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# Standing Waves for AP Physics B & C

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By McGraw-Hill Professional
Updated on Feb 12, 2011

Practice problems for these concepts can be found at:

Waves Practice Problems for AP Physics B & C

If you play a stringed instrument, you are already familiar with standing waves. Any time you pluck a guitar string or bow on a cello string, you are setting up a standing wave. Standing waves show up a bit on the AP exam, so they're worth a closer look.

A standing wave occurs when a string is held in place at both ends. When you move the string up and down at certain precise frequencies, you produce a standing wave. Let's first examine several simple standing waves.

Figure 23.8a shows the simplest standing wave, called the fundamental frequency. The wavelength of this wave, λ1, is twice as long as the distance between the two walls, L.

Notice we labeled our wave with the terms node and antinode.

Figure 23.8b shows the next simplest standing wave.

Here, the wavelength, λ2, is exactly equal to the distance between the walls, L. The next standing wave in our progression is drawn in Figure 23.8c.

Okay, we have a pattern developing here. Notice that the relationship between L, the distance between the walls, and the wavelength of the standing wave is

We can manipulate this equation to say

And we know that the velocity of a wave is v = λf. So we can do some more manipulation of variables and come up with this equation:

This equation says that the frequency of the 1st, 2nd, 3rd, or nth standing wave equals n, multiplied by the velocity of the wave, v, divided by two times the distance between the walls, L.

There are two uses of this equation. One is for stringed instruments—the illustrations on the previous pages could be viewed as a guitar string being plucked—and one is for sound in a pipe open at both ends.

If we have sound waves in a pipe that's closed at one end, the situation looks slightly different. The fundamental frequency is shown in Figure 23.9a.

And the next simplest standing wave is shown in Figure 23.9b.

Notice that we did not have a λ2. When one end of a pipe is closed, we can only have odd values for n. The frequency of the nth standing wave in a closed pipe is

When you have two sound waves of almost, but not quite, equal frequency, you may hear beats.

If you have a couple of tuning forks of similar—but not identical—frequency to play with, or if you have a couple of tone generators at your disposal, you might enjoy generating some beats of your own. They make a wonderful "wa-wa" sound.

Practice problems for these concepts can be found at:

Waves Practice Problems for AP Physics B & C

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