# Sound waves in columns of air

• Standing waves in a tube open at both ends are constrained to have specific wavelengths and frequencies
• Standing waves in a tube open at only ONE end are constrained to have slightly different specific wavelengths and frequencies
• Resonance occurs when some periodic force drives a system with a frequency close to the natural frequency of the system
• If two waves with ALMOST, but not quite, the same frequency interfere, the result is a wave with a time-varying amplitude; we call the slow variations in amplitude "beats"
• The beat frequency is simply the difference between the frequencies of the two original waves
• Complex waves are combinations of a fundamental frequency and some of its higher harmonics
• The different shapes of complex waves produced by different musical instruments are detected by our ears as different timbres
• Any complex wave can be expressed as the sum of simple sine and cosine waves; this is called Fourier analysis

Recall that we considered earlier the case of standing waves on a string:

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Today, let's focus on a different type of standing wave: the pressure wave in a column of air.

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Consider the western flute, which is (in very simple terms) a tube open at both ends.

1. The flute in the picture above has a total length of about L = 100 cm. What should the fundamental frequency of a flute be?
2. But flautists can produce much higher frequencies! How? By opening one or more holes in the body of the tube.

3. The flute has one hole at a distance 50 cm from the end of the tube. If that hole were opened, what would the frequency be?
4. But flautists can STILL produce higher frequencies! How? By opening two or more holes.

5. Suppose the flautist opens two holes which are 40 cm and 45 cm from the mouth end of the tube. What frequency would resonate now?
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