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 H 1 ( z ) as ( 2 / π ) − J 0 ( z ) + ( 2 / π ) I ( z ) , where J 0 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 c ̂ t + d ̂ , 0 ≤ t ≤ 1 , and the resulting approximated Fourier integral is readily computed explicitly in terms of sin z / z and ( 1 − cos z ) / z 2 . The same approach has been used by Maurel, Pagneux, Barra, and Lund [Phys. Rev. B 75, 224112 (2007)] to approximate H 0 ( 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 , t ̂ 0 ] and [ t ̂ 0 , 1 ] with t ̂ 0 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 H 0 and H 1 . 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 H 0 and of 2.6 for H 1 . Recursion relations satisfied by Struve functions, initialized with the approximations of H 0 and H 1 , yield approximations for higher order Struve functions.