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I. In this case, the functions are solutions (this is easy to check), for | Solutions to the wave equation for the closed string or the open pipe, shown as functions of x when t=0. Red: n=1; blue: n=2; green: n=3. |

II. In case II, the functions are solutions for | Solutions to the wave equation for the half-open pipe, shown as functions of x when t=0. Red: n=1/2; blue: n=3/2; green: n=5/2. |

For a pipe or string of length *L*, a factor of must be inserted before each of the *x* and *t* arguments.

Frequency. The pitch of the sound produced by the vibrations depends on the frequency, which can be determined from the time-dependent factor . For vibrating columns of air, the *c* in question is the speed of sound, 344 m/sec at sea level. The frequency of the sound corresponding to the *n*th harmonic is then for the open pipe and for the half-open one, which gives 344/2*L* Hz and 344/4*L* Hz as fundamental (lowest) frequencies for the open pipe and the half-open pipe respectively.

In our Song Sparrow record the lowest frequency shown is the "D" at 2325 Hz. This would correspond to an open pipe of length *L*= 7.1 cm, or a half-open pipe of length 3.7cm, kind of a stretch given the bird's length of some 15cm including the tail. The presence of the second harmonic would correspond to the "pipe" being slightly open at the closed end: an open pipe has fundamental frequency twice that of the half-open pipe of the same length. The standard reference for these problems is Crawford H. Greenewalt, *Bird Song: Acoustics and Physiology*, Smithsonian Institution Press, Washington 1968.

The Mourning Dove *Zenaida macrocoura*, common in North America, measures 30cm including a long tail but has a coo at 445 Hz, corresponding to a half-open pipe of length 19.3cm. Where could a vocal organ of that size fit in a bird so small? *Eppure canta.*

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