Handbook for Sound Engineers

634 Chapter 17

response over this range of frequencies. This mecha-
nism is the mechanical impedance due to the moving
mass of the piston, which rises with the square of
frequency. Therefore, a piston excited by a force that
does not vary with frequency responds with a velocity
that falls off as the square of frequency. But, in the low-
frequency region, the acoustic impedance rises with the
square of frequency, so the two effects effectively
cancel each other over a significant range of frequen-
cies. It is this serendipitous balance between key
mechanical and acoustic parameters that makes the cone
transducer an effective acoustic radiator.
The second regime is the region for which ka >>1
(high frequencies and/or a large piston). In this case,
because the piston resistance function approaches unity,
we get

(17-16)

Note that there is no frequency dependency in the
above expression: the radiation resistance of a piston
approaches a constant at high frequencies. Given
velocity amplitude that falls off as the square of
frequency, it is clear that, in the high-frequency regime,
the acoustic power radiated by a typical transducer
could be expected to decrease as the fourth power of the
frequency. Note also that, in the low-frequency limit,
for constant velocity amplitude, radiated power goes as
the square of the surface area of the piston. In the high-
frequency limit, however, it goes as only the first power
of the area. Therefore, all else being equal, increasing
the size of the piston has a greater effect on its low-
frequency output than on its capacity to radiate higher
frequencies.

17.10.1.2 Piston Directivity

So far, we have examined expressions for the total
power radiated by a piston. If a piston radiated identi-
cally in all directions, no further acoustic information
would be needed. Since this is not the case, it is also
worthwhile to consider the nature of this directivity.
The mathematical technique for deriving the piston
directivity function is to consider the piston as being
made up of infinitesimal differential elements, each of
which contributes to the observed radiation at a point in
space. These individual contributions are combined via
integration to yield a value for each specific point in
space.

In coming up with a manageable expression for

piston directivity, one assumption must be made: the

distance from the piston to the observation point is

much greater than the piston’s radius. The result for the

pressure amplitude is

(17-17)

The first term in the above relationship contains the

dependency of the pressure on velocity amplitude,

piston size, and distance from the source. The second

term, called the piston directivity function, is derived

from a Bessel function, J 1 (x). The value of this function

is graphed in Fig. 17-58. Note that, up to ka= 3.83, the

value of the piston directivity function is uniformly

positive. The radiation pattern of the piston will have

only a single lobe under these conditions. If ka=3.83,

the pattern will have a null at 90° off-axis. For higher

values of ka, this null will occur at successively smaller

angles. Additionally, secondary lobes will appear

outside of the main lobe, although these lobes are

smaller in magnitude than the primary one. These lobes

will alternate in sign: the first set will be negative, the

second positive, etc.

The directivity of a real loudspeaker differs from that

predicted for a rigid piston due to the fact that several of

the basic assumptions in the preceding model are not

fully satisfied. First, no real loudspeaker has a perfectly

rigid cone or diaphragm. In the case of a cone trans-

ducer, the diaphragm is excited at its center. The excita-

tion travels outward from the voice coil as an acoustic

disturbance in the cone material. The velocity of propa-

gation of this disturbance is always finite. At lower

frequencies, this effect is negligible, but at higher

frequencies not all portions of the diaphragm will

vibrate in phase.

W

1

2

#---U 0 cSa^2 U 02

1

2

=---U 0 cSU 02.

Figure 17-58. Piston directivity function.

U

U 0 ckaU 0

2 r

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

2 J 1 ka sinT

ka sinT

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

0 2 4 6 8 10

1.0

0.5

0

0.5

x

2 j 1 (x)

x