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

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