Standing Waves, Interference and Group/Phase Velocities

Standing Waves:
When an electromagnetic wave comes into contact with a surface it is reflected.  This phenomenon is analogous to a mechanical wave created in a rope or a sound wave.  The reflected wave travels in the opposite direction with a phase difference.  We can therefore write both waves as..

Notice that the waves are indeed traveling in the opposite direction due to the plus and minus signs in their respective arguments.  Also notice both amplitudes of each wave are the same.  This is because we are modeling the standing wave to have equal amplitudes in both incident and reflective waves.  This means that no energy is absorbed in the reflection process or change in frequency.  If we super impose these two waves, we get... (Using some trigonometry.)

This expression represents the equation for a standing wave where the amplitude of the reflected wave is the same as the incident wave.  It was mentioned earlier that the wave undergoes a phase shift when reflecting off the surface.  It can be shown with Maxwell's Equations that this reflection phase shift is equal to Pi.  Therefore the above equation becomes...

One can interpret this expression as having a spatial dependent amplitude.  Meaning that the amplitude is different at each spot in space.  If we solve this amplitude for when it equals to zero, we can find the spots where the wave always has zero amplitude.  These points are called nodes.

There are also times where the wave is everywhere zero. These waves occur at the times the cosine term is equal to 0.

Similarly we can find the points where the wave is at a maximum and minimum.

If we have two mirrors and have an electromagnetic wave reflecting between them,  In lasers this occurs frequently.  We require the electric field to be equal to zero at each mirror, therefore the space between the mirrors can only hold a certain amount of wave lengths. The distance between the mirrors in terms of the amount of wavelengths between the mirrors is..

Where we have used half the wavelength because the electric field can be zero at a full wave length and a half wavelength.  Solving for wavelength we can find the frequencies of the standing wave modes.

These relations for frequency and wavelengths are strictly for plane mirrors.

Beats:
We now wish to superimpose two waves with the same amplitude, but different frequency.

Where w_p and k_p are the averages of the frequency and proportion constant respectively.  These are much higher than k_g and w_g which occur in the other cosine function.  Therefore one wave is a high frequency wave while the other is a low frequency wave.  The low frequency waves maps out the envelope of their superposition while the high frequency wave traverses in this envelope.  The envelope creates "beats" in the wave which can be used to send information.  The beat frequency is the difference between the two frequency of the two superimposed waves.

Phase and Group Velocity:
The general expression for velocity of a wave is...

The phase velocity is the velocity of the fast frequency wave inside the envelope.  And the group velocity is the velocity of the envelope itself.  Which are..

Where we have used the approximation that w1 is approximately equal to w2 which is approximately equal to w in a small neighborhood.  Same logic for k.  Plugging the phase velocity into the group velocity...

When the velocity does not depend on wavelength such as in a non-dispersive medium.  This would mean the phase and group velocities are zero.  This occurs in vacuum.  In a dispersive medium, v_p=c/n, where n is the refractive index.  The variable n is a function of lambda, meaning a function of k. n(k).