chapter 11 sound radiation jean-louis...
TRANSCRIPT
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 1
Chapter 11Sound radiationJean-Louis Migeot
1. Directivity diagrams
2. Elementary directive sources: monopoles, dipoles, quadrupoles
3. Equivalent source method
4. Multipole expansion
5. Helmholtz integral equation
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 2
Chapter 11Sound radiationJean-Louis Migeot
1. Directivity diagrams
2. Elementary directive sources: monopoles, dipoles, quadrupoles
3. Equivalent source method
4. Multipole expansion
5. Helmholtz integral equation
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 3
Directivity diagram
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 4
Emission and reception directivity
Emission Reception
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 5
Directivity is actually three-dimensional
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 6
Directivity changes when the center is offset
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 7
Example: a monopole has a uniform directivity …
90°
60°
30°
0°
330°
300°
270°
240°
210°
180°
150°
120°
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 8
… unless the monopole is not located at the radiation center !
90°
60°
30°
0°
330°
300°
270°
240°
210°
180°
150°
120°
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 9
Offset effect
+ - + - + - + - + + - + - + - + - +
Centered Off-center
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 10
Directivity changes with frequency …
400 Hz – R = 1 m
600 Hz – R = 1 m
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 11
Directivity changes with distance …
400 Hz – R = 1 m 400 Hz – R = 1000 m
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 12
Sound radiation
200 Hz
500 Hz 1.000 Hz
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 13
Frequency and directivity
1000 Hz 1500 Hz 2000 Hz
2500 Hz 3000 Hz
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 14
Qualitative observations
➢ Directivity changes with:
Plane orientation
Radiation center
Frequency
Distance
Source (of course) whose behaviour itself depends on frequency
➢ The multipole expansion theory provides interesting and general results on:
Frequency dependency
Changes with distance (near and far field)
➢ Let’s first look at elementary sources:
Monopoles
Dipoles
Quadrupoles
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 15
Chapter 11Sound radiationJean-Louis Migeot
1. Directivity diagrams
2. Elementary directive sources: monopoles, dipoles, quadrupoles
3. Equivalent source method
4. Multipole expansion
5. Helmholtz integral equation
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 16
Monopole
➢ Pressure
➢ Radial velocity
➢ Radial impedance
➢ Radial intensity
➢ Power through a sphere of radius R
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 17
Monopole
© Dan Russell – Penn State
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 18
Monopole
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 19
Monopole directivity
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 20
Monopole (k=1)
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0 5 10 15
Pre
ssu
re [
Pa]
Distance r ([m])
Real Part
Imaginary Part
Amplitude
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 21
Monopole (k=2)
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
3.0
0 5 10 15
Pre
ssu
re [
Pa]
Distance r ([m])
Real Part
Imaginary Part
Amplitude
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 22
Impedance
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 5 10 15
Re
du
ced
Imp
ed
ance
Distance r ([m])
Real Part
Imaginary Part
Near field (r<5l) Far Field (r>5l)
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 23
Plane wave and spherical waves
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 24
Dipole
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 25
Speed, intensity and power
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 26
Dipole
© Dan Russell – Penn State
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 27
Dipole directivity
+1 0 -1
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 28
Lateral quadrupole
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 29
Lateral quadrupole
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 30
Lateral quadrupole
© Dan Russell – Penn State
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 31
Lateral quadrupole directivity
+1 0 -1
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 32
Linear quadrupole
➢ Sources in Q1 and Q2: +A
➢ Source in Q: -2A
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 33
Linear quadrupole
© Dan Russell – Penn State
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 34
Linear quadrupole directivity
cos2 q
+1 0 -1
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 35
Chapter 11Sound radiationJean-Louis Migeot
1. Directivity diagrams
2. Elementary directive sources: monopoles, dipoles, quadrupoles
3. Equivalent source method
4. Multipole expansion
5. Helmholtz integral equation
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 36
Any source can be described by a set of point sources
v1
q1
q2
q3
u13
u12
u11v2
v3
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 37
Chapter 11Sound radiationJean-Louis Migeot
1. Directivity diagrams
2. Elementary directive sources: monopoles, dipoles, quadrupoles
3. Equivalent source method
4. Multipole expansion
5. Helmholtz integral equation
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 38
Multipole expansion
P
Q
Pi
r
ri
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 39
Higher order terms increase with frequency …
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 40
Elementary or canonical directivity diagrams
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 41
Elementary or canonical directivity diagrams: sinpq cosqq
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 42
Elementary or canonical directivity diagrams: sin pq cos qq
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 43
Far field and near field
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 44
Generality of the multipole expansion
➢ Any source (e.g. vibrating surface) may be replaced with an arbitrary accuracy by a set of point sources with frequency dependent amplitude generating, outside a given surface, the same sound field -> any sound field may be analyzed in terms of monopole, dipole, etc … but with M, D, Q, O depending on w
➢ General principles are:
for a vibro-acoustic source, the amplitude of high order terms tends to increase with frequency (vibrations are more complex)
this effect is strengthened by the fact that the terms involved are M, kD, k2Q, k3O, …
directivity thus increases with frequency
in the far field, only the first line in the matrix remains
in the near field all components are important
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 45
Chapter 11Sound radiationJean-Louis Migeot
1. Directivity diagrams
2. Elementary directive sources: monopoles, dipoles, quadrupoles
3. Equivalent source method
4. Multipole expansion
5. Helmholtz integral equation
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 46
Green’s identity
➢ For all u and v sufficiently continuous on S and in V, n being the outward normal to S
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 47
Green function
➢ Green function associated with the Helmholtz equation:
➢ This function satisfies the non-homogeneous Helmholtz equation:
and corresponds to the free field generated by a point source at P.
-20
-15
-10
-5
0
5
10
15
20
0
1.0
15 P
Q
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 48
Helmholtz integral equation
P
P
P
P
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 49
Interior case (P V, P S)
➢ Let’s choose u=p and v=G and let’s apply Green’s identity to V-Vs:
➢ but:
➢ so that:
( ) ( ) ( )p G G p dV p G G p dS p G G p dSii ii
V V
n n
S
n n s s
− = − + −−
( ) ( ) ( )( )
( ) ( ) ( )
p G G p dV p G k G G p k G dV
G P Q P Q dV Q
ii ii
V V
ii ii
V V
V V
s s
s
− = + − +
= =
− −
−
2 2
0, ,
( ) ( )p G G p dS p G G p dSn n
S
n n s
− + − = 0
S
V
PVs s
nS
ns
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 50
Integral on s (P V, P S)
( ) ( )
( )( )
( )
lim lim
lim sin
lim sin
lim sin
s
s
q q
q q
q q
→ →
→
− −
→
−
→
−
− = − −
= −
−
= − − + −
=
=
0 0
0
2
00
2
000
2
000
2
4 4
1
41
1
4
p G G p dS p G G p dS
pe e
p d d
p ik p e d d
p e d d
p P
n n
ik ik
ik
ik
( )
400
2
q q
sin d d
p P
=
( ) ( )p P p G G p dSn n
S
+ − = 0
( ) ( )p P p G i Gv dSn n
S
+ + = w 0
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 51
P S
( ) ( ) ( )1
4q q
s
sin .d d p P c P p P =
( )c p p G G p dSP P n n
S
+ − = 0
S
V
s
nS
ns
PVs
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 52
P V, P S
S
V
P
nS
( ) ( )p G G p dS p G G p dSn n
S
n n s
− + − = 0=0
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 53
➢ Same procedure but integral on S must be handled ...
➢ … but it vanishes provided Sommerfeld conditions apply
( ) ( ) ( )p G G p dS p G G p dS p G G p dSn n
S
n n n n s
− + − + − =
0
Exterior case
S V
PVs snS
ns
n
( )
( )
( )( )
( )( )
lim
lim
lim sin
lim sin
q q
q q
→
→
→
−
→
−
−
= − −
= − − + −
= + +
p G G p dS
p G G p dS
p ik p e d d
ikp p p e d d
n n
ik
ik
1
41
1
4
00
2
00
2
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 54
Helmholtz integral equation
Pressure at any point P in V
Pressure distribution
on S
Gradientof Green
function on S
Normal vibrationacceleration
distribution on S
Greenfunction
on S
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 55
Limits of the direct approach
➢ S must be closed:
no thin plate
no stiffener
no hole
etc ...
➢ Fluid must be homogeneous
➢ The indirect method generalises the direct method and suppresses these limitations
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 56
Indirect Helmholtz integral equation
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 57
Why are loudspeakers baffled (1) ?
P
P’
Sp
Sb
Q
z
n
uz
Rigid piston
Low frequency Monopole !
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 58
Baffled piston: axial pressure distribution
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0 0.5 1 1.5 2
Pressure amplitude (Pa)
Z-coordinate (m)
EXACT 4000 Hz
EXACT 8000 Hz
ACTRAN Infinite 4000 Hz
ACTRAN Finite 4000 Hz
ACTRAN Infinite 8000 Hz
ACTRAN Finite 8000 Hz
➢ Radius: 0.10 m
➢ Unit acceleration: 1 m/s2
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 59
Baffled piston: directivity at 1m
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0 10 20 30 40 50 60 70 80 90
Pressure amplitude (Pa)
Theta angle (degree)
Exact 4000 HzACTRAN 4000 Hz
Exact 8000 HzACTRAN 8000 Hz
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 60
Why are loudspeakers baffled (2) ?
P
z
Low frequency
Dipole !
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 61
Why are loudspeaker baffled ?
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 62
Key Takeaways
➢ Radiation is the generation of sound in free field by a vibrating structure
➢ Multipole expansion is a powerful technique for understanding a describing radiated sound fields:
it presents the sound field as the linear combination of a set of standard elementary directivity patterns
it shows how directivity evolves with distance (near field / far field) and with frequency (increased directionality)
➢ Helmholtz integral equation is another important tool for studying and understanding sound radiation
➢ Diffraction may be framed as a modified sound radiation problem
© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 63
Lecture 9Sound radiationJean-Louis Migeot
1. Directivity diagrams
2. Elementary directive sources: monopoles, dipoles, quadrupoles
3. Equivalent source method
4. Multipole expansion
5. Helmholtz integral equation