Chapter 11 Sound radiation Jean-Louis...

© 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


Directivity diagram


Emission and reception directivity

Emission Reception

../../../DataMovie/Fischetti/Directivité.MOV


Directivity is actually three-dimensional


Directivity changes when the center is offset


Example: a monopole has a uniform directivity …

90°

60°

30°

0°

330°

300°

270°

240°

210°

180°

150°

120°


… 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


Offset effect

+ - + - + - + - + + - + - + - + - +

Centered Off-center


Directivity changes with frequency …

400 Hz – R = 1 m

600 Hz – R = 1 m


Directivity changes with distance …

400 Hz – R = 1 m 400 Hz – R = 1000 m


Sound radiation

200 Hz

500 Hz 1.000 Hz


Frequency and directivity

1000 Hz 1500 Hz 2000 Hz

2500 Hz 3000 Hz


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


Monopole

➢ Pressure

➢ Radial velocity

➢ Radial impedance

➢ Radial intensity

➢ Power through a sphere of radius R


Monopole

© Dan Russell – Penn State


Monopole


Monopole directivity


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


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


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)


Plane wave and spherical waves


Dipole


Speed, intensity and power


Dipole



Dipole directivity

+1 0 -1


Lateral quadrupole


Lateral quadrupole directivity

+1 0 -1


Linear quadrupole

➢ Sources in Q1 and Q2: +A

➢ Source in Q: -2A


Linear quadrupole



Linear quadrupole directivity

cos2 q

+1 0 -1


Any source can be described by a set of point sources

v1

q1

q2

q3

u13

u12

u11v2

v3


Multipole expansion

P

Q

Pi

r

ri


Higher order terms increase with frequency …


Elementary or canonical directivity diagrams


Elementary or canonical directivity diagrams: sinpq cosqq


Elementary or canonical directivity diagrams: sin pq cos qq


Far field and near field


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


Green’s identity

➢ For all u and v sufficiently continuous on S and in V, n being the outward normal to S


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


Helmholtz integral equation

P

P

P

P


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


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


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


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


➢ 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


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


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


Indirect Helmholtz integral equation


Why are loudspeakers baffled (1) ?

P

P’

Sp

Sb

Q

z

n

uz

Rigid piston

Low frequency Monopole !


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


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


Why are loudspeakers baffled (2) ?

P

z

Low frequency

Dipole !


Why are loudspeaker baffled ?


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


Lecture 9Sound radiationJean-Louis Migeot






Chapter 11 Sound radiation Jean-Louis...

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