Sound Waves in Ideal Gas
Consider a uniform ideal gas of equilibrium mass density and equilibrium pressure . Let us
investigate the longitudinal oscillations of such a gas. Such oscillations
are known as sound waves. Generally speaking, a sound wave
in an ideal gas
oscillates sufficiently rapidly that heat is unable to flow fast enough to smooth out any
temperature perturbations generated by the wave. Under these circumstances,
the gas obeys the adiabatic gas law (Reif 2008),

(5.31) 
where is the pressure, the volume, and the ratio of specific
heats (i.e., the ratio of the gas's specific heat at constant pressure to its
specific heat at constant volume) (ibid.). This ratio is for ordinary air (Haynes and Lide 2011b).
Consider a sound wave in a column of gas of crosssectional area . Let
measure distance along the column. Suppose that the wave generates an
directed displacement of the column, . Consider a small section of
the column lying between
and
. The change in volume
of the section is
.
Hence, the relative change in volume, which is assumed to be small compared to unity, is

(5.32) 
In the limit
, this becomes

(5.33) 
The pressure perturbation
associated with the volume perturbation
follows
from Equation (5.31), which yields

(5.34) 
or

(5.35) 
giving

(5.36) 
where use has been made of Equation (5.33). Here, we have neglected terms that are second order, or higher,
in the small quantities
and
.
Consider a section of the gas column lying between
and
. The mass of this section is
. The directed
force acting on its left boundary is
, whereas the
directed force acting on its right boundary is
.
Finally, the average longitudinal (i.e., directed) acceleration of the section is
.
Thus, the section's longitudinal equation of motion is written

(5.37) 
In the limit
, this equation reduces to

(5.38) 
Finally, Equation (5.36) yields

(5.39) 
where

(5.40) 
is a constant with the dimensions of velocity, which
turns out to be the sound speed in the gas. (See Section 6.2.)
Figure 5.5:
First three normal modes of an organ pipe with one open end. The plots show contours of the magnitude of the
pressure perturbation associated with each mode. Dark/light contours correspond to high/low magnitude.

As an example, suppose that a standing wave is excited in a uniform organ pipe of length .
Let the closed end of the pipe lie at , and the open end at .
The standing wave satisfies the wave equation (5.39), where represents the
speed of sound in air. The boundary conditions are that
—that is, there is
zero longitudinal displacement of the air at the closed end of the pipe—and
—that is, there is zero pressure perturbation at the open end of the pipe (because the small pressure perturbation in the pipe is not
intense enough to modify the pressure of the atmosphere external to the pipe).
Let us write the displacement pattern associated with the standing wave in the form

(5.41) 
where , , , and are constants.
This expression automatically satisfies the boundary condition
.
The other boundary condition is satisfied provided

(5.42) 
which yields

(5.43) 
where the mode number is a positive integer. Equations (5.39) and (5.41) give the dispersion relation

(5.44) 
Hence, the th normal mode has a wavelength

(5.45) 
and an oscillation frequency (in hertz)

(5.46) 
where
is the frequency of the fundamental harmonic (i.e., the normal
mode with the lowest oscillation frequency). Figure 5.5 illustrates the
characteristic displacement patterns and oscillation frequencies of the pipe's first
three normal modes (i.e., , and 3). In fact, the figure plots the magnitude of the pressure perturbation
associated with each mode.
It can be seen that the modes all
have an antinode in the pressure (which corresponds to a node in the displacement, and vice versa) at the closed end of the pipe, and a node at the open end. The
fundamental harmonic has a wavelength that is four times the length of the pipe.
The first overtone harmonic has a wavelength that is rds the length of the pipe, and a frequency that is three times that of the fundamental. Finally, the second overtone
has a wavelength that is ths the length of the pipe, and a frequency
that is five times that of the fundamental. By contrast, the normal modes
of a guitar string have nodes at either end of the string. (See Figure 4.6.)
Thus, the fundamental harmonic has a wavelength that
is twice the length of the string. The first overtone harmonic has a wavelength that
is the length of the string, and a frequency that is twice that of the fundamental. Finally, the second overtone harmonic has a wavelength that is rds the length of the
string, and a frequency that is three times that of the fundamental.