Room Acoustics – An Approach Based on Waves and Particles
45
05/10/2010 1 Room Acoustics – An Approach Based on Waves and Particles Jens Holger Rindel Odeon A/S, Denmark EuroRegio 2010 2 Outline • Introduction • Waves and normal modes in rooms • Diffuse and non-diffuse sound fields • Particles and reflections in rooms – Diffraction – scattering – curved surfaces • Computer modelling • Speech in rooms • Music in rooms • Conclusion
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Transcript of Room Acoustics – An Approach Based on Waves and Particles
05/10/2010
1
Room Acoustics – An Approach
Based on Waves and Particles
Jens Holger Rindel
Odeon A/S, Denmark
EuroRegio 2010 2
Outline
• Introduction
• Waves and normal modes in rooms
• Diffuse and non-diffuse sound fields
• Particles and reflections in rooms– Diffraction – scattering – curved surfaces
• Computer modelling
• Speech in rooms
• Music in rooms
• Conclusion
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2
Introduction
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Wallace C. Sabine (1868 – 1919)
Setup for measurement of reverberation time using four sets of organ pipes
4
Measured decay times
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3
5
Measurement resultswith 1 and 4 organ pipes:
Decay times
t1 = 8,69 s
t4 = 9,55 s
Level differences
∆L1 = 10 log 1 = 0,0 dB
∆L4 = 10 log 4 = 6,0 dB
EuroRegio 2010
s 6,810)69,855,9(0,6
60)( 14 =⋅−=−= ttT
The reverberation time is the time for a 60 dB decay; thus:
Waves and normal modes
6
Ripple tank model of an
auditorium (around 1929)
The wave length is visible
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Modes in a rectangular room
⋅
⋅
⋅=
z
z
y
y
x
xl
zn
l
yn
l
xnpp πππ coscoscos0
p : Sound pressure in the room (p0 is maximum)lx, ly, lz : Room dimensionsnx, ny, nz : Natural numbers or 0
Relative sound pressure |p/p0|
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Transfer function in a room
At low frequencies the room modes may be
identified by peaks at the natural frequencies
222
2
+
+
=
z
z
y
y
x
xn
l
n
l
n
l
ncf c : Speed of sound (344 m/s)
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5
Normal modes in a room
• Modal overlap
– Number of modes
within the
bandwidth of the
modes
TBr
2,2=
f
NBM r
∆
∆=
• Bandwidth of the modes
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Schroeder’s limiting frequency
• Modal overlap should be minimum M = 3 for statistical considerations
• This is fulfilled above the limiting frequency
V
Tf st 2000=
T : Reverberation time (s)
V : Volume (m3)
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Transfer function above fst
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Measured sine sweep, 900 – 1000 Hz, A: Reverberant room, B: Dead room
Ref.: Lyon (1969)
Tf
4max ≈∆
Average distance
between maxima in the
transfer function
depends on the
reverberation time
A: ∆fmax = 2 Hz => T = 2 sB: ∆fmax = 12.5 Hz => T = 0.5 s
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Natural frequencies in 3D
Frequency space with the modes of in rectangular room.
Volume of 1/8 sphere:(4 π f 3 / 3) / 8 = π f 3 / 6
Volume of one cell: c3 / (8 lx ly lz) = c3 / (8 V)
Total number of modes below f :
3
3
3
3
3
48
6 c
fV
c
VfN
ππ==
This is only the 3-dimensional modes
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Normal modes in a room
Modal density
c
Lf
c
Sf
c
V
df
dN
8
'
2
'42
2
3++=
ππ
V : VolumeS’ : Total surface areaL’ : Total edge lengthc : Speed of sound, 343 m/sf : Centre frequency of band ∆f
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The average modal densityincreases with f 2
2
3
4f
c
V
df
dN π≈
Particles and reflections
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Optical light beams in
section of a hall.
(Satow, 1929)
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Reflection density
A rectangular room with image sources
The average reflection densityincreases with t 2
V
cttN
3
34 )(
)(π
=
23
4 tV
c
dt
dN t π=
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1
23
1: Direct sound
2: Early reflections
3: Late reflections
Early and late reflections
Suggested
transition time(ms) 2 Vtst =
So
un
d e
ne
rgy
Time
SectionV : Volume (m3)
Early reflections can (often)
be identified
Late reflections merge together with high reflection density
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Specular and scattered reflection
Specular reflections are not a good representation after typically 3rd -4th order reflections (depends on the actual room)
Later reflections are dominated by scattered energy
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Building a room - Direct sound
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One reflecting wall - Echo
1st order image source
Time delay106 ms
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Three walls
1st and 2nd orderimage sources shown
Several echoes
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Six surfaces
Up to 3rd orderimage sourcesshown
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A sound wave represented by a ray
Mean free path
in a 3D diffuse
sound field:
lm = 4V/ S
V : Volume
S : Surface area
NB: It is assumed that all surfaces
have the same absorption
coefficient αm
Energy of sound wave
is reduced by (1- αm)
after each reflection
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EuroRegio 2010 23
Sound pressure after n reflections
)1ln(2
0
2
0
2 e)1()( mnn
m pptpαα −⋅⋅=−⋅=
m
i
i lntcl ⋅=⋅=∑t
l
cm
mptp⋅−⋅
⋅=)1ln(
2
0
2 e)(α
60
)1ln(6 )1ln()10ln(6e10
60
Tl
cm
m
Tl
cm
m ⋅−⋅=⋅−⇒=⋅−⋅
− αα
m
m
m
m
c
l
c
lT
αα ⋅
⋅≈
−⋅−
⋅=
8.13
)1ln(
8.1360
General equation for reverberation time
Total path:
⇒⋅=⇒= −62
0
2
60 10)( ptpTt
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S
Vlm
4=
U
Sl x
m
π=
xm ll =
3D:
2D:
1D:
General:
m
m
m
m
c
l
c
lT
αα ⋅
⋅≈
−⋅−
⋅=
8.13
)1ln(
8.1360
Reverberation time equations
)1ln(
3.55
mSc
V
α−⋅⋅−
⋅
m
x
c
l
α⋅
⋅8.13
mSc
V
α⋅⋅
⋅3.55
m
x
Uc
S
α⋅⋅
⋅4.43
)1ln(
4.43
m
x
Uc
S
α−⋅⋅−
⋅
)1ln(
8.13
m
x
c
l
α−⋅−
⋅
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EuroRegio 2010 25
Eyring and Sabine equations (3D)
)1ln(
3.5560
mSc
VT
α−⋅⋅−
⋅=Eyring:
Sabine:mSc
VT
α⋅⋅
⋅=
3.5560
Approximately the same as Eyring for αm < 0,3
Better when α is different for different surfaces
More correct for very high absorption, αm → 1
Special cases for 3D diffuse field and even distribution of absorption
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Non-diffuse rooms
Example: Rectangular room
Direction lm (m) αm T60 (s)
3-dim. (Sabine)
3-dim. (Eyring)
2-dim. (horizontal)
1-dim. (length)
1-dim. (width)
1-dim. (height)
5.7
5.7
10.5
20
10
5
0.30
0.30
0.10
0.10
0.10
0.45
0.76
0.63
4.21
8.02
4.01
0.45
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The scattering coefficient, s
(1-s)1
s
Ratio of reflected energy in
non-specular directions
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Simulation with ray tracing
Low scattering:s = 0.01
T30 = 1,69 s @ 1 kHz
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Simulation with ray tracing
s = 0.05
T30 = 1,40 s @ 1 kHz
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Simulation with ray tracing
s = 0.50
T30 = 0,71 s @ 1 kHz
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Simulation with ray tracing
s = 1.00
T30 = 0,71 s @ 1 kHz
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Non-diffuse rooms
• The sound decay in a room is a complicated mixture of the decay of 1-, 2-, and 3-dimensional modes
• With uneven distribution of absorption the degree of scattering is very important
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33
Finite size single reflectors
Reflector
S
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21
21* 2
aa
aaa
+=
Characteristic distance
34
Kirchhoff-Fresnel approximation
( ) Asnrnrs
Qj
A
srjk
d),cos(),cos(e
8
)(
∫ −=Φ+−
πλ
Aaa
Qj
A
srjkde
cos
4
)(
21
∫+−≅Φ
θ
πλ
Dimensions of the surface << a1 and a2
Coordinate system has Origo in the point of geometrical reflection
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35
Transformation of variables
ηθλ
ξλ
⋅⋅
+=
⋅
+=
cos112
112
21
21
aav
aau
( )jNMaa
Qj
aajk
−+
≅Φ+−
21
)( 21e
8π
Rectangular aperture
∫∫−−
=−2
1
22
1
2
dd 22
v
v
vju
u
uj
veuejNM
ππ
( )[ ] ( )[ ])()()()()()()()( 12121212 vSvSjvCvCuSuSjuCuCjNM −−−⋅−−−=−
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Cornu’s spiral
Result for an infinite large surface:
This is taken as the reference for the attenuation due to size
( ) zzvCv
dcos)(0
2
2∫= π ( ) zzvSv
dsin)(0
2
2∫= πThe Fresnel integrals:
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37
Rectangular reflector
( )21log10 KKLs =∆
Deviation from geometrical acoustics:
( ) ( )( )2
12
2
1221
1 )()()()( uSuSuCuCK −+−=
( ) ( )( )2
12
2
1221
2 )()()()( vSvSvCvCK −+−=
( ) θλ
cos22
*,1 bea
v i −=
θλ
cos2
*,2 ea
v i =
(Two orthogonal sections)
(corresponds to left edge of plate)
(corresponds to right edge of plate)
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Rectangular reflector
Low freq. � High freq.
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39
Reflection from finite size surface
Attenuation below a limiting frequency due to diffraction
Angle of incidence = 30°, a1 = 3.0 m, a2 = 3.0 m
Frequency, Hz
theory
measurement
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Attenuation due to size – simplified model
c = 344 m/s is speed of sound
a* is characteristic distance
S is area of reflector
θ is angle of incidence
θcos2
*
S
acf g =
Frequency
Design frequency:
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41
Measured directivity of reflection
Plane surface
Diffusing surface
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Ref.: M. Kleiner (1996)
Scattering due to finite size
Scattering due to finite size
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2
max,
sinsin
⋅=
Y
Y
X
X
I
I
r
r( )( )0
0
coscos
coscos
ββ
αα
−=
−=
kbY
kaX
Reflected sound intensity in the considered direction
Incident sound
Reflected sound
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43
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 30 60 90 120 150 180
Angle of radiation
ka = 1/4
ka = 1/2
ka = 1
ka = 2
ka = 4
ka = 8
ka = 16
ka = 32
Lambert
0
30
60
90
120
150
180
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Directivity for angle of incidence α0 = 90°
1/1 octave band
44
0
30
60
90
120
150
180
0
0,1
0,2
0,3
0,4
0,5
0,6
0,7
0,8
0,9
1
0 30 60 90 120 150 180
Angle of radiation
ka = 1/4
ka = 1/2
ka = 1
ka = 2
ka = 4
ka = 8
ka = 16
ka = 32
Lambert
Directivity for angle of incidence α0 = 30°
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1/1 octave band Ref.: Rindel, BNAM (2004)
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45
Examples of dimensions and ka
2aka = 1/4
ka = 1/2
ka = 1
ka = 2
ka = 4
ka = 8
ka = 16
ka = 32
0,22 m 125 250 500 1000 2000 4000 8000
0,44 m 63 125 250 500 1000 2000 4000 8000
0,88 m 63 125 250 500 1000 2000 4000
1,75 m 63 125 250 500 1000 2000
3,50 m 63 125 250 500 1000
Frequency in HzPanel size
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Specular and diffuse reflections
Scattering coefficient s: The ratio between the acoustic energy reflected in non-speculardirections and the totally reflected acoustic energy
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Defined in
ISO 17497-1:2004
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EuroRegio 2010 47
Typical results of scattering coefficients
Ref.: CSTB, France
48
Reflection based scattering coefficient
)1()1(1 sdr SSS −⋅−−=
Energy which is not scattered
due to roughness
Energy which is not scattered
due to diffraction
Resulting specular fraction i.e. not scattered due to roughness or diffraction
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49
Scattering coefficient due to
diffraction, sd
Two cutoff frequencies defined from length and width of panel, and distance from source.
Scattered energy
Attenuated specular reflection
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Reflection based scattering
Scattering depends on:
• size of reflecting surface
• distance from the source
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5151
Room Geometry and scattering
• In a detailed model the
scattering comes
automatically from the
geometry
• In a simplified model
the scattering
coefficients must be set
by the user
Odeon©1985-2008 Licensed to: Odeon A/S
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Simple model without scattering
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53
Simple model with scattering
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Detailed model
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55
Curved reflectors
plane convex concave
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Geometrical analysis
Source
Image source
Receiver
Image Receiver
+
+−=∆
121
2
d)(
d)(log10
β
β
aa
aaLk
θβθβϕ cos/dcos/dd 11 ⋅=⋅=⋅ aaR
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57
Attenuation due to curvature
R > 0 (convex)
R < 0 (concave)
R < 0 (concave)
θcos1log10
*
R
aLk +−=∆
21
21* 2
aa
aaa
+=
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Computer modelling
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Schroeder (1970)
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59
Sound reflection and image sources
One surface Two surfaces1st and 2nd order image sourcs
Potential, but not valid image source
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Image source model
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61
Schroeder (1970)
Particle Tracing Model
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Particle tracing
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6363
P1
Odeon©1985-2004
Ray tracing – highlighted one ray from source point
Ray Tracing Method
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P1
Odeon©1985-2004
Ray tracing – highlighted one ray from source point
Ray Tracing Method
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6565
P1
Secondary sources created at all reflection points.Each source has time delay and frequency dependent strength according to Ray Tracing history
Ray Tracing Method combined with Visibility Check
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P1
Receiver point collects contributions from all visible secondary sources
Ray Tracing Method combined with Visibility Check
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P2
3
1
6767
α : absorption coefficients : scattering coefficient
receiver
sourceImage source
Image Source MethodFirst order reflection - Specular part of reflection: (1-α)(1-s)
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Image Source MethodFirst order reflection - Diffuse part of reflection : (1-α)s
P2
3
1
α : absorption coefficient
s : scattering coefficient
receiver
source
Many secondary sources
distributed over the reflecting surface
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6969
P2
3
1
Odeon©1985-2005
Reflection paths including 3rd order reflections
Reflectogram
Arrival time: 61.74 ms (0.00 ms rel. direct)
Level of: -6.61 dB (0.00 dB rel. direct)
Azimuth angle: 9.61°, elevation angle: 2.22°
Reflection: 0. order, 1. reflection of 30, source:2
time (seconds rel. direct sound)
0,120,110,10,090,080,070,060,050,040,030,020,010
SP
L (
dB
)
-10
-15
-20
-25
-30
-35
-40
-45
-50
-55
-60
-65
Elevation
-20
-40
-60
-20
-40
-60
-50
-50
Azimuth
-20
-40
-60
-20
-40
-60
-50
-50
Frequency (Hz)
63 250 2000
-7
-8
Odeon©1985-2005
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Head Related Transfer Function (HRTF)
Example: Sound incident from the left
Time
Frequency
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Ref.: D. Hammershøj (1993)
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7171
AuralisationAuralisation
Anechoic recording, e.g. a trumpet
Result of the convolution
Binaural room impulse response from simulation
Left ear
time (seconds incl. filter delay)
21,81,61,41,210,80,60,40,20
p (
%)
100
50
0
-50
-100
Right ear
time (seconds incl. filter delay)
21,81,61,41,210,80,60,40,20
p (
%)
100
50
0
-50
-100
Odeon©1985-2008 Licensed to: Odeon A/S
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AuralisationAuralisation
Anechoic recording, e.g.
a trumpet
Result of the convolution
Binaural room impulse response from simulation
Left ear
time (seconds incl. filter delay)
21,81,61,41,210,80,60,40,20
p (
%)
100
50
0
-50
-100
Right ear
time (seconds incl. filter delay)
21,81,61,41,210,80,60,40,20
p (
%)
100
50
0
-50
-100
Odeon©1985-2008 Licensed to: Odeon A/S
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7373
AuralisationAuralisation
Anechoic recording, e.g. a trumpet
Result of the convolution
Binaural room impulse response from simulation
Left ear
time (seconds incl. filter delay)
21,81,61,41,210,80,60,40,20
p (
%)
100
50
0
-50
-100
Right ear
time (seconds incl. filter delay)
21,81,61,41,210,80,60,40,20
p (
%)
100
50
0
-50
-100
Odeon©1985-2008 Licensed to: Odeon A/S
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AuralisationAuralisation
Anechoic recording, e.g.
a trumpet
Result of the convolution
Binaural room impulse response from simulation
Left ear
time (seconds incl. filter delay)
21,81,61,41,210,80,60,40,20
p (
%)
100
50
0
-50
-100
Right ear
time (seconds incl. filter delay)
21,81,61,41,210,80,60,40,20
p (
%)
100
50
0
-50
-100
Odeon©1985-2008 Licensed to: Odeon A/S
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Speech in rooms
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Vocal communication and ambient noise
The Lombard effect
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Ambient noise level, dB(A)
Sp
ee
ch
leve
l (1
m)
dB
(A)
Relaxed
Normal
Raised
Loud
Very loud
Shouting
Vocal effort, ISO 9921:2003
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Theoretical model
Ambient noise level from speech, assuming a diffuse sound field, equivalent absorption area A, and number of persons speaking at the same time NS:
77EuroRegio 2010
(dB) log1045691
1,
−⋅−⋅
−=
S
ANN
Ac
cL
Where c is the Lombard slope. With c = 0.5 dB/dB we get:
(dB) log2093,
−=
S
ANN
AL
NB: Double A => - 6 dB. Double NS => + 6 dB.
Food court, V = 3133 m3, T = 0.9 s
Measurements: Navarro & Pimentel (2007), Applied Acoustics 68, pp. 364-375Calculations: Rindel, accepted for Applied Acoustics (2010)
60
65
70
75
80
85
100 200 300 400 500 600
Nois
ele
vel, d
B(A
)
Number of people
Calculated
Measured
g = 3 persons per
speaking person
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Noise level and speech level
20
30
40
50
60
70
80
90
100
110
120
1 10 100 1000
Absorption are a (m2) / Numbe r of spe aking pe rsons
dB (A)
Nois e level
S peech level, 1 m
Parameter c= 0,5 dB/dB
79EuroRegio 2010
Open plan office – new parameters
• ISO/DIS 3382-3 Acoustics — Measurement of room acoustic parameters — Part 3: Open plan spaces
• Spatial sound distribution of STI (Speech Transmission Index)
• Distraction distance: rD (distance from a speaker where STI falls below 0,50)
• Privacy distance: rP (distance from a speaker where STI falls below 0,20)
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Open plan office – new parameters
With background noise 35 dBA
Distraction distance: rD = 9.9 m
Privacy distance: rP = 21.8 m
Ref.: Vironen et al. (2009)
81EuroRegio 2010
Music in rooms
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Room acoustic parameters
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Subjective listener aspect Acoustic quantity
Subjective level of sound Sound Strength, G, in dB
Perceived reverberance Early Decay Time, EDT, in s
Perceived clarity of sound
Clarity, C80, in dB
Definition, D
Centre Time, TS, in ms
Apparent Source Width, ASW Early Lateral Energy Fraction, LF
Listener Envelopment, LEVLate Lateral Sound Level, LG, in dB
Inter Aural Cross Correlation, IACC
Measurement method: ISO 3382-1:2009
8484
DR Concert Hall, Copenhagen
Architect:
Jean Nouvel
Acoustics:
Toyota,
Nagata Acoustics
1800 seats,
28.000 m3
RT = 2,0 s @ 1 kHz
(with audience)
Opened Jan. 2009
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43
85
Musical instruments for Mahler’s
1st SymphonyInstrument Number of
sources
Number of
Recordings
Directional
characteristic
1st violin 16 2 Violin
2nd violin 14 2 Violin
Viola 12 1 Violin
Cello 10 1 Omni
Double bass 8 1 Omni
Flute 4 2 Omni
Oboe 4 4 B-Clarinet
Clarinet 5 4 B-Clarinet
Bassoon 3 3 B-Clarinet
French horn 7 7 French horn
Trumpet 4 4 Trumpet
Trombone 3 3 Trumpet
Tuba 1 1 Omni
Percussion 4 4 Omni
Total 95 39 -
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Computer model
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Odeon©1985-2008 Licensed to: Odeon A/S
Orchestra setup with 95 sources
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8787
Tutti – Position R2
Odeon©1985-2008 Licensed to: Odeon A/S
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Brass only – Position R8
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89
Acknowledgementsto orchestra simulation
• The anechoic recordings of the Mahler Symphony were
made by Jukka Pätynen, Ville Pulkki, and Tapio Lokki
from Helsinki University of Technology with musicians
from various Finnish orchestras
• The directional characteristics were measured by Felipe
Otondo (DOREMI project)
• The Odeon model of the concert hall was delivered by
Dr. A.C. Gade
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Conclusion
• Particle tracing and ray tracing are very efficient for room acoustic simulations, if combined with wave-based models for scattering and diffraction
• There is a need for good wave-based simulation models for small rooms and low frequencies (promising results have appeared with the finite difference time domain method, FDTD)
EuroRegio 2010 90
