Waves in fluid and solid media - Building Acoustics - page 73

( )

[ ]

Z = ρ c S R (2 ka ) + ⋅ j X (2 ka ) ,

r 0 0 1 1

piston

(3.57)

2J() x 2H() x

1 1

where Rx ( ) =− 1 and X x ( ) = .

1 1

x x

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

2

⋅

Wc S u ≈ ρ . (3.58)

00

ka >> 1

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.

1.4

1.2

Real part

1.0

0.8

0.6

0.4

0.2 Imaginary part

0.0

0 2 4 6 8 10 12 14 16 18 20

ka

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.

2

x 4 x

π

x << 1 x << 1

At the other extreme we get R ⎯⎯⎯→ and X ⎯⎯⎯→ , which implies that

1 1

8 3