Audio Engineering

678 Chapter 23

phase of the pressures from these elementary elements. For r large compared with the
radius of the pistona , the equation will be

P

jfp u a

v

JK

K

(sound pressure N/m )

sin

a

a

2 00

(^2221)

√ πφ
φ

⎡ ()

⎣

⎢

⎢

⎤

⎦

⎥

⎥ ee.

jtrω() (23.7)

where u 0  RMS velocity of the piston and J 1 () is Bessel function of the fi rst order. Note
that the portion of Equation (23.7) in square brackets yields the directivity pattern.

23.5 Directivity

At frequencies where the wavelength of sound ( X ) is large compared with the diameter
of the piston, the radiation is spherical. As the frequency is increased, the wavelength
becomes comparable or less than the piston diameter and the radiation becomes
concentrated into a progressively narrowed angle.

The ratio of pressure P 0 at a point set at an angleθ off the axis, to the on axis pressure PA
at the same radial distance, is given by

P

P

J

a

K

aK

φ

π

λ

φ

π

λ

A φ

a

a

sin

sin

.

2

2

2

1

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥

.

(23.8)

Figure 23.2 shows radiation patterns for different ratios of λ/D. The radiation from a
piston is directly related to its velocity, and we can compute the acoustic power radiated
and the sound pressure produced at any given distance in the far fi eld.

Piston

a r A

Infinitely large

plane wall

Figure 23.1 : Piston in infi nitely plane wall.