On reference solutions for ship waves

The classic problem of ship waves of infinitely small amplitudes is studied within the recently developed approach of reference solutions [V. G. Gnevyshev and S. I. Badulin, Moscow Univ. Phys. Bull. 72, 415 (2017)]. The evolution of narrow-banded Gaussian wavetrains of a finite volume is considered as an alternative to the conventional inherently point-wise tracking of wavetrains within the asymptotic methods of the stationary phase or the steepest descent. The approach allows for avoiding general problems of the methods: occurrence of singularities of wave fields. The non-singular solutions for a stationary ship wake are presented in an analytical form in terms of two key dimensionless quantities: the Froude number and the aspect ratio of the initial (boundary) domain of wave generation. Systems of transverse and diverging ship waves as well as amplitude and phase effects of their transformation can be analyzed separately within this approach for small Froude numbers. The essential role of dispersion of three-dimensional water waves is emphasized and detailed.