Time domain models of edge diffraction

Background

The concept of edge diffraction is used in many fields of acoustics. Edge diffraction is, for the case of scattering/reflection from objects with plane rigid surfaces, used to describe the sound field which must be added to the geometrical acoustics sound field to reach a correct total sound field. Edge diffraction is used in, e.g., room acoustics (prediction of impulse responses in rooms), electroacoustics (transducer response), noise control (noise barriers), underwater acoustics (reflections from discontinuities in the ocean bottom or from scattering objects), geophysics (reflections from wedge-shaped discontinuities between layers of different media), general scattering problems.
 
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.
 
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 for the geometry in Fig. 1 [Biot and Tolstoy 1957].

Fig. 1    The geometry of an infinite wedge, with a point source S and a receiver R.
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 [Jebsen & Medwin 1982].

Earlier time-domain models for the finite wedge

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.

New time-domain model for the finite wedge

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, see Fig. 2.
Fig. 2    The angles used for an analytic directivity function of secondary edge sources. The angle alpha is indicated in the imaginary plane going through the source point S and the crest of the wedge. In another imaginary plane, going through the receiver point R and the crest of the wedge, the angle gamma is indicated.
The impulse response can then be written as an integration along the edge, as
 
where the directivity function is
and the so/called wedge index is
The summation in the directivity function, beta, is a sum of four terms given by the four possible sign combinations indicated.
 
 

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 the references below. If you have problems to find these references, please contact svensson@tele.ntnu.no.
 

Modeling in room acoustics

A very interesting use for the edge diffraction models is in room acoustics computer modeling. Most computer models use geometrical acoustics methods and these can not represent wave phenomena correctly. This is clearly visible in the following example, which is further described in [Torres et al, Nov. 2000]. We study a simpified stage house, and calculate the impulse response along an array of receiver positions, as indicated in Fig. 3.
Fig. 3    Model of a stage house, with a source on stage, and a horizontal receiver array in front of the stage.

 

Publications

Other references on edge diffraction, arranged cronologically by method

Papers by Biot and Tolstoy, and papers on Medwin's 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).

The Kirchhoff diffraction approximation
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).

Vanderkooy's method
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).

The Geometrical theory of diffraction (GTD)
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).

Various frequency domain methods
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).

Last change October 5, 2004 by Peter Svensson