Audio Engineering

Microphone Technology 649

Ft pxt pxt pxtd S

x

() (,) (,) (,) cos

∂

∂

⎧

⎨

⎪⎪

⎩⎪⎪

⎫

⎬

⎪⎪

⎭⎪⎪

⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

θ



∂

∂

⎡

⎣

⎢

⎢

⎤

⎦

⎥

x pxtd(,) cosθ⎥S (22.8)

where S is the surface area of one side of the diaphragm and (/ )(,)∂∂x pxt is the
gradient of the acoustic pressure in the direction of increasingx.

The pressure gradient is calculated by taking the partial derivative with respect to x of
Equation (22.7) as follows:

∂

∂x m

pxt jkpetkx

jkp x t

(,)

(,).

 ()



ω

(22.9)

Upon substituting the result of Equation (22.9) into Equation (22.8), the driving force
becomes

Ft() jkp xt Sd( , ) cosθ. (22.10)

In a given sound wave of normally encountered intensities, a relationship exists between
the acoustic pressure and the acoustic particle velocity. The ratio of the acoustic pressure
to the particle velocity is called the specifi c acoustic impedance of air for the wave type
in question. This ratio for plane waves is a real number equal to the normal density of
air multiplied by the phase velocity of sound. One can then substitute for the acoustic
pressure in Equation (22.10) in terms of the particle velocity to obtain an alternative
expression for the driving force:

Ft jk cu xt Sd

juxtSd

() ( , ) cos

(,) cos





ρ

ωρ

0

0

θ

θ

(22.11)

The signifi cance of the imaginary operator j in this equation simply means that the phase
angle of the driving force leads that of the particle velocity by π /2 radians or 90°. The
amplitude of the driving force would be

FuSdmmωρ 0 cosθ, (22.12)