Reading A2 for Class 03: Introduction to Standing Light Waves

Standing Waves	Summary
Definition of a Standing Wave Creating a Standing Wave Allowed Wavelengths	Suggested Additional Reading

In this, and in all reading assignments, Discussion Questions and Activities are meant to be completed when they are reached in the reading, before continuing. Getting the "correct" answer to Discussion Questions is not important. Instead, the purpose of Discussion Questions is to address the issues, start you thinking about the material, and identify your preconceptions. Completing these assignments before continuing with the reading will aid you greatly in the learning process.

Standing Waves

Discussion Question: Have you ever sent waves down a rope that was fixed at one end by moving the other end up and down? What happened to the crests of the waves as you moved the loose end? Could you produce a set of crests and troughs that didn't travel down the rope but stayed at the same location all the time?

Definition of a Standing Wave

Waves traveling in opposite directions can produce standing waves. A standing wave is a wave that has crests and troughs at fixed points; the amplitude changes in time, but the locations of crests do not. The figure below shows a standing wave at three different times.

Click HERE to see an animation of a standing wave on a string at the Glenbrook High School's Multimedia Physics Studio.

Standing waves can be described by a sine function depending on position, with a time-varying amplitude:

E_T = E_m cos wt sin kx

Exercise: Compare this equation with the graph above to convince yourself that it does indeed describe a standing wave.

Creating a Standing Wave

A wave like this is created when two sine waves of the same frequency, wavelength, and amplitude move in opposite directions. We will consider two electric field waves described by

E₁ = (E_m/2) sin (kx - wt)

E₂ = (E_m/2) sin (kx + wt),

where the first wave is moving toward the right (toward positive x), and the second wave is moving toward the left (toward negative x). The factor of 1/2 is put in for convenience. Adding these two waves gives the total electric field at any point and time:

E_T = E₁ + E₂ = (E_m/2) sin (kx - wt) + (E_m/2) sin (kx + wt) = (E_m/2)[sin (kx - wt) + sin (kx + wt)]

The sum of two sine functions equals a cosine times a sine by trig identities, so we have

E_T = (E_m/2) [2cos 1/2{(kx-wt) - (kx + wt)} sin 1/2{(kx - wt) + (kx - wt)}]

= 2(E_m/2)[cos {1/2( -2wt)} sin {1/2(2kx)}]

E_T = E_m cos wt sin kx

In the boldface expression above, the electric field varies as sin kx with distance, but changing the time does not shift the location of peaks and troughs. It merely changes the amplitude. For example, at x=0, the electric field is always zero, no matter what the value of t. This is not the case for the individual traveling waves that add to produce this standing wave. E₁ at x=0 can take any value, depending on the value of t.

We can show the creation of a standing wave graphically too. In the graphs below, the red line is moving to the right, the green line is moving to the left, and the blue line is the sum, or total electric field. Each successive graph is at a later time. You can see from these graphs how the amplitude of the total electric field changes, but the positions of the crests and troughs (called antinodes) and places of zero field (called nodes) never change. Plotting the three sums on the same graph yields the figure shown at the beginning of this section.

Click HERE to see an animation of this process from Glenbrook High School's Multimedia Physics Studio.

Allowed Wavelengths

One way to create a standing wave is to reflect a sinusoidal signal off of something like a wall (or the end of a guitar string). Then you will have waves traveling in opposite directions. Since the reflected wave will be reflected again at the other end, there will be a continual supply of oscillations to add and create the standing wave.

Two fixed ends

For a standing wave to persist, the wavelength of the standing wave must be related to the length of whatever contains it. For example, a guitar string is fixed at both ends, so the standing wave must have zero amplitude (be at a node) at both ends. That means the length of the string must equal some multiple of a half wavelength, as illustrated below.

The allowed wavelengths are related to the length of the string/cavity by

l = 2L/n,

where n is an integer called the harmonic of the standing wave.

One fixed, one open end

If the wave is instead contained on a string with one end free to move, the free end of the string will correspond to an antinode, or position of maximum oscillation. One way to picture why this happens is to think of the waves created when you move one end of a rope up and down. The waves are never going to be higher than they are when they are created, so no point on the rope will have a displacement larger than the displacement at the end of the rope you are moving. Since the end that can move must be at an antinode, the length of string must equal some odd multiple of a quarter wavelength (see the figure below).

The allowed wavelengths are related to the length of the string/cavity by

l = 4L/n, n odd.

It is important to remember that standing waves when both ends are fixed can have any integer n, while those with only one fixed end can only create odd harmonics.

Standing waves are used in many aspects of information transfer, so it is important to get a good feel for them. In particular, recognize that standing waves of a particular wavelength will only occur if the medium producing them (string, pipe, antenna) is a certain multiple of the wavelength. For a medium with fixed ends, the shortest possible length is one-half wavelength. For a medium with open ends, the shortest possible length is one-fourth wavelength.

Summary

-	Identical waves traveling in opposite directions will produce a standing wave, with fixed nodes that always have zero amplitude.
-	Standing waves with two fixed ends a distance L apart can take wavelengths of l = 2L/n.
-	Standing waves with one fixed end and one open end a distance L apart can take wavelengths of l = 4L/n, where n must be odd.