The Struve functions are approximated in a simple, accurate form that is valid for all . The authors previously treated the case n = 1 that arises in impedance calculations for the rigid-piston circular radiator mounted in an infinite planar baffle [Aarts and Janssen, J. Acoust. Soc. Am. 113, 2635–2637 (2003)]. The more general Struve functions occur when other acoustical quantities and/or non-rigid pistons are considered. The key step in the paper just cited is to express as , where J0 is the Bessel function of order zero and the first kind and I(z) is the Fourier cosine transform of . The square-root function is optimally approximated by a linear function , and the resulting approximated Fourier integral is readily computed explicitly in terms of and . The same approach has been used by Maurel, Pagneux, Barra, and Lund [Phys. Rev. B 75, 224112 (2007)] to approximate for all . In the present paper, the square-root function is optimally approximated by a piecewise linear function consisting of two linear functions supported by and with the optimal take-over point. It is shown that the optimal two-piece linear function is actually continuous at the take-over point, causing a reduction of the additional complexity in the resulting approximations of and . Furthermore, this allows analytic computation of the optimal two-piece linear function. By using the two-piece instead of the one-piece linear approximation, the root mean square approximation error is reduced by roughly a factor of 3 while the maximum approximation error is reduced by a factor of 4.5 for and of 2.6 for . Recursion relations satisfied by Struve functions, initialized with the approximations of and , yield approximations for higher order Struve functions.
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December 2016
December 06 2016
Efficient approximation of the Struve functions Hn occurring in the calculation of sound radiation quantities
Ronald M. Aarts;
Ronald M. Aarts
a)
1
Philips Research Europe
, High Technology Campus 34, 5656 AE Eindhoven, The Netherlands
2Department of Electrical Engineering,
Eindhoven University of Technology
, 5600 MB Eindhoven, The Netherlands
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Augustus J. E. M. Janssen
Augustus J. E. M. Janssen
3
Department of Mathematics and Computer Science
, Eindhoven University of Technology
, 5600 MB Eindhoven, The Netherlands
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a)
Electronic mail: ronald.m.aarts@philips.com
J. Acoust. Soc. Am. 140, 4154–4160 (2016)
Article history
Received:
August 15 2016
Accepted:
November 12 2016
Citation
Ronald M. Aarts, Augustus J. E. M. Janssen; Efficient approximation of the Struve functions Hn occurring in the calculation of sound radiation quantities. J. Acoust. Soc. Am. 1 December 2016; 140 (6): 4154–4160. https://doi.org/10.1121/1.4968792
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