No Access Submitted: 15 August 2016 Accepted: 12 November 2016 Published Online: 06 December 2016
The Journal of the Acoustical Society of America 140, 4154 (2016); https://doi.org/10.1121/1.4968792
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  • Ronald M. Aarts
  • Augustus J. E. M. Janssen
The Struve functions Hn(z),n=0,1,... are approximated in a simple, accurate form that is valid for all z0. 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 H1(z) as (2/π)J0(z)+(2/π)I(z), where J0 is the Bessel function of order zero and the first kind and I(z) is the Fourier cosine transform of [(1t)/(1+t)]1/2,0t1. The square-root function is optimally approximated by a linear function ĉt+d̂,0t1, and the resulting approximated Fourier integral is readily computed explicitly in terms of sinz/z and (1cosz)/z2. The same approach has been used by Maurel, Pagneux, Barra, and Lund [Phys. Rev. B 75, 224112 (2007)] to approximate H0(z) for all z0. In the present paper, the square-root function is optimally approximated by a piecewise linear function consisting of two linear functions supported by [0,t̂0] and [t̂0,1] with t̂0 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 H0 and H1. 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 H0 and of 2.6 for H1. Recursion relations satisfied by Struve functions, initialized with the approximations of H0 and H1, yield approximations for higher order Struve functions.
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