. The collected papers of Sir Thomas Havelock on hydrodynamics. Ship resistance; Water waves; Hydrodynamics. Ship Waves. 46d carried out on an actual model, would no doubt involve other changes which would have to be considered in a theory capable of taking exact account of actual dunensions ; but meantime we may isolate the eSect of this particular change. For convenience we consider separately the eSect on the local disturbance and on the wave motion to the rear. Taking the former, we see from (6) and (10) that the difference amounts to replaciug j Qq (kq^^) by ^Kfl (?2 - ^i) {Qi ('<o?2)
. The collected papers of Sir Thomas Havelock on hydrodynamics. Ship resistance; Water waves; Hydrodynamics. Ship Waves. 46d carried out on an actual model, would no doubt involve other changes which would have to be considered in a theory capable of taking exact account of actual dunensions ; but meantime we may isolate the eSect of this particular change. For convenience we consider separately the eSect on the local disturbance and on the wave motion to the rear. Taking the former, we see from (6) and (10) that the difference amounts to replaciug j Qq (kq^^) by ^Kfl (?2 - ^i) {Qi ('<o?2) - Qi (^o?!)}- (13) This can be shown in a form applicable to various velocities and to various ranges of x^ — x-^ by graphing the quantity on a base p, for several values of Ic. These curves are shown in fig. 2. (14). -3-2-1 0 I Z Fig. 2.—Curves of {Q {j) -\- h) — Qi {p)}j4:k for different values of k. In applying these curves to actual distances along the ship model, we note that p = KqX = gxju^, where u is the velocity ; and similarly h = gdju^, where d is the range over which the original sudden change in slope has been distributed. Thus the relative importance of the effects depends upon the ratio gd/u^, or upon the ratio of d to X, the wave-length of straight water waves for velocity u. In the diagram, k = 0 denotes the curve for the sharp corner ; the bow of the model is to the right of the diagram and the stern to the left. Apart from the general smoothing effect, the chief point to notice in these curves is the raising of the profile forward of the point in question and a lower- ing to the rear of it. This is due to takuig the range d entirely to the rear of the original sharp corner. If, on the other hand, the comer is taken at the middle of the range d in each case, by a suitable relative displacement of the curves, it is easily seen that the smoothing of the corner does not make any appreciable difference to the local disturbance except within the range d itsel
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