### Abstract

Original language | English |
---|---|

Pages (from-to) | 717-731 |

Number of pages | 15 |

Journal | Mathematica Æterna |

Volume | 5 |

Issue number | 5 |

Publication status | Published - 2015 |

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*Mathematica Æterna*,

*5*(5), 717-731.

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*Mathematica Æterna*, vol. 5, no. 5, pp. 717-731.

**A note on an integral associated with the Kelvin ship-wave pattern.** / Paris, R. B.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A note on an integral associated with the Kelvin ship-wave pattern

AU - Paris, R. B.

PY - 2015

Y1 - 2015

N2 - The velocity potential in the Kelvin ship-wave source can be partly expressed in terms of space derivatives of the single integral F(x, ρ, α) = ʃ ∞ −∞ exp [−1/2ρ cosh(2u − iα)] cos(x cosh u) du, where (x, ρ, α) are cylindrical polar coordinates with origin based at the source and −1/2π ≤ α ≤ 1/2π. An asymptotic expansion of F(x, ρ, α) when x and ρ are small, but such that M ≡ x2/(4ρ) is large, was given using a non-rigorous approach by Bessho in 1964 as a sum involving products of Bessel functions. This expansion, together with an additional integral term, was subsequently proved by Ursell in 1988. Our aim here is to present an alternative asymptotic procedure for the case of large M. The resulting expansion consists of three distinct parts: a convergent sum involving the Struve functions, an asymptotic series and an exponentially small saddle-point contribution. Numerical computations are carried out to verify the accuracy of our expansion.

AB - The velocity potential in the Kelvin ship-wave source can be partly expressed in terms of space derivatives of the single integral F(x, ρ, α) = ʃ ∞ −∞ exp [−1/2ρ cosh(2u − iα)] cos(x cosh u) du, where (x, ρ, α) are cylindrical polar coordinates with origin based at the source and −1/2π ≤ α ≤ 1/2π. An asymptotic expansion of F(x, ρ, α) when x and ρ are small, but such that M ≡ x2/(4ρ) is large, was given using a non-rigorous approach by Bessho in 1964 as a sum involving products of Bessel functions. This expansion, together with an additional integral term, was subsequently proved by Ursell in 1988. Our aim here is to present an alternative asymptotic procedure for the case of large M. The resulting expansion consists of three distinct parts: a convergent sum involving the Struve functions, an asymptotic series and an exponentially small saddle-point contribution. Numerical computations are carried out to verify the accuracy of our expansion.

M3 - Article

VL - 5

SP - 717

EP - 731

JO - Mathematica Æterna

JF - Mathematica Æterna

SN - 1314-3344

IS - 5

ER -