Efficient approximation of the Struve functions H_n occurring in the calculation of sound radiation quantities

The Struve functions H_n(z), n=0, 1, ...  are approximated in a simple, accurate form that is valid for all z≥0. 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 [(1-t)/(1+t)]^1/2, 0≤t≤1. The square-root function is optimally approximated by a linear function ĉt+d̂, 0≤t≤1, and the resulting approximated Fourier integral is readily computed explicitly in terms of sin z/z and (1-cos z)/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 z≥0. 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.