Acoustics Sound Field Calculations

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


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back

introduction


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



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


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back

Kirchhoff - Helmholtzintegral


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



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)


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



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


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back



introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back


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


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back

Rayleigh Integral


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



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


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back

Rayleigh Integral

I replace the effect of the wall by a mirror source

I oscillating piston → pulsating piston


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



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!


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



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


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back



introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



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?


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back


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


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back


problemsexample: sound field of a plane wave behind an openingof 25 cm diameter

sound field behind opening with Kirchhoff’s aproximation


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back


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


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back


problemsFresnel zones in case of circular opening:


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back


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


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



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


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back

finite elements


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



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


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



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


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



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


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back

FDTD:finite differences in the time

domain


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



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


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back


2D-formulation:

vnewx = v old

x − α (pright − pleft)

pnew = pold − β (vxright − vx left)− β (vytop − vybottom)


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back


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


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back


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


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back

2-/3-D simulations

I mapping of 3-dimensional geometries onto 2independent coordinates:

I translation invariant situationI rotation invariant situation


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back

2-/3-D simulations


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



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??


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



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


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back


example: road traffic situation

road traffic noise situation


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back

finite diff. in the time domain (FDTD)example: Dreirosenbrucke, sound radiation through 50cm wide slits


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back


Dreirosenbrucke


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back

finite diff. in the time domain (FDTD)example: Hardbrucke, effect of absorbing layer at thebottom of bridge


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back


reflecting bridge:

Hardbrucke - reflecting


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back


absorbing bridge:

Hardbrucke - absorbing


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back

finite diff. in the time domain (FDTD)example: railway line cutting


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back


example: railway line cutting

E-Einschnitt-KB4KH8KW0KA0GL2


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back

acoustical holography


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back


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


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back


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


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back


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


introduction



Rayleigh Integral


finite elements

FDTD



Hardbrucke



back

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Acoustics Sound Field Calculations

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