Meghan Cannon, Brenan DayMathematics 309  the elementary geometry of wave motionPart I  Scaling and ShiftingOne way of obtaining one graph from another is by scaling and shifting either x or y or both, so that the graph y = f(x) becomes the graph y = af(bx + c) + d. The way to see what happens to the graph is to understand what happens in simple steps with just one change at a time.
Only the last one is tricky, since it involves a sequence of substitutions. In order to see which ones, we unravel the process.
Part II  Wave Functions in 1DThe two main types of waves are mechanical and electromagnetic. Mechanical waves are created through the disturbance of a medium, such as water or air, while electromagnetic waves do not need a medium, such as light, radio and xrays, We will focus on light waves and examine some of their major properties. A wave can be visualized by the graph y=cosx. Their basic parameters are velocity, frequency, wavelength and amplitude.
The highest point on the cosine graph, which represents maximum displacement, is called the crest.
 
 
The wave equation As described in Part I, functions can be scaled and shifted to form new graphs. The function y = cosx is widely used to portray a wave and can be altered to depict different wave characteristics.The wave equation is y=Acos[ (2 / ) x  t] To understand how this "wave equation" works, it may be useful to formulate it from the original equation y=cosx.
 
N.B. Some write the equation of the wave using the y=sinx instead of y=cosx which gives the same representation as long as the sine equation is shifted an extra /2 horizontally to the left. Also, the wave number, k, can be substituted into the wave equation. k=2/, thus the wave equation can also be written as y = Acos[kx  t] Part III  Affine Functions in 2DA linear function, F(x), is a function that:
An affine function is a function A(x) such that A(x, y, z) = a L(x, y, z) + b where a and b are constants and x, y, z are vectors.
Multidimensional affine functions: A 2D affine function has the form Ax + By + C. In higher dimensions, the equation simply has more variable terms, i.e. a 3D affine function has equation Ax + By + Cz = D.A level line is a curve where a function equals a constant. For instance, the "curve" Ax + By = 0 has several important properties including:
Similarly, we know the line Ax + By = C is perpendicular to [A B]. One important quality to notice is that this affine function is a multiple of 1 / [ (A^2 + B^2) ^ (1/2)] away from the line Ax + By = 0. In fact, the distance between Ax + By = d and Ax + By = f, where d and f are constants, can be found with the formula:  

Consider, in 2D, the function Ax + By + C .
The lines Ax + By + C = constant are drawn in the graph below:
Now, for Ax + By + C. Set C = 0 . We want a uniform way to describe the level lines of Ax + By without seperating into cases: B equals zero and B doesn't equal zero.
The dot product of [A,B] and [X,Y] must equal zero if [A,B] is perpendicular to [X,Y] .
A level line is a curve where a function has an angle value. The "curve" Ax + By = 0 is a line through the origin perpendicular to [A,B].
The lines ...x + y = 2, x + y = 1, x + y = 0, x + y = 1, x + y = 2... are shown in the graph above. The red distance is the signed distance between x + y = 3 and x + y = 4. The line Ax + By = C is perpendicular to [A,B] and at signed distance from the line Ax + By = 0. The signed distance between two lines Ax + By = C1 and Ax + By = C2 is equal to:
The above graph shows the phases 4 , 2 , 0, 2 and 4 as solid dark lines. At t = 0, the wave height = Acos(ax + by + c) . The wave length vector(in red) is and is perpendicular to the level lines.
= a multiple of (a,b,c)
The signed distance from level 0 to level 2 =   =
= K[a,b,c]
  = K , Where K == [a,b,c]
At time t = 0, wave height = Acos(ax + by + cz) where (a,b,c) is related to , the wave length vector
Where = radian frequency.