Acoustics I: sound eld calculations - ETH Zisistaff/courses/ak1/acoustics-sound-field... ·...
Transcript of Acoustics I: sound eld calculations - ETH Zisistaff/courses/ak1/acoustics-sound-field... ·...
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
Acoustics I:sound field calculations
Kurt Heutschi2013-01-25
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
introduction
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
introductionsound field calculations:
I calculation of a situation specific location- and timedependency of sound field variable (often p)
I conditions for a valid solution:I fulfillment of the wave equation or Helmholtz
equationI fulfillment of the boundary conditions:
I sourcesI boundaries (borders of space)
I analytical solutions only for special cases
I often simplifications and approximations
I numerical solutions for the general case:I finite elementsI boundary elementsI time domain methods such as FDTD
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
Kirchhoff - Helmholtzintegral
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
Kirchhoff - Helmholtz integral
Green’s theorem:Helmholtz equation y Kirchhoff - Helmholtz integral:
p(x , y , z , ω) =
=1
4π
∫S
(jωρ0vS(ω)
e−jωr/c
r+ pS(ω)
∂
∂n
e−jωr/c
r
)dS
S : closed surfacevS : sound particle velocity on and normal to SpS : sound pressure on Sr : distance of the surface point to the receiver point(x , y , z)
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
Kirchhoff - Helmholtz integral
I Kirchhoff-Helmholtz integral KHI is valid:I in the interior of SI in the exterior of SI on the surface S with a correction factor of 2
I Kirchhoff-Helmholtz integral → wave field synthesis
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
Boundary Elements Method
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
Boundary Elements Method
typical radiation problem:
I surface velocity is given as boundary condition
I search for sound pressure in the surroundings
I solution with the Boundary Elements Method:I discretisation of the radiator surface in n elementsI with KHI: pS,i =
∑nj=1 f (pS,j , vS,j)
I solve the system of equations with n unknowns →pS,i
I calculate sound pressure at any point with the KHI
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
Rayleigh Integral
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
Rayleigh Integral
I radiation of an oscillating piston →Kirchhoff-Helmholtz Integral
I special case: oscillating piston mounted in a hardwall
I wall introduces boundary condition: vn = 0
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
Rayleigh Integral
I replace the effect of the wall by a mirror source
I oscillating piston → pulsating piston
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
Rayleigh Integral
evaluation of the Kirchhoff Helmholtz Integral:
p(x , y , z , ω) =
=1
4π
∫S
(jωρ0vS(ω)
e−jωr/c
r+ pS(ω)
∂
∂n
e−jωr/c
r
)dS
contribution of sound pressure = 0!
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
Rayleigh Integral
Kirchhoff Helmholtz Integral simplifies to theRayleigh Integral:
p(x , y , z , ω) =jωρ0
2π
∫S
vn(x , y , ω)e−jkr
rdS
S : ”visible” piston surface (front)vn: piston velocity
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
Kirchhoff’s approximations
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
Kirchhoff’s approximations: diffraction
problemsI screen with opening:
I plane wave hits the opening in a hard screenI sound pressure field behind the screen?
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
Kirchhoff’s approximations: diffraction
problems
I solution: application of the Rayleigh IntegralI necessary: sound particle velocity in the openingI Kirchhoff’s approximation:
I assume sound particle velocity as if no screenwere present
I → ignore boundariesI error decreases with decreasing ratio wavelength
/ diameter
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
Kirchhoff’s approximations: diffraction
problemsexample: sound field of a plane wave behind an openingof 25 cm diameter
sound field behind opening with Kirchhoff’s aproximation
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
Kirchhoff’s approximations: diffraction
problems
Rayleigh Integral:
p(x , y , z , ω) =jωρ0
2π
∫S
vn(x , y , ω)e−jkr
rdS
I approximation with Fresnel zones for receivers nottoo close:
I ignore small changes of rI distinguish phase in classes + (0 degrees) and -
(180 degrees) onlyI corresponding regions in the opening: Fresnel
zones
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
Kirchhoff’s approximations: diffraction
problemsFresnel zones in case of circular opening:
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
Kirchhoff’s approximations: diffraction
problems
p ∼ A1
r1− A2
r2+
A3
r3− A4
r4. . .
Ai : area of the i-th Fresnel zoneri : average distance to the i-th Fresnel zone
for large openings:
p ∼ A1
2r1
if opening = 1. Fresnel zone → amplification of +6 dBre. free field
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
Fresnel zones for reflection problems
I reflection at inhomogeneous or finite surfaces:I half of the 1. Fresnel zone defines the relevant
region on a reflectorI concept allows for the estimation of situations
with:I small reflectors F (prefl ≈ 2F
A1prefl∞)
I inhomogeneous reflectors
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
finite elements
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
finite elements
I common method to solve differential equations bydiscretization of the field volume
I well suited for:I bounded field regions such as vehicle interiorsI coupled structure/fluid systems, e.g. simulation of
airborne sound insulation in the laboratoryI simulation of inhomogeneous properties of the
medium (c, density)
I not well suited for:I radiation in unbounded space
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
finite elements
I starting point: Helmholtz equation with givenboundary conditions on the surface S
I approach: approximative solution for the sound fieldp′
I measurement of the quality of p′ by residues:I RV deviation of the Helmholtz equation in the
simulation volumeI RS deviation of boundary conditions on the
surface of S
I solution requirement:∫V
WRVdV +∫S
WRSdS = 0
I W : suitable weighting function
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
finite elements
I discretization of the field volume in finite elements
I establish one equation per element and node
I Assembly of the system of equations
I solve the system of equation for each frequency ofinterest
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
FDTD:finite differences in the time
domain
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
finite diff. in the time domain (FDTD)
I standard method to find solutions of differentialequations numerically
I usage of the fundamental acoustical partialdifferential equations in the time domain:
I grad(p) = −ρ∂~v∂t
I −∂p∂t = κP0div(~v)
I strategy:I discretization in space and timeI replacement of derivatives by differences of
neighbor points (space and time)I → updating equations in time
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
finite diff. in the time domain (FDTD)
2D-formulation:
vnewx = v old
x − α (pright − pleft)
pnew = pold − β (vxright − vx left)− β (vytop − vybottom)
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
finite diff. in the time domain (FDTD)
I typical calculation:I impulse-like pressure distribution as starting
conditionI time-stepwise updating of the field variables at the
grid points
I advantage: no system of equation that has to besolved, impulse response as a result containsinformation about all frequencies
I disadvantage: implementation of boundaryconditions is not straight forward
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
finite diff. in the time domain (FDTD)
I computational effort:I 2D-simulation of a region of 200 m × 40 mI fmax = 2 kHz → discretization in space: 0.02 mI mesh size 10’000 × 2’000 = 20·106 grid pointsI calculation time → a few hours
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
2-/3-D simulations
I mapping of 3-dimensional geometries onto 2independent coordinates:
I translation invariant situationI rotation invariant situation
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
2-/3-D simulations
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
2-/3-D simulations
I translation invariant situationI cartesian coordinate systemI situation geometry does not change in y -directionI all derivatives of the sound field equations in
y -direction are set to 0I simulated source = coherent line source with
extension in y -directionI coherent - incoherent line source??
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
2-/3-D simulations
I rotation invariant situationI cylindrical coordinate systemI situation geometry does not change with angle φI all derivatives of the sound field equations inφ-direction are set to 0
I simulated source = point source in the originI caution: reflections lead to focusing effects at the
source position → only strictly propagating wavesallowed
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
finite diff. in the time domain (FDTD)
example: road traffic situation
road traffic noise situation
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
finite diff. in the time domain (FDTD)example: Dreirosenbrucke, sound radiation through 50cm wide slits
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
finite diff. in the time domain (FDTD)
Dreirosenbrucke
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
finite diff. in the time domain (FDTD)example: Hardbrucke, effect of absorbing layer at thebottom of bridge
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
finite diff. in the time domain (FDTD)
reflecting bridge:
Hardbrucke - reflecting
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
finite diff. in the time domain (FDTD)
absorbing bridge:
Hardbrucke - absorbing
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
finite diff. in the time domain (FDTD)example: railway line cutting
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
finite diff. in the time domain (FDTD)
example: railway line cutting
E-Einschnitt-KB4KH8KW0KA0GL2
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
acoustical holography
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
acoustical holography
I Kirchhoff-Helmholtz integral is valid for all kind ofsurfaces
p(x , y , z , ω) =
=1
4π
∫S
(jωρ0vS(ω)
e−jωr/c
r+ pS(ω)
∂
∂n
e−jωr/c
r
)dS
I further simplifications are possible for specialsurfaces
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
acoustical holography
for a plane S that closes in infinity
sound pressure in the right half space is given as:
p(x , y , z , ω) = j
∫S
pS(ω) cosφ
(1− j
kr
)e−jkr
λrdS
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
acoustical holography
I equation from above describes p in 3D-space by a prepresentation on a 2D-plane
I → principle of holography
I holography in practical applications:I simultaneous determination of sound pressure
distribution (amplitude and phase) at discrete gridpoints on a suitable plane
I usage of microphone arraysI sequential sampling by using a fixed reference
(phase)I → complete information about the 3D field
Acoustics: SoundField Calculations
introduction
Kirchhoff -Helmholtz integral
Boundary Elements Method
Rayleigh Integral
Kirchhoff’s approximations
finite elements
FDTD
example: road trafficsituation
example: Dreirosenbrucke
Hardbrucke
example: railway linecutting
acousticalholography
back
eth-acoustics-1