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

*E*_{T} = *E*_{m}
cos w*t* sin *kx*

*E*_{1} = (*E*_{m}/2)
sin (*kx* - w*t*)

*E*_{2} = (*E*_{m}/2)
sin (*kx* + w*t*),

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} = *E*_{1}
+ *E*_{2} = (*E*_{m}/2)
sin (*kx* - w*t*) + (*E*_{m}/2)
sin (*kx* + w*t*) = (*E*_{m}/2)[sin
(*kx* - w*t*)* *+ sin (*kx*
+ w*t*)]

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*-w*t*) - (*kx*
+ w*t*)} sin 1/2{(*kx* - w*t*)
+ (*kx* - w*t*)}]

= 2(*E*_{m}/2)[cos {1/2( -2w*t*)}
sin {1/2(2*kx*)}]

*E*_{T} = *E*_{m}
cos w*t* 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

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.

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

*l* = 2*L*/*n*,

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

*l* = 4*L*/*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.

- | 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
= l2L/n. |

- | Standing waves with one fixed end and one
open end a distance L apart can take wavelengths of
= l4, where L/nn
must be odd. |

Any introductory physics text, such as *Fundamentals of Physics*
by Halliday, Resnick and Walker.

The entire wave
module from Glenbrook
High School's Physics ClassRoom.

**Copyright
© 2001 Doris Jeanne Wagner. All Rights Reserved.**