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After originating at the bow (and stern), the trajectory of the rays depends upon and may be calculated in terms of the “displacement” flow about the hull. The deformation by the displacement flow of the ray propagating waves, with amplitudes calculated according to linear theory, was studied by Inui and Kajitani(1977) and later, in the same spirit, by Yim(1981). Subsequently, Tulin(1985) produced a general ray analysis of the generation and propagation of waves around a ship of finite beam.

In Keller ray theory it was proposed to base the displacement flow on a naive Froude number expansion. Tulin showed that the calculation of the amplitude of waves propagating on rays is dependent on the pressure gradient at the stagnation point in the low Froude number flow, and the theory gives rise to unbounded amplitudes for wedge bows. Then he raised the question, “is the naive Froude number expansion even applicable (uniformly convergent) in the neighborhood of the point of the bow?” He continued, “there exists a good chance that it is not. I say that because in nature it is normal on wedge models, see Standing (1974), to find the highest point on the free surface at some distance aft of the point of the bow (as Michell's theory predicts!); is it possible that this behavior is reflected at all Froude numbers on a scale near the bow which increases with speed, perhaps as U2, creating an inner flow at the point of bow for which the naive Froude number expansion is an outer flow.”

The simulation results presented here clearly show that the side splash on the hull is crucial for the generation of divergent waves. Furthermore, we shall show that it scales exactly as the “high speed ” inner flow suggested by Tulin. Our calculations indicate that the relaxation of the splash is the prime source of divergent waves. One of the consequences of this discovery is that conventional ray theory, which is conceived as asymptotic to zero Froude number, cannot predict the generation of divergent waves. Whether it is relevant to wave propagation is another question.

Eventually we conclude that neither ray theory nor linear theory is relevant to the understanding of bow divergent waves. The key to their understanding lies in the spectral wave number content in the splash, which for typical ships is governed by the ship's beam. This is a non-linear effect. Furthermore the waves are typically very steep and sometimes breaking.

3.
The wedge flow

The free surface flow about the inclined wedge, for which a length scale is absent, offers an important opportunity for understanding the morphology of bow waves produced by fine ships. Its consideration leads immediately to understanding of the inner flow (side splash) and to essential scaling relations. These, in turn, allow us to understand some of the effects of speed on the divergent waves, particularly, their spatial pattern.

For an inclined wedge (wedge angle α, stem angle β) at speed U, the wave height η, at any longitudinal location measured from the wedge-still water intersection, must on dimensional grounds obey the scaling law:

(1)

This means that the flow speed U simply serves(through κ) as a factor scaling both η and the actual position on the free surface where certain flow features appear. Therefore high speed features such as the origination of splash, appear closer and closer to the bow as the speed decreases; equivalently low speed features (like disappearance of waves) will always appear if we look far enough behind the wedge. And in between the splash at the bow and the wave free field to the rear, we can expect to find observable waves. These are the divergent waves. A chronology of events near the bow can be constructed, based on the scaling of eq. (1) and using other results, see Figure 6.

The flow near the bow begins as a spatially self similar spray, just as in gravity-free water entry, and every line originating at and lying on the free surface is a straight line, since in the absence of κ;

(2)

This is confirmed in the 2D+T calculations, which then show that for sufficiently large values of along the wedge, the spray sheet is maintained at high levels by the continually expanding wedge, and does not fall. However, divergent waves do originate in the sheet and propagate outwards, Figure 7. We note that the



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