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 .
Wedge geometry and edge-aligned cylindrical coordinate system is used to describe the source and receiver positions.
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.
Relative error as a function of frequency in case C. 2, 4, 6 and 8 quadrature points.
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.
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.
The three geometry cases that are studied.
Error in the computation of the cases A, B and C with 5, 10, 15 and 20 Gaussian quadrature points.
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.
Timing of the brute-force method for different edge lengths, . Compare with constant a 2–3 ms for the new method.
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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