Sources of Sound: The Physics of the Complex Sound Wave 123Figure 5.2 Diagram of what one might see if air molecules were pho-
tographed as a sound source vibrated. The rarefaction and condensation are
shown, as well as the direction (grey arrows above the source) in which the
molecules were moving at the instant the picture was taken. The wave moves
out in circular manner (actually as a sphere in the three-dimensional real
world). As the wave moves out from the source it occupies a greater area,
and thus the density of molecules at rarefactions and condensations lessens.
The area around the border of the figure represents the static air motion
before the propagated wave reaches this area. Source: Adapted from
Yost (2000).Static
Air PressureRarefaction Source Condensationthe complex vibration to its frequency. The phase spectrum
provides the starting phases of each frequency component.
That is, a complex vibration is the sum of sinusoidal vibrations.
The amplitude spectrum describes the amplitudes of each sinu-
soid and the phase spectrum the starting phase of each
sinusoidal component. When the instantaneous amplitudes of
each sinusoidal component of the complex vibration are added
point for point in time, the time domain description is deter-
mined. The time domain and the frequency domain descrip-
tions of complex vibrations are transforms of each other, with
each completely describing the vibration. Simple vibrations
are sinusoidal vibrations and complex vibrations are the sum of
simple or sinusoidal vibrations.
Several different complex signals are described in this
chapter. Transient (click) signals are brief (usually less then
1 msec) signals that come on suddenly, stay on at a fixed
level, and then go off suddenly. Transients have very broad
amplitude spectra, with most of the spectral energy lying in
the spectral region less than 1/T, where Tis the duration of
the transient expressed in seconds (thus, 1/Thas the units of
frequency). Noise stimuli have randomly varying instanta-
neous amplitudes and contain all frequencies (within a cer-
tain range). If the instantaneous amplitudes vary according to
the normal (Gaussian) distribution, the noise is Gaussian
noise. If the average level of each frequency component in
the noise is the same, the noise is white noise. Noises can be
generated (filtered) to be narrow band, such that a narrow-
band noise contains frequency components in a limited fre-
quency range (the bandwidth of the noise). The amplitudes or
frequencies of a signal can vary as a function of time. For
instance, a sinusoidal signal can have its amplitude modu-
lated:A(t) sin(2 ft); or it can have its frequency modulated:
Asin(2 F(t)t), where A(t) is the amplitude-modulation pat-
tern and F(t) is the frequency-modulation pattern. In general,
any signal [x(t)] can be amplitude modulated: A(t)x(t). In this
case,A(t) is often referred to as the signal envelope and x(t)
as the signal fine structure. Such amplitude- and frequency-
modulated sounds are common in nature.
Sound Propagation
Objects vibrate and the effects of this vibration travel through
the medium (e.g., air) as a sound wave that eventually reaches
the ears of a listener. Air consists of molecules in constant
random motion. When an object vibrates in air, it causes the
air molecules to move in the direction of the vibrating ob-
ject’s outward and inward movements. An outward motion
causes the air molecules to propagate from the source and to
condense into areas of condensation where the density of
molecules is greater than the average density of air molecules
in the object’s surrounding environment. Thus, at a conden-
sation, the air pressure is greater than the average static air
pressure, because pressure is proportional to the density of
molecules. When the object moves inward, rarefaction areas
of lower density are produced, generating lower pressure.
These areas of condensation and rarefaction propagate away
from the source in a spherical manner as the object continues
to vibrate. Figure 5.2 is a schematic depiction of these areas
of condensation and rarefaction at one instant in time. Even-
tually, the pressure wave of alternating areas of condensations
and rarefactions cause the eardrum (tympanic membrane) to
vibrate, and the process of hearing begins.
The distance between successive condensations (or suc-
cessive rarefactions) is the wavelength () of sound. Wave-
length is proportional to the speed of sound in the medium (c)
and inversely proportional to frequency (f): =c/f. The
pressure of the sound wave decreases as a function of the
square of the distance from the source, and this relationship is
called the inverse square law of sound propagation.