Mathematics 309 - the elementary geometry of wave motionPart III - waves in 2D & 3DNow 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.
![]() 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) 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.
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:
=> |
Similarly, in 3D
At time t = 0, waveheight = Acos(ax + by + cz) where (a,b,c) is related to
wave height = Acos( ax + by + cz -
|