Handbook for Sound Engineers

638 Chapter 17

(17-31)

17.10.3 Horns

Although there are numerous mathematical treatments
of horns in the texts on acoustics, they all suffer from a
common set of inadequacies: the models developed in
the literature account for energy transmission inside the
horn, but there are no closed-form solutions to the
problem of horn directivity—i.e., the behavior of a
horn’s radiation outside the boundaries of the horn
walls, where listeners are located. Modern horn
designers have been far less concerned with optimizing
acoustic loading than with creating desirable directivity
characteristics, and the designs have without exception
been derived empirically rather than analytically.
In an exponential horn, the cross-sectional area is
given by

(17-32)

where,
S 0 is the cross-sectional area at the horn’s throat, or
entry,

mx is called the flare constant.

The radiation impedance of an exponential horn,
assumed to be infinitely long for our purposes, is

. (17-33)

The first term in the brackets is the radiation resistance,
and the second is the radiation reactance. Of interest is
the frequency at which the value of expression inside
the square root becomes zero

(17-34)

or

(17-35)

This is known as the horn cutoff frequency. The
above theory predicts that no sound will be transmitted
through the horn below this frequency. Clearly, this is
not the case with real horns, so the theory contains one
or more assumptions that are not met in practice. Note

also that the second term in the brackets, the radiation

reactance, goes to zero in the high-frequency limit.

17.11 Loudspeaker Testing and Measurement

As with most other devices that transmit or process a

signal containing information, measurement techniques

have been developed specifically for the testing and

evaluation of loudspeakers. Before the early 1980s,

accurate, comprehensive testing of loudspeakers gener-

ally required expensive anechoic chambers or large

outdoor spaces. Since that time, the advent of

computer-based time-windowed measurements has

revolutionized the field of acoustic instrumentation,

particularly as regards the testing of loudspeakers.

17.11.1 Linear Transfer Function

One objective in testing a loudspeaker is to determine

the linear portion of its characteristic transfer function

(or, equivalently, impulse response). The most common

means for acquiring this data is a spectrum analyzer. A

spectrum analyzer applies a signal with known spectral

content to the input of a system and processes the signal

that appears at the output of the device to acquire the

system’s transfer function.

17.11.1.1 Spectrum Analysis Concepts

All spectrum analysis techniques are subject to a set of

general constraints imposed by the mathematical rela-

tionship between time and frequency. It is useful to have

a feeling for these constraints when gathering or evalu-

ating loudspeaker data. Time and frequency are the

mathematical inverses of each other. A signal that has

only one frequency must exist for all time and,

conversely, a signal that exists for a finite amount of

time must contain multiple frequencies. A signal that

exists only within a known time interval—i.e., at all

times before time t 0 the value of the signal is zero and at

all times after time t 1 the value is zero—can only

contain frequencies given by the expression:

(17-36)

where,

N is an integer.

The frequency corresponding to N= 1 gives the best

(lowest) frequency resolution that is possible in a test

conducted for that precise time interval. All other

Q 1 Qm

Re

Zmax

= ©¹§·-----------

SS 0 e

mx

=

Zr

U 0 c

S 0

-------- 1 m

(^2) c 2
4 Z
----------- 2 -– j
mc
2 Z
= + ------ -
Zc mc
2
------ -=
fc mc
4 S
------ -= fN
1
t 1 – t 0
= -------------------