Audio Engineering

Loudspeakers 679

23.6 Sound Pressure Produced at Distance r ................................................................

Low frequencies. When the piston diameter is less than one-third wavelength (AT a  1.0)
it is essentially nondirectional and can be approximated by a hemisphere whose RMS
volume velocity ux equals:

uSduau 1 (diaphragm velocity)  ccπ^2 (23.9)

and the RMS sound pressure at distance r is:

p( )r^00 N/m

sdfp

r

afp

r



π^22

(23.10)

and total power radiated Wt is:

W

p

c

t      Sd f u W

4 π (^02)
( c). (23.11)
Medium frequencies. At frequencies where the radiation from the piston becomes
directional but still vibrates as one unit, the pressure produced at a distance r depends on
the power radiated and the directivity factor Q:
p(r^0

WQPC

ar

) r

π^2

⎛

⎝

⎜⎜

⎜

⎞

⎠

⎟⎟

⎟⎟ (23.12)

D



λ/4

D



λ/2

D



λ D

2 λ

Figure 23.2 : Directivity of piston as a function of piston diameter and wavelength.