The (onedimensional) wave equation is the partial differential equation
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 vibrationtype 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
aircolumn 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 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 halfopen 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
timedependent 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 nth harmonic is then
for the open pipe and
for the halfopen one,
which gives
344/2L
Hz and
344/4L
Hz as
fundamental (lowest) frequencies for
the open pipe and the halfopen 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 halfopen
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 halfopen 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 halfopen pipe of length 19.3cm. Where could a vocal organ of that size fit in a bird so small? Eppure canta.