Building Acoustics

Waves in fluid and solid media 71

3.8. The diameter 2⋅a is chosen equal to 250 mm and the sound pressure level
distribution is shown for three frequencies: 1000, 2000 and 5000 Hz. To facilitate the
comparison the maximum sound pressure is arbitrarily set to 40 dB for all frequencies.
At the higher frequencies, i.e. when the product ka becomes much larger than one, the
directivity pattern will be very complicated (ka will approximately be equal to 11.5 at
5000 Hz), whereas the pattern at low frequencies will be little different from the ball-
shaped pattern of a monopole.
Thus, assuming that the wavelength is much larger than the dimensions of the
piston by setting ka << 1, we may show that [2J 1 (ka sinφ)/ ka sinφ)] ≈ ½. Then we may
write

2

00 j( )

00 j( )

(,) j ˆ e

4

or ( , ) j e.

22

tkR

tkR

cka

pRt u

R

ckQ

pRt

R

ω

ω

ρ

ρ

π

−

−

=⋅

=⋅

 (3.52)

The latter expression is as expected identical to the one giving the sound pressure from a
monopole having source strength Q. (Show that Equation (3.37) gives the same
expression when setting ka << 1 and r >> a.)

Figure 3.8 Directivity pattern of a piston in a baffle. The maximum sound pressure level is arbitrarily set to 40
dB for all frequencies.

3.4.4 Radiation impedance

In the previous chapter, the concepts of mechanical impedance and mobility were
introduced to facilitate the calculation of the response of a mechanical system to a given

0 10203040

Relative level (dB)

Diameter of piston - 250 mm

1000 Hz

2000 Hz

5000 Hz