Time domain models of edge diffraction

What is edge diffraction?

Edge diffraction is the sound field component which must be added to the geometrical acoustics sound field to achieve the correct total sound field, for scattering from objects that consist of surfaces that are plane and rigid.

How can it be used?

In computational room acoustics, to improve the accuracy in general at low frequencies, or for stage reflectors, near balcony edges, and in orchestra pits [Svensson, Medwin, Torres 2000], [Torres, Svensson, Kleiner 2001].

In studies of transducers (loudspeakers, microphones), in order to take the influence of the cabinet into account, [Vanderkooy 1991], [Svensson, Wendlandt 2000].

Scattering from rough surfaces, if these are modelled as deterministic "wedge assemblages" [Keiffer, Novarini 2000].

Noise barriers - the classical edge diffraction problem [Maekawa 1968], [Medwin 1981].

Geophysics: reflections from wedge-shaped discontinuities between layers of different media [Trorey 1977].

Underwater acoustics: both surface scattering and for wedge-shaped discontinuities as above.

What analysis methods are available?

Frequency-domain methods: a large number of studies have developed exact (Macdonald 1915) and approximate methods for infinite edges. Examples of the latter are the Geometrical Theory of Diffraction (GTD), [Keller 1962], and the Uniform Asymptotic Theory of Diffraction (UATD), [Ahluwalia et al 1968]. Others can be found in [Maekawa 1968], [Bowman et al 1969], [Pierce 1974], [Kouyomjian, Pathak 1974], [Cox, Lam 1993].

Time-domain methods: Biot-Tolstoy presented the time-domain solution for the infinite wedge in 1957. Later Medwin derived a "Huygens interpretation" from this, which could be extended to finite wedges and multiple diffraction [Medwin 1982]. Svensson et al then derived an analytical directivity function for secondary edge sources, which gives the exact impulse response for finite wedges, even curved ones [Svensson et al 1999].

A classical method is based on the Kirchhoff diffraction approximation, which can give elegant time-domain line integral expression for the edge diffraction [Sakurai, Nagata 1981]. It is often believed to be asymptotically correct for high frequencies. However, as shown by Jebsen and Medwin, the Kirchhoff diffraction approximation is not asymptotically correct and it can give large errors at high frequencies [Jebsen, Medwin 1982].

Matlab toolbox for the time domain models of edge diffraction

A Matlab toolbox is available as free-ware for non-commercial use under www.tele.ntnu.no/users/svensson/Matlab.html. It calculate impulse responses inclduing low-order specular reflections and edge diffraction for geometries defined in a simple CAD-format.

Example: Computational room acoustics

In room acoustics it is clear that geometrical acoustics models can work well at mid to high frequencies, especially if surface scattering is included, as shown by Round-Robin comparisons between computer predictions and measurements by M. Vorländer [Vorländer1995]. At low frequencies, however, geometrical acoustics is insufficient for accurate predictions. Also, near balcony edges and seat edges edge diffraction might be very significant also at higher frequencies as shown by R. Torres et al [Torres 1997] using auralization.
One example of the edge diffraction method in room acoustic impulse response prediction is shown below: a simple model of a stage house [taken from Torres et al 2001]. A single source is placed on the stage, and there is a lateral array of 100 receiver positions in front of the stage.

 

Impulse responses were calculated for these 100 receiver positions. The are plotted below, on top of each other but displaced vertically. Thereby the wavefronts are clearly visible. Results are shown for pure specular reflections, and with edge diffraction added. The impulse responses were low-pass filtered with a single Hanning window, corresponding to a cut-off frequency of approx. 350 Hz.

                            

When only specular reflections are included, typical truncated wavefronts result. This means that as we move the receiver position just past the point where the image source becomes invisible, the reflection suddenly disappears. These truncated wavefronts are clearly unphysical. When the edge diffraction components are added, the wavefronts become perfectly smooth and continuous. Furthermore, there are more wavefield components since many diffraction or specular-diffraction combinations are possible.

Background/theory

The basic problem in edge diffraction is that of the diffraction from an infinite wedge which is irradiated by a point source. This problem was given a solution for the harmonic source signal in 1915 (frequency domain solution). In 1957, the transient problem, i.e. the impulse response was solved by Biot and Tolstoy [Biot and Tolstoy 1957]. No analytical solution has, however, been presented for the finite wedge, neither in the frequency-domain, nor the time-domain. The case of the infinite truncated wedge, on the other hand, was solved for plane wave incidence by Tolstoy in 1982 [Tolstoy 1982]. A number of approximate methods are known such as the Geometrical Theory of Diffraction (GTD) and the Uniform Asymptotic Theory of Diffraction (UATD). Both are frequency-domain methods which are correct, asymptotically, at high frequencies. The common Kirchhoff approximation can lead to very simple and efficient time-domain formulations but give only first-order diffraction, and might be erroneous at both low and high frequencies for certain cases.

H.Medwin used the exact Biot-Tolstoy solution to suggest an edge source interpretation of the edge diffraction phenomenon [Medwin 1981]. This was quite successful and gave very good agreement between calculations and measurements for many different cases. However, a theoretical derivation wasn't offered so the model was qualitatively reasonable but its quantitative accuracy was somewhat unsure. J.Vanderkooy explored an asymptotic frequency-domain formulation for the infinite wedge and managed to transform it into a very elegant and simple time-domain formulation. Comparisons with boundary element calculations for a point source on a rigid loudspeaker cabinet showed good agreement at mid and high frequencies but erroneous results at low frequencies, and including higher-order diffraction components did not improve the results [Vanderkooy 1991]. It had been shown already in the 20s that the so-called Maggi-Rubinowicz transformation when applied to the Kirchhoff approximation gives the scattering from a finite plane in terms of a line integral which is a very simple expression. This expression was exploited by Sakurai in the 80s [Sakurai and Nagata1981]. However, it was shown by Medwin and his colleagues that this Kirchhoff approximation leads to erroneous results both at low and high frequencies for certain radiation angles [Jebsen and Medwin 1982]. Consequently it is not suitable for general diffraction modelling.

In 1997, a new model was suggested by U. P. Svensson, R. I. Andersson (later R.I. Fred), and J. Vanderkooy for the finite wedge diffraction [Svensson et al 1997]. By assuming the existence of an analytic directivity function for edge sources along the edge of an infinite wedge, it was possible to derive the form of this directivity function. This can be seen as an analytic extension to the model used by Medwin et al. With access to such a directivity function, curved edges can be studied as well and very good agreement has been shown between the new method and accurate frequency-domain calculations for the axisymmetric backscattering of a circular thin disc. More about this new method can be read in [Svensson et al [1999] or [Torres et al 2001]. If you have problems to find these references, please contact svensson@tele.ntnu.no.

References to the new method by Svensson, Fred and Vanderkooy

U. P. Svensson, R. I. Fred, J. Vanderkooy, "Analytic secondary source model of edge diffraction impulse responses," J. Acoust. Soc. Am., 106, 2331-2344 (1999).

R. R. Torres, U. P. Svensson, M. Kleiner, "Computation of edge diffraction for more accurate room acoustics auralization," J. Acoust. Soc. Am., 109, 600-610 (2001).

P. Svensson, R. Andersson, J. Vanderkooy, "Time-domain approaches to the edge diffraction problem - applications to boxed loudspeakers," presented at the Third Joint Meeting of the Acoustical Society of America and the Acoustical Society of Japan, J. Acoust. Soc. Am. (Abstracts) 100, 2749 (1996). No proceedings available.

R. R. Torres, M. Kleiner, U. P. Svensson, "Computation of impulse response for stage-house geometries with use of edge-diffraction models," J. Acoust. Soc. Am. 101, 3134 (1997). (Abstracts).

U. P. Svensson, R. I. Andersson, J. Vanderkooy, "A new interpretation of the edge diffraction phenomenon," in Proceedings of the Autumn meeting of the Acoustical Society of Japan, Sapporo, Japan, Sept. 17-19, 813-814 (1997).

U. P. Svensson, R. I. Andersson, J. Vanderkooy, "A time-domain model of edge diffraction based on the exact Biot-Tolstoy solution," in the Technical Report of the Institute of Electronics, Information and Communication Engineers, EA97-39, Tokyo, Japan, Sept. 26, 9-16 (1997). (Contact the first author if you are interested in a copy of this paper)

R. R. Torres, M. Kleiner, "Audibility of edge diffraction in auralization of a stage house," J. Acoust. Soc. Am. 103, 2789 (1998). (Abstracts).

U. P. Svensson, R. Andersson, J. Vanderkooy, "An analytical time-domain model of edge diffraction," in Proceedings of NAM98, Nordic Acoustical Meeting, Stockholm, Sweden, Sept. 7-9, 269-272 (1998). (Contact the first author if you are interested in a copy of this paper)

U. P. Svensson, R. Andersson, J. Vanderkooy, "An analytical decomposition of the Biot-Tolstoy edge diffraction model," J. Acoust. Soc. Am. (Abstracts) 104, 1858 (1998).

R. R. Torres, M. Kleiner, "Considerations for including surface scattering in room-acoustics auralization," J. Acoust. Soc. Am. 105, 1198 (1999). (Abstracts).

U. P. Svensson, " Modelling scattering impulse responses with analytic edge sources," presented at the 23rd Scandinavian Symposium on Physical Acoustics, Ustaoset, Norway, Jan. 30 - Feb. 2 (2000). In Proc., 33-36, NTNU, Dept. of Telecommunications, Trondheim, Norway (2000).

P. Svensson, K. Wendlandt, U. Kristiansen, F. Goyard, P. Auguereau, "Efficient method for calculating the sound radiation of vibrating structures," pres. at Baltic Acoustic 2000, Vilnius, Lithuania, Sept. 17-21 (2000).

P. Svensson, K. Wendlandt, "The influence of a loudspeaker cabinet’s shape on the radiated power," pres. at Baltic Acoustic 2000, Vilnius, Lithuania, Sept. 17-21 (2000).

R. R. Torres, U. P. Svensson, M. Kleiner, "Edge diffraction in room acoustics computations," pres. at the EAA symposium on Architectural Acoustics, Madrid, Spain, Oct. 16-20 (2000).

U. P. Svensson, R. R. Torres, H. Medwin, "The Color of Early Sound Arrivals in an Auditorium," presented at the Joint Meeting: 140th Meeting of the Acoustical Society of America and Noise-Con 2000, Newport Beach, CA, USA, Dec. 3-8 (2000). J. Acoust. Soc. Am. (Abstracts) 108, 2648 (2000).

U. P. Svensson, "Acoustic rendering beyond geometrical acoustics," presented at the Campfire: Acoustic rendering for virtual environments, Snowbird, Utah, USA, May 26-29 (2001). Abstract on http://www.bell-labs.com/topic/conferences/campfire/program.html

R. R. Torres, M. Kleiner, U. P. Svensson, "Edge diffraction and scattering in the early room impulse response," presented at the 141st Meeting of the Acoust. Soc. Am., Chicaho, Illinois.

U. P. Svensson, L. Savioja, T. Lokki, U. R. Kristiansen, "Low-frequency models for room acoustic prediction," presented at the 17th International Congress on Acoustics, Rome, Italy, Sept. 2-7, (2001).

R. R. Torres, M. Vorländer, U. P. Svensson, M. Kleiner, "Studies of scattering from faceted room surfaces," presented at the 17th International Congress on Acoustics, Rome, Italy, Sept. 2-7, (2001).

Other references on edge diffraction, arranged by method

M. A. Biot, I. Tolstoy, "Formulation of wave propagation in infinite media by normal coordinates with an application to diffraction," J. Acoust. Soc. Am. 29, 381-391 (1957).

H. Medwin, "Shadowing by finite noise barriers," J. Acoust. Soc. Am. 69, 1060-64 (1981).

H. Medwin, E. Childs, G. M. Jebsen, "Impulse studies of double diffraction: A discrete Huygens interpretation," J. Acoust. Soc. Am. 72, 1005-1013 (1982).

G. M. Jebsen, H. Medwin, "On the failure of the Kirchhoff assumption in backscatter," J. Acoust. Soc. Am. 72, 1607-11 (1982).

C. S. Clay, W. A. Kinney, "Numerical computations of time-domain diffractions from wedges and reflections from facets," J. Acoust. Soc. Am. 83, 2126-2133 (1988). Misprints in eqs 24 and 25 and 16.

I. Tolstoy, "Exact, explicit solutions for diffraction by hard sound barriers and seamounts," J. Acoust. Soc. Am. 85, 661-669 (1989).

G. V. Norton, J. C. Novarini, R. S. Keiffer, "An evaluation of the Kirchhoff approximation in predicting the axial impulse response of hard and soft disks," J. Acoust. Soc. Am. 93, 3049-3056 (1993).

R. S. Keiffer, J. C. Novarini, G. V. Norton, "The impulse response of an aperture: Numerical calculations within the framework of the wedge assemblage method," J. Acoust. Soc. Am. 95, 3-12 (1994).

J. P. Chambers, Y. H. Berthelot, "Time-domain experiments on the diffraction of sound by a step discontinuity," J. Acoust. Soc. Am. 96, 1887-1892 (1994).

R. S. Keiffer, J. C. Novarini, "A time domain rough surface scattering model based on wedge diffraction: Application to low-frequency backscattering from two-dimensional sea surfaces," J. Acoust. Soc. Am. 107, 27-39 (2000).

W. Trorey, "A simple theory for seismic diffraction," Geophysics 35, 762-784 (1970).

F. J. Hilterman, "Amplitude of seismic waves - a quick look," Geophysics 40, 745-762 (1975).

J. R. Berryhill, "Diffraction response for nonzero separation of source and receiver," Geophysics 42, 1158-1176 (1977).

W. Trorey, "Diffraction for arbitrary source receiver locations," Geophysics 42, 1177-1182 (1977).

Y. Sakurai, K. Nagata, "Sound reflections of a rigid plane and of the "live end" composed by those panels," J. Acoust. Soc. Jpn (E) 2, 5-14 (1981).

G. M. Jebsen, H. Medwin, "On the failure of the Kirchhoff assumption in backscatter," J. Acoust. Soc. Am. 72, 1607-11 (1982).

Y. Sakurai, K. Nagata, "Practical estimation of sound reflection of a panel with a reflection coefficient," J. Acoust. Soc. Jpn (E) 3, 7-19 (1982).

Y. Sakurai, K. Ishida, "Multiple reflections between rigid plane panels," J. Acoust. Soc. Jpn (E) 3, 183-190 (1982).

J. Vanderkooy, "A simple theory of cabinet edge diffraction," J. Aud. Eng. Soc. 39, 923-933 (1991).

S. Rasmussen, K. B. Rasmussen, "On Loudspeaker Cabinet Diffraction," J. Aud. Eng. Soc. 42, 147-150 (1994).

J. B. Keller, "The geometrical theory of diffraction," J. Opt. Soc. Am. 52, 116-130 (1962).

D. L. Hutchins, R. G. Kouyomjian, "Calculation of the field of a baffled array by the geometrical theory of diffraction," J. Acoust. Soc. Am. 45, 485-492 (1969).

R. M. Bews, M. J. Hawksford, "Application of the Geometric Theory of Diffraction at the Edges of Loudspeaker Baffles," J. Aud. Eng. Soc. 34, 771-779 (1986).

D. S. Ahluwalia, R. M. Lewis, J. Boersma, "Uniform asymptotic theory of diffraction by a plane screen," S.I.A.M. J. Appl. Math. 16, 783-807 (1968).

Z. Maekawa, "Noise reduction by screens," Appl. Acoust. 1, 157-173 (1968).

J. J. Bowman, T. B. A. Senior, "The wedge", Chap. 6 in ' Electromagnetic and acoustic scattering by simple shapes'. Eds. J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, North-Holland, Amsterdam, (1969).

R. M. Lewis, J. Boersma, "Uniform asymptotic theory of edge diffraction," J. Math. Phys. 10, 2291-2305 (1969).

R. G. Kouyomjian, P. H. Pathak, "A uniform geometrical theory of diffraction for an edge in a perfectly conducting surface," Proc. of the IEEE 62, 1448-1461 (1974).

A. D. Pierce, "Diffraction of sound around corners and over wide barriers," J. Acoust. Soc. Am. 55, 941-955 (1974).

W J. Hadden, Jr., A. D. Pierce, "Sound diffraction around screens and wedges for arbitrary point source locations," J. Acoust. Soc. Am. 69, 1266-1276 (1981).
Erratum in J. Acoust. Soc. Am. 71(5). may (1982, p. 1290 98-06-14

P. Saha, A. D. Pierce, "Geometrical theory of diffraction by an open rectangular box," J. Acoust. Soc. Am. 75, 46-49 . (1984).

T. J. Cox, Y. W. Lam, "Evaluation of Methods for Predicting the Scattering from Simple Rigid Panels," Appl. Acoust. 40, 123-140 (1993).

K. B. Rasmussen, "Model experiments related to outdoor propagation over an earth berm," J. Acoust. Soc. Am. 96, 3617-3620 (1994).

Other references

M. Vorländer, "International round robin on room acoustical computer simulations," in Proc. of the Internat. Congress on Acoust., Trondheim, Norway, 26-30 June 1995, 689-692 (1995).

B.-I. L. Dalenbäck, "Room acoustic prediction based on a unified treatment of diffuse and specular reflection," J. Acoust. Soc. Am. 100, 899-909 (1996).