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An entire thin sheet is formed in this process and appears as a splash or spray on either side of the hull, Figure 2. This spray as viewed in successive vertical sections aft of the stem, resembles the rise or jet formation which occurs during two dimensional water entry; indeed the mechanism of water entry and of the fine bow flow are closely related.

In Michell's thin ship theory, this splash appears as a result of the calculation so that it is not entirely of nonlinear origin. Following the splash, an oblique wave then appears on each side of the hull, often followed by others. These waves were observed by Froude, who called them “divergent” waves, Figure 3, and described them in the context of the entire wave system, including the broad crested transverse waves. Subsequently, Lord Kelvin, who knew about Froude's observations deduced the wave pattern behind an infinitesimal disturbance. A set of nested divergent waves appeared, Figure 4. Since that time, actual divergent bow waves have with little exception been linked with the Kelvin wave pattern, and simply considered part of it. In fact, the waves drawn by Froude, Figure 3, differ substantially and in a basic way from Kelvin's picture, Figure 4; we will account for this difference later. We owe to Inui(1970) and his colleagues in Tokyo, Inui, et at(1979), Miyata(1981), the earliest attention to little understood aspects of divergent bow waves. Miyata claimed, following experimental studies that these waves were non-dispersive and non-linear; he called them “surface shock waves”. After their work, the subject rested until now, perhaps because for conventional Froude numbers the divergent waves do not contribute in a major way to the wave resistance. Of course, at higher speeds, they are important and eventually merge with spray. Recently, there has arisen an interest in the very long narrow wake behind ships which can be observed remotely. For naval combatant ships, photographs show substantial wakes created abeam the ship by the extensive breaking of divergent bow and stern waves; the same can be observed in early photographs in the Tokyo tank using aluminum powder on the surface, see Figure 5. This has created an interest in divergent waves for their own sake.

How to predict these waves? The numerical computation of ship wave resistance and patterns has evolved through Michell, Guilloton, Neuman-Kelvin, and Dawson techniques to fully non-linear codes [Daube, Raven(1993)]. No doubt, the best of these latter codes can provide useful information on the divergent waves, provided they can resolve the thin bow splash and steep waves which sometimes appear. Very often bow waves are seen to break, sometimes strongly, and the 3-dimensional codes may then have trouble. For sufficiently fine ships, we know that the divergent wave pattern, both at the bow and stern, emerges from numerical calculations made on the basis of nonlinear slender body theory, Tulin and Wu(1994). The calculation is carried out in two dimensions, vertical and transverse, and successively in time (2D+T). It has the advantage of high resolution, sufficient to define breaking, and even to trace out overturning jets. At the same time, it neglects certain three dimensional effects which become increasingly important as the ship becomes full. For instance, it does not allow upstream influence, so it can not predict breaking before the bow. However, it has much to recommend it for fine ships like combatants, and especially for the higher Froude numbers where massive breaking of divergent waves is observed. Furthermore, the theory can be applied to divergent stern waves, about which we know little.

There is much to understand about divergent waves, both bow and stern: their source or origin; the shapes they take and the patterns they make; the influence of the beam, draft, and the speed on their strength and propagation, the onset and type of breaking, and even their post breaking behavior.

Our principal purpose here is to report and comment on the calculations of some nonlinear 2D+T numerical studies of divergent waves and to illuminate their basic mechanics. We begin with a discussion of some past ideas.

The inadequacy of ray theory; the splash as a high speed inner flow

For fine ships without sharp corners except at the bow and stern, at normal Froude numbers, the Michell theory shows that the ship 's wave pattern is equivalent to two Kelvin patterns originating at the bow and stern. The ray theory according to Keller(1974,1979) also concludes that all the rays must originate at discontinuities in the waterline shape, which normally occur only at the bow and stern.

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