Handbook for Sound Engineers

Loudspeakers 633

17.10.1 Acoustics of Radiators

An understanding of direct sound radiation from a
piston in space, a baffle, or a box can be approached by
analyzing two distinct but directly related quantities,
radiation resistance and directivity. Radiation resistance
is the measurement of the capacity of an acoustic radi-
ator to convert vibratory motion into sound energy. It is
the ratio of pressure to the volume velocity due to the
piston’s motion. At high frequencies, all pistons have
the same capacity per unit surface area to produce
acoustic power. However, as the size of the wavelength
of sound being produced approaches the size of the
piston, the radiation resistance decreases as the square
of frequency, i.e., at approximately 12 dB/octave.

17.10.1.1 Piston in an Infinite Baffle

A piston in a wall of infinite extent (half space) is the
model most commonly employed to develop predictive
equations. Even though this model is not representative
of the majority of actual loudspeakers, its simplicity and
mathematical manageability make it useful for instruc-
tional and comparative purposes.

A piston in an infinite baffle will see an acoustic
load that depends on its size relative to a wavelength of
sound at the frequency of interest. The radiation resis-
tance, which is the part of the acoustic impedance that
accounts for transmission of sound energy, is given by

(17-11)

where,

p 0 is the equilibrium density of air,

c is the velocity of sound in air,

a is the radius of the piston,

k is the wave number, 2Sf/c,

is the surface area of the piston,

R 1 [2ka] is the piston resistance function, given by

(17-12)

The value of the piston resistance function

approaches unity for values of 2ka above 6. For

example, in the case of a piston with an effective radius

of 6 inches, the radiation resistance will be approxi-

mately constant above 1100 Hz.

The acoustic power radiated by a flat piston is given

by

(17-13)

where,

U 0 is the amplitude of the piston’s velocity.

Two regimes of interest may be derived from the

above equation. If we first consider 2ka < 1—i.e., a

small piston and/or low frequency—we can neglect the

higher-order terms in the expression for the piston resis-

tance function

(17-14)

and the power radiated by a flat piston becomes

(17-15)

Note that, for constant velocity amplitude, the

acoustic power rises as the square of the frequency.

Clearly, there must be a compensating mechanism, as a

typical cone transducer has relatively flat amplitude

Figure 17-57. Three-dimensional representation of octave
band isobar. Courtesy Frazier Loudspeakers.

Rf=U 0 cSa^2 >@R 12 ka

=U 0 cS R>@ 12 ka

S=Sa^2

R 1 x x

2

24 x

----------- x

4

24

2

x x 6

---------------------- x

6

24

2

x 6

2

x x 8

= – +----------------------------------–}

W

RrU 02

2

---------------=

U 0

2

U 0 cSa

2

R 1 >@ 2 ka

2

=------------------------------------------------

=U 02 U 0 cSR 1 >@ 2 ka

R 1 x x

2

8

|---- -

W

U 0 ck^2

4 S

=-------------- S^2 U 02