Handbook for Sound Engineers

Loudspeakers 635

A second difference between real loudspeakers and
our theoretical piston is that practical diaphragms are
very seldom flat. Most often, they are in the shape of a
concave cone, but convex dome shapes are also
employed. In many instances, the shape of the
diaphragm is chosen so as to minimize the effect of
finite-velocity wave propagation in the diaphragm
material on the device’s on-axis response.
Generally speaking, the directivity of real-world
cone or dome transducers is qualitatively similar to that
of a rigid, flat piston. The nonideal behavior of real
transducers can actually create beneficial effects in that
the frequency at which secondary lobes appear can be
higher than the theory predicts.

17.10.2 Direct Radiator Enclosure Design

A woofer is not effective as a freestanding radiator. If it
were to be employed in this fashion, radiation from the
rear of the diaphragm, which is out of phase with that
from the front, would cause cancellation, particularly at
low frequencies. Consequently, woofers are always
housed. Two types of enclosures are widely used: sealed
and vented.

17.10.2.1 Sealed-Box Systems

The low-frequency response of a sealed-box system
may be modeled as a second-order high-pass filter. The
effect of the enclosure is to add stiffness to the woofer
suspension, which will modify the free-air resonant
frequency of the woofer. The contribution made by the
air in the enclosure to the stiffness of the diaphragm is
given by

(17-18)

where
kb is the box effective spring constant,
p 0 is the equilibrium density of air,
c is the speed of sound in air,
SD is the diaphragm area,
VB is the enclosure volume.

The spring constant of the enclosure simply adds to
that of the woofer suspension, so

(17-19)

The air mass that effectively adds to the moving
mass of the diaphragm is given by

(17-20)

where,

S is the surface area of diaphragm,

a is the radius of diaphragm.

Again, this mass is additive, so

. (17-21)

Note the dependence of the enclosure spring

constant and the effective mass on two properties of air:

its equilibrium density and the speed of sound. Both of

these quantities are subject to significant variations with

atmospheric conditions, so the degree of accuracy with

which one can predict the response of an enclo-

sure/transducer combination in actual use is intrinsically

limited.

The resonant frequency of the woofer/enclosure

system is given by

(17-22)

where,

is the angular resonant frequency.

The expression for the low-frequency farfield pres-

sure response of a sealed-box woofer when driven by a

constant-voltage source may be written as

(17-23)

where:

Em is the amplitude of the applied voltage,

Re is the dc resistance of the voice coil,

B is the flux density in the magnet gap,

l is the length of the voice coil conductor in the gap, and

r is the distance from the source to the observation

point.

Qt is given by

(17-24)

where,

Rm is the woofer mechanical resistance (damping).

kb

U 0 c^2 SD^2

VB

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

kc kd+= kB

ma

U 08 Sa

3 S

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

mc md+= ma

Z 0 kc

mc

----- -=

Z 0 = 2 Sf

p

EmBlU 0 Sd

2 SRemcr

-------------------------

• Z
  Z 02

--------

1 jZ

QtZ 0

------------ Z

2

Z 02

–+ --------

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

Qt kcmc

Rm E

2

l

2

Re

----------+

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