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

The Struve functions $Hn(z), n=0, 1, ...$ 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.