Chapter 8 Sound reflection and room acoustics Jean-Louis …homepages.ulb.ac.be/~aderaema/vibrations/08-Reflection.pdf© Jean-Louis Migeot –MSC Software –Free Field Technologies

© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 1

Chapter 8Sound reflection and room acousticsJean-Louis Migeot

1. Reflection of plane waves

2. Reflection of spherical waves

3. Ray tracing method

4. Reverberation time








Rigid wall - Time domain approach


Rigid wall - Image source


Rigid wall – Frequency domain approach

➢ The staring point is the general 1D solution to the Helmholtz equation:

➢ The velocity is calculated using Euler’s equation:

➢ Velocity at the wall must be zero:

the amplitude of the reflected wave is therefore the same as the amplitude of the incident wave: perfect reflection !

➢ The pressure field is:

The pressure at the wall is twice the incident pressure.

Incident p+

Réflected p-

X

Mouvements vibratoiresEchelle de perroquet


Rigid wall – Frequency domain approach

➢ Velocity distribution:

➢ Impedance:

➢ Impedance is purely imaginary → there is no active intensity:

Intensity carried to the right by the incident wave is perfectly compensated by intensity carried to the left by the reflected wave.

Incident p+

Réflected p-

X


Standing waves

Progressive wave

Regressive wave

Standing wave


Standing waves

© Dan Russell – Penn State


Reflection - Image source

X

X

Real source

Image sourceReflected

Incident


Reflection on a free surface


Reflection on a free surface: inverted image


Analogy with strings


Zero reaction force

Zero acoustic velocity

Zero displacement

Zero acoustic pressure


Normal incidence, absorbing surface

➢ Always the same staring point ☺

➢ Pressure and velocity at the wall (x=0):

➢ Normal impedance at the wall:

➢ Let’s introduce the reflection factor:

Incident p+

Réflected p-

X


Limit case

➢ Perfectly rigid wall (v=0): Zn= → R=1 → p+=p-

➢ Perfectly soft wall (p=0): Zn=0 → R=-1 → p+=-p-

➢ Total absorption: Zn=rc → R=0 → p-=0


Reflection and absorption coefficients

➢ Let’s think in terms of intensity: incident, reflected and absorbed:


Oblique incidence: Descartes law

qq

X

Y

René Descartes

1596-1650


Reflection - Image source

X

Y

X

Y


Reflection - Oblique incidence, absorbing surface

➢ The pressure field takes a slightly more complex form. We can write it in terms of the components kx and ky of the wave vector:

➢ Or using the wave number k and the angle of incidence q:

➢ Pressure and velocity at the wall yield the normal impedance:

➢ From where we get the reflection factor and the absorption coefficients (some calculations required for the latter … a good exercise …):


Variation of absorption (a) with incidence

0

0.2

0.4

0.6

0.8

1

1.2

0 10 20 30 40 50 60 70 80 90 100

Z=rc

Z=4rc

Z=2rc+2jrc

Z=rc/4


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0.01 0.1 1 10 100

Reduced impedance modulus |z| (log scale)

Variation of absorption (a) with impedance (normal incidence)

a (absorption coefficient)

j = 0°

j = 30°

j = 45°

j = 60°

j = 70°

j = 80°

j = 85°


0

20

40

60

80

100

120

140

160

180

200

0.01 0.1 1 10 100

Reduced impedance modulus |z| (log scale)

Phase shift between incident and reflected wave (normal incidence)

(phase shift)

j = 0°

j = 30°

j = 45°

j = 60°

j = 70°

j = 80°

j = 85°


Locally and non-locally reacting impedances

v=0

continuity of p and q

Air

Foam

p=Z.v

Air

p=Z.v

Air

A

B

C

P P’ Q

D


Vibrating and absorbing panels

vfvs


Absorption and impedance: a summary

➢ Impedance is the ratio of pressure and velocity spectra. Admittance is the ratio of velocity and pressure spectra.

➢ The absorption of the material covering a wall is characterized by its normal impedance: the material enforces a certain ratio between pressure and velocity (complex) amplitudes.

➢ A plane wave of amplitude A impinging with incidence q on a plane covered with a material of impedance Z is reflected with a reduced amplitude R.A with:

➢ The ratio of reflected to incident intensity is called the reflection coefficient:

➢ The ratio of absorbed to incident intensity is called the absorption coefficient:

Normalincidence


A modern Kundt’s tube








Monopole

➢ Pressure

➢ Radial velocity

➢ Radial impedance

➢ Radial intensity

➢ Power through a sphere of radius R


Monopole



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


Z

P(X,Y,Z)

Q(x,y,z)

Z

P(X,Y,Z)

Q(x,y,z)

P’(X,Y,-Z)

r r

r’

Z

P(X,Y,Z)

Q(x,y,z)

P’(X,Y,-Z)

r

r’

(a) (b) (c)

S S

SR

Reflection and image source


r

r’

r

r’

t1=r/c t2=r’/ct

Source above a reflecting plane: reflectogram

➢ Three complementary analysis:

in terms of wavefronts

in terms of sound rays

in terms of image sources


Source in a corner: wavefronts


P

S1

S2

Q

Source in a corner: sound rays

t0=r/c

t1=r1/c

tt2=r2/c t12=r12/c


P P1

P12P2

S1

S2

Source in a corner: image sources (1)


P P1

P12P2

S1

S2

Q

rr1

r2

r12

Source in a corner: image sources (2)


Image sources: generalization


Order 1 images

Rectangular room (order 1)


Order 2 images



Order 3 images



Direct vs. diffuse

Direct field

Réflexions précocesEarly reflections

Champ diffusDiffuse field

‘Sound field in which the time average of the mean-square sound pressure is everywhere the same and the flow of acoustic energy in all directions is equally probable’ INCE


Importance of the diffuse field

© Antonio Fischetti – Initiation à l’acoustique

Direct sound

Diffuse field








Wavefronts and rays


Ray tracing method

➢ A noise source is modelled by a large number of rays (typically several thousand) each carrying a fraction of the source energy.

➢ A receptor area is defined and we track the time at which a ray crosses the receptor and what energy it carries at that time -> reflectogram

➢ Each ray propagates until it reaches a wall where it is specularly reflected.

➢ At each reflection, the ray loses a fraction a of its energy

➢ During its propagation, the ray also loses energy due to

attenuation

spherical divergence


Ray tracing method

© Odeon Room Acoustics Software


Ray tracing method



Ray tracing method



Ray tracing method



Ray tracing method



Ray tracing method









Reverberation time

Build-up ExtinctionI

L

60 dB

TR


TR(f)


Sabine law

Wallace Sabine (1868-1919)

( )aa

a

−

−=→=

=→=

1ln

16.0

16.0

S

VT

S

A

A

VTSA

REyring

RSabine

i

ii

http://hyperphysics.phy-astr.gsu.edu/hbase/acoustic/revmod.html


It’s all about surface …


Reverberant room© IAC Acoustics


Anechoic room© IAC Acoustics


Semi-anechoic room© IAC Acoustics


Measuring a in a reverberation chamber


Alpha Cabin

➢ Bounded Cavity (walls are not parallel)

➢ Foam sample on one face

➢ Several acoustic spherical sources inside to simulate a diffuse sound field (random positions and random phases)

Reverberation time Absorption coefficient


Key Takeaways

➢ Acoustic waves reflect on surfaces like light rays on mirrors: Descartes’ law

➢ Part of the intensity is absorbed. Absorption is characterized by the material’snormal impedance Zn which defines the reflection factor R and the absorption andreflection coefficients a and r. Zn and R are complex, a and r are real. All are frequency dependent.

➢ The angle of incidence influences the absorption on a given wall.

➢ Room acoustics:

Spherical sources

reflectograms

ray tracing

reverberation time and Sabine’s law

➢ Acoustic test facilities







Documents

Chapter 8 Sound reflection and room acoustics Jean-Louis …homepages.ulb.ac.be/~aderaema/vibrations/08-Reflection.pdf© Jean-Louis Migeot –MSC Software –Free Field Technologies