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:

(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 functions:

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.