Chapter 8 Sound reflection and room acoustics Jean-Louis...
Transcript of Chapter 8 Sound reflection and room acoustics Jean-Louis...
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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
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© Jean-Louis Migeot – MSC Software – Free Field Technologies – Université Libre de Bruxelles – Conservatoire Royal de Musique de Liège – IJK Numerics 2
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
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Rigid wall - Time domain approach
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Rigid wall - Image source
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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
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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
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Standing waves
Progressive wave
Regressive wave
Standing wave
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Standing waves
© Dan Russell – Penn State
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Reflection - Image source
X
X
Real source
Image sourceReflected
Incident
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Reflection on a free surface
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Reflection on a free surface: inverted image
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Analogy with strings
© Dan Russell – Penn State
Zero reaction force
Zero acoustic velocity
Zero displacement
Zero acoustic pressure
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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
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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
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Reflection and absorption coefficients
➢ Let’s think in terms of intensity: incident, reflected and absorbed:
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Oblique incidence: Descartes law
X
Y
René Descartes
1596-1650
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Reflection - Image source
X
Y
X
Y
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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 …):
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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
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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°
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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°
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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
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Vibrating and absorbing panels
vfvs
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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
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A modern Kundt’s tube
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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
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Monopole
➢ Pressure
➢ Radial velocity
➢ Radial impedance
➢ Radial intensity
➢ Power through a sphere of radius R
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Monopole
© Dan Russell – Penn State
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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
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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
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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)
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Plane wave and spherical waves
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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
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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
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Source in a corner: wavefronts
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P
S1
S2
Q
Source in a corner: sound rays
t0=r/c
t1=r1/c
tt2=r2/c t12=r12/c
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P P1
P12P2
S1
S2
Source in a corner: image sources (1)
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P P1
P12P2
S1
S2
Q
rr1
r2
r12
Source in a corner: image sources (2)
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Image sources: generalization
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Order 1 images
Rectangular room (order 1)
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Order 2 images
Rectangular room (order 2)
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Order 3 images
Rectangular room (order 3)
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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
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Importance of the diffuse field
© Antonio Fischetti – Initiation à l’acoustique
Direct sound
Diffuse field
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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
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Wavefronts and rays
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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
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Ray tracing method
© Odeon Room Acoustics Software
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Ray tracing method
© Odeon Room Acoustics Software
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Ray tracing method
© Odeon Room Acoustics Software
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Ray tracing method
© Odeon Room Acoustics Software
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Ray tracing method
© Odeon Room Acoustics Software
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Ray tracing method
© Odeon Room Acoustics Software
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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
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Reverberation time
Build-up ExtinctionI
L
60 dB
TR
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TR(f)
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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
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It’s all about surface …
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Reverberant room© IAC Acoustics
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Anechoic room© IAC Acoustics
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Semi-anechoic room© IAC Acoustics
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Measuring a in a reverberation chamber
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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
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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
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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