Theory
Basic equations for room acoustic simulations
ELEC-E5630
Acoustic seminar
February 22, 2016
Pierre Chobeau
Variables of interest
•
Pressure (scalar) in Pascal (Pa) equivalent to Newton per square
meter (N.m-2):
, , , ,
•
Particle Velocity (vector) in meter per second (m.s-1):
, , , ,
•
Velocity potential (scalar):
,
.
Example: =1 Pa ~ 94 dB SPL, with =2.42 mm/s at 20°C.
Variation in air pressure
www.physics.byu.edu
www.acs.psu.edu/drussell
Sound level meter calibration
www.bksv.com
- 3D Cartesian coordinate system: , , , ,
- time-domain: in seconds (s),
- frequency-domain: in Hertz (Hz) or in radian per second (rad.s-1),
- wavenumber-domain: in (rad.m-1).
Room acoustic systems:
February 22, 2016
Acoustic seminar
ELEC-E5630
1/4
The conservation equations
• Conservation of mass:
+ .
0,
(1)
• Consevation of momentum (Euler eq.):
+
0,
(2)
• Equation of state(adiabatic sound speed):
! " , with!
'
,
χ)
Variation in air pressure and corresponding waveform
www.mediacollege.com
SATP conditions (T=25°C and patm=1bar):
c ~ 343 m.s-1
February 22, 2016
Acoustic seminar
ELEC-E5630
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Time-Domain wave equation
Time-domain wave
equation:
• From the consevation equations (1) and (2):
*+ , -,.,/,0
*0+
1+ 2+ , -, ., /, 0
3. (3)
General solution in 1D for , -, 0 :
, -, 0
,4 - 10 + ,5 - 6 10
• Finite-Difference Time-Domain (FDTD),
• Finite-Volume Time-Domain (FVTD),
• ...
FDTD model of the Elmia concert hall.
[Sheaffer and Fazenda-ForumAc2014]
February 22, 2016
Acoustic seminar
ELEC-E5630
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Frequency-Domain wave equation
Frequency-domain
wave (Helmholtz)
equation:
2+ P -, ., / 6 7+ P -, ., /
3.
(4)
Assuming an harmonic behavior for the solution:
, -, ., /, 0
83 9:; <0=7--=7..=7// ,
where
"
>
"
6
?
"
6
@
"
A "
• Finite-Element Method (FEM),
• Boundary-Element Method (BEM),
• ...
.
FEM calculation of modes in the control room of a recording studio.
www.akustik.rwth-aachen.de
February 22, 2016
Acoustic seminar
ELEC-E5630
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What is next?
Next lecture on Monday, 29 February:
- Finite-Difference Time-Domain method,
- Finite-Volume Time-Domain method.
February 22, 2016
Acoustic seminar
ELEC-E5630