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Efficient evaluation of edge diffraction integrals using the numerical method of steepest descent
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10.1121/1.3479545
/content/asa/journal/jasa/128/4/10.1121/1.3479545
http://aip.metastore.ingenta.com/content/asa/journal/jasa/128/4/10.1121/1.3479545

Figures

Image of FIG. 1.
FIG. 1.

The contours of the imaginary part of the oscillator in the complex plane and the corresponding paths of steepest descent. A single path emerges from each of the end points and , and they are connected by a path passing through the stationary point at .

Image of FIG. 2.
FIG. 2.

Wedge geometry and edge-aligned cylindrical coordinate system is used to describe the source and receiver positions.

Image of FIG. 3.
FIG. 3.

Example of the analytical structure of the oscillator function, , for the edge diffraction integral. Contours of the imaginary part of the oscillator are indicated, and steepest descent paths are drawn as continuous curves, the approximations of these as dashed curves. Branch cuts are marked with bold lines.

Image of FIG. 4.
FIG. 4.

Relative error as a function of frequency in case C. 2, 4, 6 and 8 quadrature points.

Image of FIG. 5.
FIG. 5.

Relative error as a function of angle in case C. 5, 10, 15 and 20 quadrature points. Frequency 10 kHz. Zone boundaries are marked with vertical lines.

Image of FIG. 6.
FIG. 6.

Relative error as a function of angle in case C. 5, 10, 15 and 20 quadrature points. Frequency 10 kHz. Integration interval . Zone boundaries are marked with vertical lines.

Tables

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Generic image for table
Generic image for table
TABLE I.

The three geometry cases that are studied.

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TABLE II.

Error in the computation of the cases A, B and C with 5, 10, 15 and 20 Gaussian quadrature points.

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TABLE III.

Timing for the brute-force method applied to the different cases for frequencies 0.5 1, 10 and 20 kHz with a tolerance of . The new method clocks between 1 and 2 ms depending on the number of quadrature points.

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TABLE IV.

Timing of the brute-force method for different edge lengths, . Compare with constant a 2–3 ms for the new method.

Generic image for table
TABLE V.

Relative error for a classical asymptotic approximation compared with the numerical steepest descent with few, i.e., 1, 2 and 3, quadrature points.

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/content/asa/journal/jasa/128/4/10.1121/1.3479545
2010-10-18
2014-04-23
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: Efficient evaluation of edge diffraction integrals using the numerical method of steepest descent
http://aip.metastore.ingenta.com/content/asa/journal/jasa/128/4/10.1121/1.3479545
10.1121/1.3479545
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