The mathematics of piano tuning

4. The wave equation and bird song

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

\begin{displaymath}\frac{\partial^2 u}{\partial x^2} = \frac{1}{c^2}\frac{\partial^2 u}{\partial t^2}\end{displaymath}

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 $\pi$, 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 $u=\sin(nx)\cos(nct)$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 $\frac{\textstyle\partial u}{\textstyle\partial x}=0$ 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 $u=\cos(nx)\cos(nct)$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 $\pi/L$ 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 $\cos(\frac{\textstyle\pi}{\textstyle L}nct)$. 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 nth harmonic is then $\frac{\textstyle nc}{\textstyle 2L}$ for the open pipe and $\frac{\textstyle nc}{\textstyle 4L}$ for the half-open one, which gives 344/2L Hz and 344/4L 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.