Evaluation of the Wavelike Disturbance in the Kelvin Wave Source Potential

Abstract

This paper discusses the numerical evaluation of the characteristic Kelvin wavelike disturbance trailing downstream from a translating submerged source. Mathematically the function describing the wavelike disturbance is expressed as a single integral with infinite integration limits and a rapidly oscillatory integrand. Numerical integration of such integrals is both cumbersome and time-consuming. Attention is therefore focused on two complementary Neumann-series expansions which were originally derived by Bessho [1].2 Numerically stable algorithms are presented for the accurate and efficient evaluation of the two series representations. When used in combination with the Chebyshev expansions for the nonoscillatory near-field component which were recently obtained by Newman [2], the present algorithms provide an effective solution to the numerical difficulties associated with the evaluation of the Kelvin wave source potential.

Developments in the calculation of the wavemaking resistance of ships

“…The singularity can be isolated by decomposing the function W into singular and regular components. Recently we have shown that the singular behaviour is mathematically related to Dawson's integral function (see Baar & Price 1988). The theoretical and practical implications of this decomposition remain the subject of further investigation.…”

“…It has been shown by Newman (1984) that the Bessel functions Jn and Yn(n = 0, 1) can be computed by means of polynomial expansions and similar approximations for the modified functions I n, K n were derived by Allen (1956). For n^ 2, Baar & Price (1988) developed a numerical procedure based on the work of Gautschi (1967). This is briefly outlined as follows and commences from the general three term recurrence relation where fn = fn (x),n^l, and c ( -2 n/x,l) («".&") ={ ( 2n / x ,-1) if fn = | In(x) \{-2 n/x,-\) which is a result given Abramowitz & Stegun (1972).…”

“…A brief description of this numerical approach is given in the Appendix and full details of the method are given by Baar & Price (1988). It is shown that simple recursion schemes can be derived for the accurate and efficient evaluation of the JK series in ( 27) and the YI series in (28).…”

“…In particular great emphasis has been placed on the derivation of accurate and efficient algorithms for the evaluation of the Kelvin wave source potential. The major advancements in the numerical techniques are discussed in this paper, and additional details are given by Baar (1987) and Baar & Price (1988).…”

On the theory of the Kelvin ship-wave source: asymptotic expansion of an integral

“…[ 81 ] the decomposition preferred by Bessho (1964, equation (4.1)), Newman (1987, equation (1)) and Baar & Price (1988) involves space derivatives of e"^z j exp (-\z cosh 2 u + |iy sinh 2 u )c os (x cosh u) d…”