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

Waves have the form

y = A cos ( k x - w t + d).

The (dimensionless) argument of the cosine function is called the "phase", and is of the form b (x - c t); we therefore see that it describes a wave which translates to the right (for positive c) in time. Let us take a more in-depth look at this wave equation.

A is the amplitude of the function, or light wave.

d is called the "phase angle", and effectively allows us to specify the relative "starting point" of the wave at time zero. By experimenting with various values of d (ie., 0, p / 2, p, 3 p / 2, 2p), we see that we can produce waves which have any given initial value (between - A and A) at time zero

Let us assume that if the amplitude is maximum at x = 0 this has the functional form:

### y(x) = A cos ((2/)x)

Now, if this is moving to the right with speed v it will be described by:

### y(x,t) = A cos (2/(x-vt))

So we see that a simple harmonic wave moving with speed v in the x direction is described as above. A plot is as follows:

With c being the velocity and T as the period, by using c=/T=/2 as mentioned before, and by defining k=2/, we can write this as:

y(x,t) = A cos (kx-t)

F(x-ct) represents a wave of shape F(x) (at time t=0) moving in the direction of increasing x at constant speed c. That is, F(x-ct) represents the same shape as F(x), but with the origin moved to x=ct, ie the same shape as F(x) translated by ct units in the positive x direction.

F(x+ct) represents a wave of shape F(x) (at time t=0) moving in the direction of decreasing x at constant speed c. That is, F(x+ct) represents the same shape as F(x), but with the origin translated to x = -ct.

By: Edmund Lai and Leo Cheng