Audio Engineering

680 Chapter 23

where

Q

P

pd

 x

4

2

2

2

π

πφφφ

ax

∫ 0 sin.

(23.13)

The mechanical impedance in MKS mechanical ohms (Newton-seconds/meter) of the air load
upon one side of a plane piston mounted in an infi nite baffl e and vibrating sinusoidally is

Z

j

mmR mRjXac ^0 ,

JK

K

c

k

a KK

a

a

2

0

1

(^121)

2

2

πρ 2

()πρ

()

⎡

⎣

⎢

⎢

⎤

⎦

⎥

⎥ (23.14)

where Zm is mechanical impedance in Newton seconds/meter, α is radius of piston in
meters,ρ 0 is density of gas in kg/cubic meter, c is velocity of sound in meters/second,
RmR is mechanical resistance in Newton seconds/meter (this component varies with
frequency), X is mechanical reactance in Newton seconds/meter, K is ∞/c  2 π / λ 5 wave
number, and J 1 K 1 is two types of Bessel function given by the series:

JW

WW W W

1

3

2

5

22

7

()22 42 4 62 4 68   222          (23.15)

KW

WW W

1

35

2

7

22

2

3 35357

() ,



π

⎛

⎝

⎜⎜

⎜⎜

⎞

⎠

⎟⎟

⎟⎟^ (23.16)

where W is 2 Ka.

Figure 23.3 shows graphs of the real and imaginary parts of this equation:

ZR jxmmRm as a function of ka

It will be seen that for values of Ka  \ , the reactance X varies as the fi rst power of frequency,
while the resistive component varies as the second power of frequency. At high frequencies
(i.e.,Ka  5 ) the reactance becomes small compared with resistance, which approaches a
constant value. The graph can be closely approximated by the analogue ( Figure 23.4 ), where

Rac

Rc

m0

m

MKS mechanical ohms

MKS mechanical

1

2

2 2 0

1 386



. ρ
πρ oohms
/ metrea/Newton
/.

m0

m

Cac

Ma Kg

1

2

1 3 0

06

83





. ρ
ρ