Handbook of Psychology, Volume 4: Experimental Psychology

122 Audition

of sound; follows with a description of the anatomy and
physiology of the auditory system, especially the auditory
periphery; and concludes with a discussion of auditory detec-
tion, discrimination, and segregation.

SOURCES OF SOUND: THE PHYSICS OF THE
COMPLEX SOUND WAVE

Simple Vibrations

An object that vibrates can produce sound if it and the
medium through which sound travels has mass and the prop-
erty of inertia. A simple mass-and-spring model can be used
to describe such a vibrating system, with the spring repre-
senting the property of inertia. When the mass that is attached
to the spring is moved from its starting position and let go,
the mass will oscillate back and forth. A simple sinusoidal
function describes the vibratory oscillation of the mass after
it is set into motion: D(t)=sin[(s/m)t+], where D(t) is
the displacement of the mass as a function of time (t),mis a
measure of mass, and sa measure of the spring forces. In gen-
eral, such a sinusoidal vibration is described by D(t)=
Asin(2 ft+), where fis frequency (f=s/m) and Ais
peak amplitude. Thus, a sinusoidal vibration has three mutu-
ally independent parameters: frequency (f), amplitude (A),
and starting phase (). Figure 5.1 shows two cycles of a
sinusoidal relationship between displacement and time. Fre-
quency and amplitude (also level and intensity) are the phys-
ical parameters of a vibration and sound. Pitch and loudness
are the subjective and perceptual correlates of frequency and
amplitude, and it is often important to keep the physical

descriptions separated from the subjective. Pitch and loud-

ness are discussed later in this chapter.

In addition to describing the vibration of the simple mass-

and-spring model of a vibrating object, sinusoidal vibrations

are the basic building blocks of any vibratory pattern that can

produce sound. That is, any vibration may be defined as the

simple sum of sinusoidal vibrations. This fact is often re-

ferred to as the Fourier sum or integral after Joseph Fourier,

the nineteenth-century French chemist who formulated this

relationship. Thus, it is not surprising that sinusoidal vibra-

tions are the basis of most of what is known about sound and

hearing (Hartmann, 1998).

Frequency is the number of cycles competed in one sec-

ond and is measured in hertz (Hz), in which ncycles per sec-

ond is nHz. Amplitude is a measure of displacement, with A

referring to peak displacement. Starting phase describes the

relative starting value of the sine wave and is measured in

degrees. When a sinusoid completes one cycle, it has gone

through 360 (2 radians) of angular velocity, and a sinusoid

that starts at time zero with an amplitude of zero has a zero-

degree starting phase (= 0

 ). The period (Pr) of a sine wave

is the time it takes to complete one cycle, such that period and

frequency are reciprocally related [F= 1 /Pr,Prin seconds

(sec), or F= 1000 /Pr,Prin milliseconds (msec)]. Thus, in

Figure 5.1, frequency (f) is 500 Hz (Pr=2 msec), peak am-

plitude (A) is 10, and starting phase () is 0o.

Complex Vibrations

Almost all objects vibrate in a complex, nonsinusoidal man-

ner. According to Fourier analysis, however, such complex

vibrations can be described as the sum of sinusoidal vibra-

tions for periodic complex vibrations:

D(t)



n 1

ansin(2 nfot)bncos(2 nfot),

whereanandbnare constants and sinandcosare sinusoidal

functions.

Or as the complex integral for any complex vibration:

f(t)(1 2 )f(w)eiwtdt,

wherew= 2 f,f(t) is a function of time, and f(w) is a func-

tion of frequency.

Anycomplexvibrationcanbedescribedineitherthetimeor

the frequency domain. The time domain description provides

the functional relationship between the amplitude of vibration

and time. The frequency domain description contains the am-

plitude and phase spectra of the vibration. The amplitude spec-

trum relates the amplitude of each frequency component of

Figure 5.1 Two cycles of sinusoidal vibration, with a frequency of 500 Hz,
period (Pr) of 2 ms, peak amplitude (A) of 10 mm, and 0 starting phase.

Time - ms

Amplitude (mm)

 10

 5

0

01234

5

10

A

Pr