**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 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.**