Light, it was discovered in the last century, is an example of a larger phenomenon wave motion. We can see other, clearer, examples of wave motion in the world around us.

A rock thrown into a quiet pool disturbs the equilibrium of its surface and sends out ripples.

A stretched string, plucked at one end, transmits a pulse down its entire length.

Within a noiseless room, the once stationary air ripples from the shock of a hand-clap.

In general, any elastic medium (rubber, Jell-O, a water surface, a block of steel, air, etc.), when stretched from its equilibrium position and released, will ripple in response, sending the initial energy away from the point of disturbance. Notice that it is the energy, not the material itself that travels. If the waves are repeated, as in a sine wave, it is possible to identify a wavelength:

Note that wavelength is denoted by the Greek letter £, pronounced "lambda".

1. From the sketches and dimensions below, find the length of one wavelength.

a)	b)
c)	d)
a) The wave train is 5£ long, so: 5£ = 35 cm £ = 7.0 cm	b) 3£ = 1.7 m £ = .567 m
c) 2£ = 11.2 ft £ = 5.6 ft note: wave begins and ends in center	d) 3.5£ = 54 cm £ = 15.4 cm

2. Again, find the length of a single wavelength.

a)	b)
c)	d)
a) 1.5£ = 3.8 m £ = 2.53 m	b) 3.25£ = 41 cm £ = 12.6 cm
c) (1/2)£ = 1.8 m £ = 3.6 m	d) 2.25£ = 22.5 cm £ = 10.0 cm

3. Given the wavelength in each case, find the length of each segment shown.

a)	b)	c)
a) L = 4.25£ L = (4.5)(12) L = 54.0 cm	b) L = 1.25£ L = (1.25)(6.7) L = 8.38 ft	c) L = 2.75£ L = (2.75)(4.07) L = 11.2 m

When a continuous train of waves is sent down a string, it will be reflected at the end. The returning waves will combine with the initial waves to produce patterns of standing waves. Here are some standing waves that can be set up with a string tied down at both ends. Try to think of the pictures as frames of a movie.

Initial position:		Animation
Later:
Still later:
Later yet:

This is the kind of wave that is normally set up in a guitar string when it is plucked, or a violin string that is bowed. By changing the initial condition, other standing wave patterns are possible. (Think of how you can set up standing waves by wiggling one end of a rope whose other end is tied to a tree.)

Two crest (second mode)
Three crest (third mode)
Four crest (fourth mode)

These are called the second, third, and fourth modes of vibration in the string. In music they are called the second, third and fourth harmonics, OR the first, second and third overtones. (Note that the second harmonic is the first overtone.) Higher harmonics are, of course, possible.

It is easier to visualize standing waves than it is to visualize traveling waves. One need only show the two extreme positions of the vibrating medium.

4. What is the wavelength of each of these?

a)	b)	c)
a) £ = 2.6 m	b) 2.5£ = 71.3 cm £ = 28.5 cm	c) 3.5£ = .633 ft £ = .181 ft

5. A stretched string vibrates in its 3rd mode with a wavelength of 1.2 meters. What is the length of the string?

1st mode: In the third mode: L = 1.5(1.2) L = 1.8 m 2nd mode: 3rd mode:

It turns out that there are other kinds of standing waves. Consider what happens in a bathtub. If you gently slosh back and forth in a half-filled tub, the water will rise and fall like this:

Animation	We picture it like this

Higher modes look like this:

6. The third mode in a bathtub has a wavelength of 1.2 m. What is the length of the tub?

L = (3/2)£ L = (3/2)(1.2) L = 1.8 m

The difference between the string and the waver is that the string is tied down at the end while the water is left free to vibrate. This turns out to be an important difference. Look at what happens when a string is tied down, and when it is free to move:

Now look at what happens when the string is not tied down:

Note how the tied string reflects with the phase of the wave reversed, while the loose end reflects an unreversed wave.

It's also possible to have situations where one end of the medium is restricted while the other is free to move.

First mode	Second mode	Third mode

7. A string, 1.8 m long, is held at one end and free to vibrate at the other. What are the wavelengths of its second and fourth mode?

1st: (1/2)£1 2nd: (3/4)£2 3rd: (1.25)£3 4th: (1.75)£4 2nd mode: (3/4)£2 = 1.8 m £2 = 2.4 m 4th mode: (1.75)£4 = 1.8 m £4 = 1.03 m