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

➢ Two main theories:

Multipole expansion

Boundary integral equation


Monopole

© Dan Russell – Penn State


Monopole directivity


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)


Dipole


Dipole directivity

+1 0 -1


Lateral quadrupole


Lateral quadrupole directivity

+1 0 -1


Linear quadrupole

-A -A2A


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


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


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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