A determines the amplitude of the wave. The (ax + by) / (ax + by + cz) terms inside the brackets are called affine functions and they determine the level lines/planes.

Affine Functions:

Consider in 2D the affine function ax + by + c , and draw the picture of the level lines.

y = (c - ax)/b

b = 0, x = c / a



b not equal to 0, y = -(a/b)x + c

We want a uniform way of describing the level lines of ax + by without separating into cases b equal or not equal to 0.

Set c = 0.

ax + by = 0 <=> (a,b).(x,y) = 0 <=> (a,b) orthogonal to (x,y)

A level line is a curve where a function has a single value. The curve ax + by = 0 is just the line through the origin that is perpendicular to the vector (a,b).

Similarly, we can describe any level line of the form ax + by = c as

a(x - x*) + b(y - y*)= 0

(x*,y*) is the point where vecotr (a,b) intersects the level curve.

note: the signed distance between the line ax + by = c and the line ax + by = 0 is given by the formula c/(a^2 + b^2)^(1/2)

Back to wave equations:

In 2D, at t = 0 wave height = Acos(ax + by)

= wavelength vector ( perpendicular to the level lines)

= signed distance from level 0 to level 2 = a multiple of vector (a,b)

=> || = 2 /(a^2 + b^2)

= || (a,b) = {2 /(a^2 + b^2)}(a,b)

Similarly, in 3D

={2 /(a^2 + b^2 + c^2)}(a,b,c)

At time t = 0, waveheight = Acos(ax + by + cz) where (a,b,c) is related to , the wavelength vector. Now plugging back in the time variable t

wave height = Acos( ax + by + cz - t)

= frequency in radians