piano4 ## **The mathematics of piano tuning**

## 4. The wave equation and bird song

The (one-dimensional) wave equation is the partial differential equation

where *u* is an unknown function of *x* and *t*. For example, *u* can be the vertical elongation of a horizontally stretched string, as a function of distance *x* along the string, and of time *t*. Or *u* can represent the pressure at a point inside a column of air, with *x* the distance along the column.

In general any function *f*(*x*+ *ct*) or *f*(*x*- *ct*) satisfies this equation. If we plot *f*(*x*+ *ct*) as a function of *x* for various values of *t*, the graph moves to the left with speed *c* as *t* varies. It can be thought of as a wave moving along the medium (string or column). Similarly *f*(*x*- *ct*)moves to the right. Hence the "wave equation" and the interpretation of *c* as velocity.

The wave equation has vibration-type solutions when it is supplemented by boundary conditions. There are two types that will interest us (to lighten the notation, we will consider a string or a pipe of length , and in the air-column cases, *u*(*x*, *t*) will be the difference between the pressure at (*x*, *t*) and the outside pressure):

I. *u* = 0 at both ends, for all values of *t*. This corresponds to a string with both ends fixed, or a pipe open at both ends. In this case, the functions are solutions (this is easy to check), for *n*=1, 2, 3, .... | 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. *u *= 0 at one end, and at the other. This corresponds to a pipe open at the end where *u*=0 and closed at the other. In case II, the functions are solutions for *n*=1/2, 3/2, 5/2, ... (odd numerators only), assuming that the closed end is at *x*=0. | 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.*