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Spatial Sound Synthesis with Loudspeakers

Spatial Sound Synthesis with Loudspeakers

Talk I gave at the EAA Winter School 2013 in Meran, Italy on the theoretical basis of Sound Field Synthesis techniques like higher-order Ambisonics (HOA) and Wave Field Synthesis (WFS).

Sascha Spors

March 16, 2013
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  1. Spatial Sound Synthesis
    with Loudspeakers
    Sascha Spors 1 and Franz Zotter 2
    1Universität Rostock, Institute of Communications Engineering
    2University of Music and Performing Arts Graz,
    Institute of Electronic Music and Acoustics

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  2. Problem Statement
    Perfect synthesis of the virtual source s(x, t) within the listening area V
    by loudspeakers (secondary sources) placed on the border ∂V
    s(x, t) ∂V
    V
    virtual
    source
    • assumed to result in authentic reproduction of original scene
    • frequency dependent weighting of secondary sources ⇒ driving signal
    Spors, Zotter | Sound Field Synthesis | Overview 1

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  3. Overview
    [from W. Snow, Basic Principles of Stereophonic Sound, 1955]
    • thorough review of the underlying physical problem
    • Higher-Order Ambisonics and Wave Field Synthesis
    Spors, Zotter | Sound Field Synthesis | Overview 2

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  4. Conventions
    Fourier-Transformation with respect to time t
    S(ω, x) =

    −∞
    s(t, x)e−iωt dt
    Fourier-Transformation with respect to position vector x = [x y z]T
    ˜
    S(ω, k) =

    −∞
    S(ω, x)e+i k,x dxdydz
    Example: 3D free-field Green’s function (acoustic point source)
    g0,3D
    (x|x0
    , t) =
    1

    δ(t − x−x0
    c
    )
    x − x0
    G0,3D
    (x|x0
    , ω) =
    1

    e−i ω
    c
    x−x0
    x − x0
    ˜
    G0,3D
    (k|x0
    , ω) =
    1
    k 2 − (ω
    c
    )2
    ei k,x0
    Caution: Different definitions of the spatio-temporal Fourier-Transformation
    in the literature
    Spors, Zotter | Sound Field Synthesis | Overview 3

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  5. The Kirchhoff-Helmholtz Integral
    The Kirchhoff-Helmholtz integral provides the solution of the interior problem
    for a bounded source free region V with smooth boundary ∂V
    ∂V
    P(x0
    , ω)
    ∂G(x|x0
    , ω)
    ∂(n, x0
    )
    − G(x|x0
    , ω)
    ∂P(x, ω)
    ∂(n, x)
    x=x0
    dS0
    =





    P(x, ω) x ∈ V
    1
    2
    P(x, ω) x ∈ ∂V
    0 x ∈ ¯
    V
    0
    ∂V
    V
    ¯
    V
    n
    x
    x0
    G(x|x0
    , ω)
    P(x, ω)
    Spors, Zotter | Sound Field Synthesis | Theoretical Basis 4

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  6. The Kirchhoff-Helmholtz Integral
    The Kirchhoff-Helmholtz integral provides the solution of the interior problem
    for a bounded source free region V with smooth boundary ∂V
    ∂V
    P(x0
    , ω)
    ∂G(x|x0
    , ω)
    ∂(n, x0
    )
    − G(x|x0
    , ω)
    ∂P(x, ω)
    ∂(n, x)
    x=x0
    dS0
    =





    P(x, ω) x ∈ V
    1
    2
    P(x, ω) x ∈ ∂V
    0 x ∈ ¯
    V
    Definition of the directional derivative
    ∂P(x, ω)
    ∂(n, x)
    x=x0
    = ∇x
    P(x, ω), n(x)
    x=x0
    External problem
    • exchange results for V and ¯
    V on right hand side
    • reverse normal vector n or reverse signs on the right hand side
    Spors, Zotter | Sound Field Synthesis | Theoretical Basis 4

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  7. Sound Field Synthesis
    The field of the primary source S(x, ω) within the region V is given by its pressure and
    pressure gradient on the boundary ∂V
    ∂V
    S(x0
    , ω)
    Ddipole(x0,ω)
    ∂G(x|x0
    , ω)
    ∂(n, x0
    )
    − G(x|x0
    , ω)
    ∂S(x, ω)
    ∂(n, x)
    x=x0
    Dmonopole(x0,ω)
    dS0
    = S(x, ω) x ∈ V
    0
    S(x, ω) ∂V
    V
    ¯
    V
    virtual
    source
    n
    x
    x0
    G(x|x0
    , ω)
    P(x, ω)
    Spors, Zotter | Sound Field Synthesis | Theoretical Basis 5

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  8. Synthesis with Secondary Monopoles
    P(x, ω) =
    ∂V
    D(x0
    , ω)G(x|x0
    , ω)dS0
    • aim P(x, ω) = S(x, ω) for x ∈ V
    • solution of integral equation with respect to driving signal D(x0
    , ω)
    Theretical basis
    • acoustic scattering and radiation
    • Boundary Element Method (BEM)
    • Sound Field Synthesis (SFS)
    Spors, Zotter | Sound Field Synthesis | Theoretical Basis 6

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  9. Overview – Single Layer Solutions
    1. Equivalent Scattering Approach
    2. explicit solution of the single layer potential
    3. high-frequency approximation of the Kirchhoff-Helmholtz integral
    Spors, Zotter | Sound Field Synthesis | Equivalent Scattering Approach 7

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  10. Equivalent Scattering Approach
    Scattering of the virtual source S(x, ω) at a scatterer with boundary ∂V
    0
    S(x, ω) ∂V
    V
    ¯
    V
    virtual
    source
    n
    x
    x0
    G(x|x0
    , ω)
    P(x, ω)
    • incident field (virtual source) is not affected by the presence of the scatterer
    • scattered sound field Psc
    (x, ω)
    • total sound field Pt
    (x, ω) = S(x, ω) + Psc
    (x, ω)
    Spors, Zotter | Sound Field Synthesis | Equivalent Scattering Approach 8

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  11. Equivalent Scattering Approach (cont’d)
    The total sound field is then given as [Williams, eq.(8.37)]
    S(x, ω) +
    ∂V
    G(x|x0
    , ω)
    ∂Pt
    (x, ω)
    ∂(n, x)
    x=x0
    − Pt
    (x0
    , ω)
    ∂G(x|x0
    , ω)
    ∂(n, x0
    )
    dS0
    =





    0 x ∈ V
    1
    2
    Pt
    (x, ω) x ∈ ∂V
    Pt
    (x, ω) x ∈ ¯
    V
    Pressure release (Dirichlet) boundary ∂V ⇒ Pt
    (x, ω) = 0 for x ∈ ∂V
    ∂V
    G(x|x0
    , ω)
    ∂Pt
    (x, ω)
    ∂(n, x)
    x=x0
    −D(x0,ω)
    dS0
    =





    −S(x, ω) x ∈ V
    −S(x, ω) x ∈ ∂V
    Psc
    (x, ω) x ∈ ¯
    V
    • accurate synthesis of virtual source within listening area V
    • driving function is given by directional derivative of total sound field Pt
    • requires knowledge of the scattered field Psc
    (x, ω)
    Spors, Zotter | Sound Field Synthesis | Equivalent Scattering Approach 9

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  12. Spherical Secondary Source Distribution
    Plane wave with traveling direction (βpw
    , αpw
    ) as virtual source
    Spw
    (x, ω) = 4π

    n=0
    n
    m=−n
    (−i)njn
    (
    ω
    c
    r)Ym
    n
    (βpw
    , αpw
    )∗Ym
    n
    (β, α)
    Scattering of plane wave at sphere with pressure release boundaries [Gumerov et al. 2004]
    Psc,pw
    (x, ω) = −4π

    n=0
    n
    m=−n
    (−i)n
    jn

    c
    R)
    h(2)
    n

    c
    R)
    h(2)
    n
    (
    ω
    c
    r)Ym
    n
    (βpw
    , αpw
    )∗Ym
    n
    (β, α)
    Calculation of driving function
    • superposition of virtual source and scattered sound field
    • directional derivative with inward pointing radial unit vector
    ⇒ Analytic (Near-Field Compensated) Higher-Order Ambisonics [Ahrens et al. 2008]
    Spors, Zotter | Sound Field Synthesis | Equivalent Scattering Approach 10

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  13. Example – Equivalent Scattering Approach
    Scattered sound field Psc,pw
    (x, ω)
    x (m)
    y (m)
    −3 −2 −1 0 1 2 3
    −3
    −2
    −1
    0
    1
    2
    3
    3D ESA, f = 500 Hz, βpw = 90o, αpw = 90o, R = 1.5 m, z = 0
    Spors, Zotter | Sound Field Synthesis | Equivalent Scattering Approach 11

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  14. Example – Equivalent Scattering Approach
    Scattered and incident sound field Psc,pw
    (x, ω) + Spw
    (x, ω)
    x (m)
    y (m)
    −3 −2 −1 0 1 2 3
    −3
    −2
    −1
    0
    1
    2
    3
    3D ESA, f = 500 Hz, βpw = 90o, αpw = 90o, R = 1.5 m, z = 0
    Spors, Zotter | Sound Field Synthesis | Equivalent Scattering Approach 11

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  15. Example – Equivalent Scattering Approach
    Synthesized sound field P(x, ω)
    x (m)
    y (m)
    −3 −2 −1 0 1 2 3
    −3
    −2
    −1
    0
    1
    2
    3
    3D ESA, f = 500 Hz, βpw = 90o, αpw = 90o, R = 1.5 m, z = 0
    Spors, Zotter | Sound Field Synthesis | Equivalent Scattering Approach 11

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  16. Is the Solution Unique?
    The scattering problem could degenerate to a trivial problem if S(x, ω) = 0 for x ∈ ∂V
    ∂V
    D(x0
    , ω)G(x|x0
    , ω)dS0
    = 0
    Uniqueness of solution can be ensured by additional constraints
    • CHIEF point method [Schenk 1968]
    • Burton-Miller method [Burton et al. 1971]
    • define pressure on symmetry axis of rotationally symmetric problems
    e.g. analytic NFC-HOA [Ahrens et al. 2008]
    Spors, Zotter | Sound Field Synthesis | Equivalent Scattering Approach 12

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  17. Overview – Single Layer Solutions
    1. Equivalent Scattering Approach
    2. explicit solution of the single layer potential
    3. high-frequency approximation of the Kirchhoff-Helmholtz integral
    Spors, Zotter | Sound Field Synthesis | Explicit Solution 13

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  18. Explicit Solution of the Single Layer Potential
    Synthesis equation
    P(x, ω) =
    ∂V
    D(x0
    , ω)G(x|x0
    , ω)dS0
    • Fredholm operator of index zero
    • solution by expansion into series of orthogonal basis functions
    Expansion of Green’s function
    G(x|x0
    , ω) =
    N
    n=1
    ˜
    G(n, ω) ¯
    ψn
    (x0
    ) ⊗ ψn
    (x)
    Driving function
    D(x, ω) =
    N
    n=1
    ˜
    S(n, ω)
    ˜
    G(n, ω)
    ψn
    (x)
    Spors, Zotter | Sound Field Synthesis | Explicit Solution 14

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  19. Overview – Single Layer Solutions
    1. Equivalent Scattering Approach
    2. explicit solution of the single layer potential
    3. high-frequency approximation of the Kirchhoff-Helmholtz integral
    Spors, Zotter | Sound Field Synthesis | High-Frequency Approximation 15

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  20. High-Frequency Approximation
    Introducing the 3D free-field Green’s function and its directional derivative into the
    Kirchhoff-Helmholtz integral results in
    ∂V
    S(x0
    , ω) ·
    1 + j ω
    c
    r
    r
    · cos φ −
    ∂S(x, ω)
    ∂(n, x)
    x=x0
    G(x|x0
    , ω) dS0
    = S(x, ω)
    for x ∈ V
    0
    S(x, ω) ∂V
    V
    ¯
    V
    virtual
    source
    n
    x
    x0
    r
    P(x, ω)
    φ
    r = x − x0
    φ = ∠(n, (x − x0
    ))
    Spors, Zotter | Sound Field Synthesis | High-Frequency Approximation 16

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  21. High-Frequency Approximation
    Introducing the 3D free-field Green’s function and its directional derivative into the
    Kirchhoff-Helmholtz integral results in
    ∂V
    S(x0
    , ω) ·
    1 + j ω
    c
    r
    r
    · cos φ −
    ∂S(x, ω)
    ∂(n, x)
    x=x0
    G(x|x0
    , ω) dS0
    = S(x, ω)
    for x ∈ V
    Stationary phase approximation
    1. (x − x0
    )||n ⇒ cos φ = 1
    2. ω
    c
    r ≫ 1 ⇒ 1+j ω
    c
    r
    r
    → j ω
    c
    ∂V
    j
    ω
    c
    S(x0
    , ω) −
    ∂S(x, ω)
    ∂(n, x)
    x=x0
    D(x0,ω)
    G(x|x0
    , ω) dS0
    ≈ S(x, ω)
    Approximation is known as the high-frequency BEM [Herrin et al. 2003]
    Spors, Zotter | Sound Field Synthesis | High-Frequency Approximation 16

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  22. Wave Field Synthesis
    Plane wave with traveling direction npw
    Spw
    (x, ω) = e−j ω
    c
    npw,x
    Driving function for the synthesis of a plane wave
    Dpw
    (x0
    , ω) = 1 + npw
    , n · j
    ω
    c
    · e−j ω
    c
    npw,x0
    Selection of secondary sources
    1 + npw
    , n ≈
    2 npw
    , n for npw
    , n > 0
    0 for npw
    , n < 0
    ∂V
    V
    n
    x0
    npw
    plane wave
    Spors, Zotter | Sound Field Synthesis | High-Frequency Approximation 17

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  23. Wave Field Synthesis
    Plane wave with traveling direction npw
    Spw
    (x, ω) = e−j ω
    c
    npw,x
    Driving function for the synthesis of a plane wave
    Dpw
    (x0
    , ω) = 1 + npw
    , n · j
    ω
    c
    · e−j ω
    c
    npw,x0
    Selection of secondary sources
    1 + npw
    , n ≈
    2 npw
    , n for npw
    , n > 0
    0 for npw
    , n < 0
    Driving function of 3D WFS [Spors et al. 2008]
    Dpw,3D
    (x, ω) = apw
    (x0
    ) · 2 npw
    , n · j
    ω
    c
    · e−j ω
    c
    npw,x
    • same procedure for a virtual point source
    • WFS constitutes a high-frequency approximation of the BEM
    Spors, Zotter | Sound Field Synthesis | High-Frequency Approximation 17

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  24. Example – Wave Field Synthesis
    Synthesized sound field P(x, ω)
    x (m)
    y (m)
    −3 −2 −1 0 1 2 3
    −3
    −2
    −1
    0
    1
    2
    3
    3D WFS, f = 500 Hz, βpw = 90o, αpw = 90o, R = 1.5 m, z = 0
    Spors, Zotter | Sound Field Synthesis | High-Frequency Approximation 18

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  25. Comparison of Approaches
    Equivalent Scattering Approach (e.g. Higher-order Ambisonics)
    • global dependency of driving function
    • analytic driving functions for regular geometries only
    • exact reproduction within entire listening area possible
    Approximation of Kirchhoff-Helmholtz Integral (e.g. Wave Field Synthesis)
    • local dependency of driving function
    • analytic driving functions for arbitrary (convex) geometries
    • exact reproduction only for planar systems,
    minor degradations for curved systems
    Spors, Zotter | Sound Field Synthesis | High-Frequency Approximation 19

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  26. Spatial Dimensionality
    The explicit form of the Green’s function depends on the dimensionality of the problem:
    Reproduction in a volume (3D)
    G0,3D
    (x|x0
    , ω) = 1

    e−j
    ω
    c
    |x−x0
    |
    |x−x0
    |
    ⇒ secondary point sources
    Reproduction in a plane (2D)
    G0,2D
    (x|x0
    , ω) = − j
    4
    H(2)
    0

    c
    |x − x0
    |)
    ⇒ secondary line sources
    2.5-Dimensional Synthesis
    • secondary point sources are used for the synthesis in a plane
    • amplitude deviations in the listening area V
    • minor spectral deviations
    Spors, Zotter | Sound Field Synthesis | Dimensionality 20

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  27. Spatially Discrete Secondary Source Distribution
    Secondary source distribution is spatially sampled in practice
    S(x, ω) ∂V
    V
    virtual
    source
    • high impact on synthesis with full audio-signal bandwidth
    • may (perceptually) degrade the synthesis of the virtual source
    ⇒ sampling has to be considered in the evaluation of novel approaches
    Spors, Zotter | Sound Field Synthesis | Spatial Sampling 21

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  28. Spatial Sampling of Secondary Source Distribution
    ˜
    G0
    (x, ω)
    D(x0
    , ω) DS
    (x0
    , ω) PS
    (x, ω)
    n ∆x
    spatial sampling
    • sampling of driving function, interpolation by secondary sources
    • description by spatio-temporal spectra of driving function/secondary sources
    • intuitive anti-aliasing condition ∆x < λ
    2
    does not hold in general
    Sampling for linear/circular/spherical boundaries [Spors et al. 2006, Ahrens et al. 2012]
    • repetitions (and overlap) of spatial spectrum of driving function
    • band-limitation of spatial spectrum improves sampling artifacts (e.g. HOA)
    Spors, Zotter | Sound Field Synthesis | Spatial Sampling 22

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  29. Comparison of 2.5-dimensional NFC-HOA/WFS
    Synthesis of a monofrequent plane wave (fpw
    = 500 Hz)
    NFC-HOA WFS
    (2.5D, R = 1.50 m, N = 56, αpw = 270o)
    Spors, Zotter | Sound Field Synthesis | Spatial Sampling 23

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  30. Comparison of 2.5-dimensional NFC-HOA/WFS
    Synthesis of a monofrequent plane wave (fpw
    = 3000 Hz)
    NFC-HOA WFS
    (2.5D, R = 1.50 m, N = 56, αpw = 270o)
    Spors, Zotter | Sound Field Synthesis | Spatial Sampling 23

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  31. Comparison of 2.5-dimensional NFC-HOA/WFS
    Synthesis of a broadband plane wave (spatial impulse response)
    NFC-HOA
    x -> (m)
    y -> (m)
    -2 -1 0 1 2
    -2
    -1
    0
    1
    2 dB
    -30
    -25
    -20
    -15
    -10
    -5
    0
    WFS
    x -> (m)
    y -> (m)
    -2 -1 0 1 2
    -2
    -1
    0
    1
    2 dB
    -30
    -25
    -20
    -15
    -10
    -5
    0
    (2.5D, R = 1.50 m, N = 56, αpw = 270o)
    Spors, Zotter | Sound Field Synthesis | Spatial Sampling 24

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  32. Comparison of 2.5-dimensional NFC-HOA/WFS
    Synthesis of a broadband plane wave (spatial impulse response)
    NFC-HOA
    x -> (m)
    y -> (m)
    -2 -1 0 1 2
    -2
    -1
    0
    1
    2 dB
    -30
    -25
    -20
    -15
    -10
    -5
    0
    WFS
    x -> (m)
    y -> (m)
    -2 -1 0 1 2
    -2
    -1
    0
    1
    2 dB
    -30
    -25
    -20
    -15
    -10
    -5
    0
    (2.5D, R = 1.50 m, N = 56, αpw = 270o)
    Spors, Zotter | Sound Field Synthesis | Spatial Sampling 24

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  33. Comparison of 2.5-dimensional NFC-HOA/WFS
    Synthesis of a broadband plane wave (spatial impulse response)
    NFC-HOA
    x -> (m)
    y -> (m)
    -2 -1 0 1 2
    -2
    -1
    0
    1
    2 dB
    -30
    -25
    -20
    -15
    -10
    -5
    0
    WFS
    x -> (m)
    y -> (m)
    -2 -1 0 1 2
    -2
    -1
    0
    1
    2 dB
    -30
    -25
    -20
    -15
    -10
    -5
    0
    (2.5D, R = 1.50 m, N = 56, αpw = 270o)
    Spors, Zotter | Sound Field Synthesis | Spatial Sampling 24

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  34. Conclusions
    • theory of sound field synthesis is well grounded
    • spatial sampling reduces achievable accuracy considerably
    • monochromatic sound fields are not sufficient for evaluation
    • perception of synthetic sound fields plays an inportant role
    Reproducible Reserach
    • software implementations have high impact on results
    • Sound Field Synthesis (SFS) Toolbox http://github.com/sfstoolbox
    • SoundScape Renderer (SSR) http://www.spatialaudio.net/ssr
    Spors, Zotter | Sound Field Synthesis | Conclusions 25

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  35. Thanks for your attention!
    www.spatialaudio.net

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  36. Simple Source Formulation [Williams]
    Interior Problem (n points inward)
    ∂V
    S(x0
    , ω)
    ∂G(x|x0
    , ω)
    ∂(n, x0
    )
    − G(x|x0
    , ω)
    ∂S(x, ω)
    ∂(n, x)
    x=x0
    dS0
    =





    S(x, ω) x ∈ V
    1
    2
    S(x, ω) x ∈ ∂V
    0 x ∈ ¯
    V
    Exterior Problem (n points inward)
    ∂V
    Psc
    (x0
    , ω)
    ∂G(x|x0
    , ω)
    ∂(n, x0
    )
    − G(x|x0
    , ω)
    ∂Psc
    (x, ω)
    ∂(n, x)
    x=x0
    dS0
    =





    0 x ∈ V
    −1
    2
    Psc
    (x, ω) x ∈ ∂V
    −Psc
    x ∈ ¯
    V
    (Interior)+(Exterior) together with S(x, ω) + Psc
    (x, ω) = 0 for x ∈ ∂V

    ∂V
    G(x|x0
    , ω)
    ∂(S(x, ω) + Psc
    (x, ω)
    ∂(n, x)
    x=x0
    dS0
    =





    S(x, ω) x ∈ V
    S(x, ω) x ∈ ∂V
    −Psc
    (x, ω) x ∈ ¯
    V
    Spors, Zotter | Sound Field Synthesis | Conclusions 26

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