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Standing Waves

So far we have been talking of waves that are unconstrained by boundaries, or traveling waves. Standing waves are waves constrained inside a cavity. The main difference is that traveling waves may have any frequency at all, whereas standing waves have only certain allowed frequencies, and there is a lowest frequency.

Think about a wave on an infinite string. There is nothing to constrain your imagination to think of waves of any wavelength. Now think about a wave on a guitar string. The fixed ends now constrain your imagination, so that the longest wavelength you can imagine would look something like that shown in Figure 11-23. How could it be any longer, with the ends forced to be at equilibrium points?

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Figure 11-23


Waves trapped in a cavity may have only certain allowed frequencies. Generally there are an infinite number of allowed frequencies, but there is a lowest possible frequency, a next lowest, and so on. These are standing waves.

A mode of motion in which every part of the medium moves back and forth at the same frequency is called a normal mode. The example of the guitar string will help make this clear.



A guitar string has a wave velocity 285 m/s, and it is 0.65 m long. What is the lowest frequency that can be played on it? (The lowest frequency corresponds to the normal mode with no nodes except at the ends.)



Figure 11-23 shows the normal mode for the lowest frequency.

A full wavelength looks like

..\art 11 jpg\figure 11-tz.jpg

So, Figure 11-23 is half a wavelength. We write

This frequency is called the fundamental.



What is the second lowest frequency that can be excited on the same guitar string?


Figure 11-24 shows the normal mode with one node between the ends. That is, the midpoint of this string experiences destructive interference and thus no motion at all. For this mode the wavelength is λ = 0.65 m. We write 

f = v/λ
= 440 Hz

This frequency is called the second harmonic (the first harmonic being the fundamental, but that is always called the fundamental).

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Figure 11-24



What is the third lowest frequency that can be excited on the guitar string?


Figure 11-25 shows the normal mode with two nodes between the ends.  The wavelength is given by


This is the third harmonic.


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Figure 11-25

When you pluck a guitar string, you hear all these frequencies with varying amplitudes, so the guitar string is not in a normal mode but in a mixed state.

These normal modes are all sine waves on a string. Here are some general rules for drawing normal modes:

  1. It is important to get the ends correct (for example, the ends of a guitar string are nodes).
  2. For the fundamental there are as few nodes as possible, usually none, excluding the end points.
  3. Each following harmonic adds one node (only rarely more).

In this chapter we explored springs and waves. To solve problems involving stationary springs, it is important to draw accurate force diagrams and remember the spring equation Fspring = kx. To solve problems involving moving springs, it is often more important to follow the energy flow. The potential energy of a spring stretched (or compressed) by a distance x is Ep = kx2/2. You should practice doing this sort of problem.

Waves involve a small movement of a medium which propagates to great distances, transporting energy. Often waves have a characteristic frequency f and a wavelength λ, and these are connected by the wave velocity v = . When waves come together they exhibit interference. If they interfere constructively, they create a wave with large amplitude. If they interfere destructively, they create a wave with small amplitude. Whenever you encounter bands of light and dark (light waves) or loud and soft (sound waves), it is likely that interference is part of the explanation as to why the bands formed.

Standing waves are waves trapped in a cavity. They have the distinguishing characteristic that only certain frequencies are allowed: a lowest, and a next lowest, and so on.


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