Building Acoustics

Waves in fluid and solid media 73

( r0)piston 01 [ 1 ]

11

11

(2 ) j (2 ) ,

2J() 2H()

where ( ) 1 and ( ).

ZcSRkaXka

x x

Rx X x

xx

=+ρ ⋅

=− =

(3.57)

As stated above J 1 is a Bessel function of order one, whereas H 1 is a Struve function of
order one. Concerning the definition and properties of these functions we may refer to
Abramowitz and Stegun (1970).
The functions R 1 and X 1 are shown in Figure 3.9 as a function of ka going from 0 to
a value of 20. For the piston used as an example in Figure 3.8 this implies going up to a
frequency of approximately 8700 Hz. As shown the function R 1 will approach the value
of 1.0 at the higher frequencies, which means that the radiated power will be given by the
expression

00 2
1

.

ka

WcSuρ

>>

≈ ⋅ (3.58)

We shall later on use this expression as a reference when defining the so-called radiation
factor (or radiation efficiency) applying it to all types of sound radiating surface. This
will be treated in section 6.3.1.

Figure 3.9 Relative radiation impedance of a piston in a baffle. Real part, R 1 , and imaginary part, X 1 , of the
impedance function.

At the other extreme we get

2

11

11

4

RandX,

83

xxx x

π

⎯⎯⎯<<→⎯⎯⎯<<→ which implies that

0 2 4 6 8 10 12 14 16 18 20

ka

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Z

/ρ

c 0

S 0

Real part

Imaginary part