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The Wave Equation\vs

The wave equation is the partial differential equation

$$\frac{\partial^2 u}{\partial x^2} = \frac{1}{c^2}\frac{\partial^2 u}{\partial t^2}$$
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.\vs

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.\vs

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):\vs



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.\vs

In this case, the functions $u=\sin(nx)\cos(nct)$
are solutions (this is 
easy to check), for $n=1, 2, 3, ...$. 
\vs

II. $u=0$ at one end, and $\frac{\textstyle\partial u}{\textstyle\partial x}=0$ at
the other. This corresponds to pipe open at
the end where $u=0$ and closed at the other.\vs



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 $x=0$. 
\vs

For a pipe or string of length $L$, a factor of $\pi/L$ must be inserted before
each of the $x$ and $t$ arguments. \vs

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,
approximately 330 m/sec at sea level. The frequency of the
sound corresponding to the $n$th 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 $\frac{\textstyle 330}{\textstyle 2L}$ Hz and $\frac{\textstyle 330}{\textstyle 4L}$ Hz as
fundamental (lowest) frequencies for
the open pipe and the half-open pipe respectively. \vs

In the Song Sparrow 
on page 1,
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 3.5cm, the latter not completely implausible given 
the bird's length of some 15cm including the tail. The presence of the
second harmonic can be explained by the ``pipe'' being slightly
open at the closed end: an open pipe of length 3.5cm would have
fundamental frequency 4714 Hz.\vs


On the other hand the Mourning Dove,
common in North America, measures 30cm including a long tail
but has a note at 445 Hz, corresponding to a half-open pipe of
length 18.5cm. Clearly some other kind of physics is involved.

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<i>Zenaida macrocoura</i>,