11.4 Standing Waves

Again, think of waves on a rope as illustrated in Figure 11.13. If a wave is sent along the rope, from the left end when it reaches the other end of the rope it will be reflected and come back along the rope. If another wave is generated at the left end at just the right time it will superpose and reinforce the reflected wave. If we continue to generate new waves like this, all at just the right times, they will each reinforce each other and we will find a wave with large amplitude. This is an example of resonance, a large amplitude motion that occurs only at a particular frequency.

Figure 11.13 The wave is reflected when it reaches the end of the rope. If a new wave is generated at just the right time it can superpose and reinforce this reflected wave leading to a large amplitude wave.

If we wiggle one end of the rope at different frequencies, we will find there are only certain, particular frequencies that give much of a pattern or provide waves with large amplitude. We will find large amplitudes only for the situations sketched in Figure 11.14. Each of these patterns is called a standing wave for the wave does not move along the rope in these cases. We can fit one "loop" or two loops or three loops or any whole number of loops between the ends of our rope. Each of these loops is one half a wavelength. The areas between the loops-where the rope does not move at all-are called "nodes" and the areas of maximum amplitude-where the rope moves the most-are called "antinodes". This is shown in Figure 11.15. When a guitar string is plucked or a violin string is bowed or a piano string is struck, standing waves are produced. This restricts the wavelengths-or the frequencies-that may be produced and gives an instrument its characteristic sound or voice. The same will be true of organ pipes or other musical instruments.

Figure 11.14 Standing waves occur for only certain, particular frequencies. A whole number of "loops" will fit between the ends of a rope when standing waves are present.

Figure 11.15 For standing waves, the area between the loops that does not move is known as a node. The area of maximum amplitude is an antinode.

Figure 11.A Both transverse waves and longitudinal waves may be standing waves.

For standing waves on a string-like standing waves on a rope in a Physics demonstration or standing waves on a guitar string when it is plucked or standing waves on a violin string when it is bowed-this restriction is that a whole number of "loops" fit on the length of the string. Each loop is one-half a wavelength. That means that the a whole number times half a wavelength must equal the length of the string. We can write that as:

(whole number) x (half wavelength) = length of string

or

n (half wavelength) = L

where n is a whole number (or integer) and L is the length of the string.