A uniform expansion is derived in terms of correction factors from the nonuniform expansion of the Kelvin wave near the cusp point.
When the principle of stationary phase is applied to integrals for ship waves, the resulting ship wave is the well known Kelvin wave which is expressed as the sum of the two waves called the transverse and divergent waves, respectively[1,2]. Because the second derivatives of the phases of the transverse and divergent waves become zero at the cusp point, the amplitudes of the transverse and divergent waves resulting from the application of the principle of stationary phase become infinite at the cusp point and the wave elevation thus obtained is not valid near it. Hence, even though the Kelvin wave obtained from the principle of stationary phase is in simple form, it is hardly useful for practical purposes of computations. In order to circumvent non-uniformity near the cusp point, Keller and Ahluwalia introduced a uniform expansion. In the present paper, the authors introduce correction factors for the treatment of the non-uniformity so that the simple form of the Kelvin wave from the application of the principle of stationary phase can be usefully utilized. We apply the present treatment to the Kelvin wave induced by a moving pressure point over the free surface in the negative x-axis and compute the wave elevation behind the pressure point with the correction factors. Then, the free surface configurations are graphically illustrated.
We take the Kelvin wave induced by the moving pressure point over the free surface at a small speed U for example. The first term on the right side of (3) is called the transverse wave and the second term is called the divergent wave. The Kelvin wave is symmetrical about the positive x-axis.