Image Processing Reference
In-Depth Information
sound source array, wave fronts within a volume can be synthesised. Kirchhoff and
Helmholtz formulated the sound pressure
P
as an integral equation, Eq. (
11.1
)
based on the Huygens
principle [
7
].
'
2
4
3
5
þ
e
jk
r
r
s
e
jk
r
r
s
j
j
P r
s
,
ð
ω
Þ
j
j
1
4
Þ
∂
∂
∂
P
ð
~
r
; ω
Þ¼
P
ð
~
r
s
,
ω
dS
;
ð
11
:
1
Þ
ˀ
n
j
~
r
~
r
s
j
rj j
|
{z
}
monopoles
n
~
r
~
∂
|
{z
}
dipoles
S
where
k
is a wave number,
ω
is the angular frequency of a wave,
c
is the speed of
sound.
r
denotes a position vector of a listening position inside the domain
surrounded by an arbitrary surface
S
.
~
~
r
s
is a position vector on the surface
S. n
is
the internal normal on
S
. In reproduction of the sound field at
~
r
, the former out of the
two main terms on the right-hand side of Eq. (
11.1
) can be implemented as acoustic
dipoles and the latter as monopoles situated on
S
. For practical implementation of
the Eq. (
11.1
) we can choose only the monopole source term in the Kirchhoff-
Helmholtz integral via modification of Green
s function [
8
]. Theoretically
Eq. (
11.1
) allows us to implement the WFS system with various combinations of
a monopole and dipole. However a monopole is generally chosen as a secondary
source for WFS in realisation due to the simple construction, more reliable con-
trollability and relatively smaller in size than a dipole.
Usually known as Rayleigh I and Rayleigh II integral Eqs. (
11.2
) and (
11.3
)
describe solutions for monopoles and dipoles respectively [
9
].
'
dS
ðð
e
jk r r
s
j
j
Þ¼ˁ
0
c
jk
P r
ð
; ω
V
n
r
s
,
ω
;
ð
11
:
2
Þ
2
ˀ
j
~
r
~
r
s
j
s
and
dS
ðð
1
e
jk r r
s
jk
2
þ
jk
~
r
~
r
s
j
j
P
ð
~
r
; ω
Þ¼
P
~
r
s
,
ω
cos
ˆ
;
ð
11
:
3
Þ
ˀ
j
~
r
~
r
s
j
j
~
r
~
r
s
j
s
where
ˁ
0
denotes the air density, and
V
n
the velocity normal on the boundary
S
surrounding the domain,
c
the speed of sound.
denotes the relative angle
between the vector from a position on
S
to the reference position L and the normal
vector
V
n
in Fig.
11.1
.
Equations (
11.2
) and (
11.3
) satisfy their solutions through the entire volume
inside
S
. However, in practical approach the valid solutions for WFS can be found
only on a horizontal plane [
1
,
10
]. In that case, the surface integral can be reduced to
a line integral [
11
]. In implementation of the solutions a continuous distribution of
secondary sound sources, i.e. ideally a line source, has been supposed to be used.
The practical limitation requires the equations described in discrete form. The
ˆ
