Audio Engineering

650 Chapter 22

where um is the particle velocity amplitude.

The more often encountered case is where the source is nearby to the microphone
location. In such an instance the appropriate wave description is that of a spherical wave
propagating along a radial line from the sound source. Mathematically, such a wave is
described by

prt

A

r

(,) ejtkr()ω , (22.13)

where Pm Ar is the pressure amplitude that is now position dependent and A is a
constant determined by the sound source.

The pressure gradient is now more complicated as the space variable r appears in both the
denominator and the exponent of the expression for the acoustic pressure.

∂

∂

⎛

⎝

⎜⎜

⎜

⎞

⎠

⎟⎟

⎟⎟

r

prt

r

(,) jk p r t(,).

1

(22.14)

If the center of the diaphragm is located at a distance r from the sound source, then the
driving force on the diaphragm for the spherical wave becomes

Ft

r

()jk p r t Sd(, ) cos

⎛ 1

⎝

⎜⎜

⎜

⎞

⎠

⎟⎟

⎟⎟ θ. (22.15)

The driving force now has two components, one of which is in phase with the acoustic
pressure while the other leads the acoustic pressure by 90°. The specifi c acoustic
impedance of air for spherical waves is not as simple as was the plane wave case. The
ratio of the acoustic pressure to the particle velocity is now

prt

urt

j

r

jk

(,)

(,)

.



ωρ 0

1

(22.16)

Upon solving Equation (22.16) for the acoustic pressure in terms of the particle velocity
and substituting into Equation (22.15), one obtains the very important result