HARMONIC WAVES

I have chosen Harmonic Waves as the title of the first assignment because y = a sin ( b x + c ) represents the most general expression of one non-traveling harmonic wave. Therefore, in the following discussion, all the curves are snapshots of the wave at time t = 0.

A harmonic wave has a sinusoidal shape, as shown in the figure below:

In the following, I use P for pi, and L for lambda.

Before attempting y = a sin ( b x + c ) , let's have a look at y = a sin ( 2Px / L ). When it has this form, the constant a represents the maximum value of the displacement. This is the reason why the constant a is called AMPLITUDE of the wave. The other constant L, called WAVELENGTH, represents the distance between ANY TWO successive maxima.

(1) WHAT HAPPENS WHEN THE CONSTANT b CHANGES?

Let's have a close look at y = a sin ( 2Px / L). When the wave function has this form, we can say that b = 2P / L, and c = 0. Therefore, since b = 2P / L, increasing b is equivalent to decreasing L, namely the wavelength. This can be seen in the following figure:

More to say about this constant b ? Sure! b = 2P / L is called WAVE NUMBER. Let's analyze the figure above then: If b = 1 , then L = 2P is the wavelength. Similarly if b = 2 , then L = P , etc. The following little table summarizes this:

 b 1 2 3 4 b L 2P P 2P / 3 P / 2 2P / b

(2) WHAT HAPPENS WHEN THE CONSTANT a CHANGES?

The constant a, called the AMPLITUDE of the wave, represents the maximum value of the displacement. Therefore, we can deduce that changing a will result in a change of amplitude, namely the maximum value of a harmonic wave. With the help of the following figure, we can see how increasing a results in an increase in the amplitude:

(3) WHAT HAPPENS WHEN THE CONSTANT c CHANGES?

Let's have a close look at y = a sin ( 2Px / L). Suppose that we increase x by L. Then we have y = a sin ( 2P (x+L) / L ) = a sin ( 2Px / L + 2P ) = a sin ( 2Px / L ). This is equivalent to say that the displacement repeats itself when we increase x by one wavelength L. In fact, this is the case when x is increased by ANY integer multiple of L.

The wave function y = a sin ( b x ) is zero at x = 0. But y is not always zero. Therefore we have to use a more general relation for a harmonic wave, namely y = a sin ( b x + c). In this equation, c is called the PHASE CONSTANT. The meaning is in its name. Namely, it is a variable that shows whether different waves are in phase. If they are in phase, then what is the phase constant? The following figure describes phase relation among the wave functi
ons:

From the figure above, we can see such relations among the wave functions. For example, the figure with c = 0 is identical to the one with c = 2P. Similarly the one with c = P / 3 is identical to the one with c = 7P / 3, etc. How about the one with c = 0 and the one with c = P ? The phase difference is P, namely 180 degrees: THEY ARE OUT OF PHASE.