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This is a Quantitative Study study.

Numerical evaluation of the Kirchhoff–Helmholtz integral outside a sphere

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Numerical testing shows improved accuracy and speed in evaluating the transient acoustic field, indicating efficiency.

The method achieves near machine-precision accuracy in evaluating the transient acoustic field.
Using Lebedev quadratures results in a significant speed-up in evaluation time for acoustic fields.
The approach balances accuracy and computational efficiency based on the order of the quadrature rule used.
Field evaluations become efficient with a number of points about 1.15 times the surface quadrature nodes.

Michael Carley (2025) studied this question.

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