Civil Engineering Reference
In-Depth Information
⎡
( )
2
⎤
2
ka
8
ka
()
Z
≈
ρ
c S
⎢
+ ⋅
j
⎥
.
(3.59)
r
0
0
piston
1
8
3
π
⎢
⎥
⎣
⎦
ka
<<
This expression may be written as
22
ρω
S
8
a
⎛
⎞
()
0
Z
≈
+ ⋅
j
ωρ
S
⎠
.
(3.60)
⎜
⎟
r
0
piston
2
π
c
3
π
⎝
0
ka
<<
1
As expected the real part will, apart from a factor of two, be equal to the impedance of a
monopole given in Equation
(3.56)
. The imaginary part will represent the mass type
impedance equal to the mass of air contained in a cylinder having the same area as the
piston and a height
h
equal to 8
a
/3π. This mass impedance will decrease with increasing
frequency in accordance with the decrease of
X
1
when increasing the frequency.
3.5
SOUND FIELDS AT BOUNDARY SURFACES
The assumption we have used up to now is that the wave propagation is taking place in
an infinite space which is homogeneous and isotropic. When dealing with the acoustics
inside buildings, however, we shall definitely be more concerned with what is happening
at the boundaries between different media, e.g. such as the interface between air and a
flexible surface of some kind. When waves are impinging at such a boundary it normally
will be diffracted in some way, a part of the energy in the wave will go in another
direction. The phenomenon is normally referred to as
reflection
when the boundary
surface is much larger than the wavelength. In the opposite case, where the wavelength is
much larger than the dimensions of the surface, we shall use the word
scattering
. We
shall in this topic mainly be concerned with boundary surfaces fulfilling the first
condition but certainly when dealing with room acoustics scattering phenomena will be
an important aspect.
When dealing with boundary surfaces we shall be interested in the reflected energy
as well as the energy transmitted through the surface. The task of designing sound
absorbers is to minimize the reflected energy, whereas designing for high sound
insulation the aim is to reduce the transmitted energy.
Boundary surfaces will, irrespective of being fixed or set in motion by the sound
waves, produce changes in the sound field, which means that some boundary conditions
are given. We shall introduce the relevant boundary conditions when they are needed, an
example is where the boundary surface is a solid, non-porous wall vibrating due to an
outer sound field. The particle velocity of sound normal to the wall surface must then
everywhere be equal to the velocity of the wall. If this were not the case, the local density
of the fluid would become abnormally high or low, which is highly unlikely.
We shall introduce a complex
pressure reflection factor R
p
giving the ratio, both in
amplitude and phase, between the sound pressure in the reflected and the incident wave.
We shall write it as
p
ˆ
j
δ
r
i
(3.61)
R
(,)
ωϕ== ⋅
R
e .
p
p
p
ˆ
As indicated, the reflection factor will in general be a function of the frequency and the
angle of incidence ϕ of the wave. One will also find a reflection factor defined on the
