Building Acoustics

Sound absorbers 169

()

()

1

1

0

0

2 Jj

1

jJj

s

s s

ρρ.

−

⎛⎞−

=−⎜

⎜− −

⎝⎠

⎟

⎟

(5.30)

Assuming that the length d of the tube in question is much shorter than the wavelength
we may express the specific acoustic impedance as

()

()

1

1

s0

0

2 Jj

j1

x jJj

p s

Zd

v s s

ωρ.

−

⎛⎞−

Δ ⎜

== −

⎜− −

⎝⎠

⎟

⎟

(5.31)

At low frequencies, setting s in Equation (5.28) less than ≈ 2.0, a very good
approximation for the term enclosed in the parenthesis is 4/3 - j⋅8/s^2 , which gives

(^) s 2

84

j

3

d

Z 0 d,

a

μ

≈+ωρ (5.32)

or expressed by the acoustic impedance

(^) a 40

84

j

3

d d

Z

aa^2

.

μ ωρ

ππ

≈+ (5.33)

The viscosity then gives us two effects. First, we get a resistive part being analogous to
the mechanical damping coefficient, and we shall note the relationship with a quantity to
be introduced later, the airflow resistance. This is a very important material parameter
for all porous media. Using the definitions found in the international measurement
standard, ISO 9053, which is treated further in section 5.6.1, the quantity 8μ/a^2 will be
the flow resistivity of the tube, having symbol r and dimension Pa⋅s/m^2. Second, the
viscosity also affects the mass term in the expression for the impedance. We get an
increase of one-third compared with our earlier calculations (see Equation (5.22)).
For a panel, either a slatted one, i.e. an assemblage of parallel beams, or perforated
by thin slits, we shall need an expression for the equivalent viscous losses in a single
long slit (see Figure 5.10). We shall assume that the input pressure is the same along the
whole length of the slit, the length being long in comparison with the wavelength.
Furthermore, the pressure p in the slit varies only in the x-direction, and the velocity in
this direction is only dependent on the z-coordinate. It may be shown (see e.g. Vigran
and Pettersen (2005)), that the effective density of the air in the slit of width b may be
written

1

0

0

tan( )

1,w^2 here

j

2

kb

k

kb

.

ωρ

ρρ

μ

−

⎛⎞′

⎜⎟

=−⎜⎟ ′=

′

⎜⎟

⎝⎠

(5.34)

For an (infinitely) long slit in a plate of thickness d the specific impedance will then be