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Ship Waves: The Relative Efficiency of Bow and Stern 
By T. H. HAVELOCK, F.R.S. 
(Received January 11, 1935) 
1. It seems fairly certain that one of the main causes of differences 
between theoretical and experimental results is the neglect of fluid friction 
in the calculation of ship waves, and further that the influence of fluid 
friction may be regarded chiefly as one which makes the rear portion of 
the ship less effective in generating waves than the front portion. The 
process may be pictured, possibly, in terms of a friction belt or boundary 
layer whose more important effect is equivalent to smoothing the lines 
of the model towards the rear. Some calculations were made from this 
point of view in a previous paper,* the purpose then being to show how 
such an asymmetry, fore and aft, reduced the magnitude of interference 
effects between bow and stern waves. We may also describe the frictional 
effect as a diminution in the effective relative velocity of the model and 
the surrounding water as we pass from bow to stern. This is not very 
satisfactory from a theoretical point of view; but, on the other hand, it 
leads to a comparatively simple modification of expressions for the 
waves produced by the model. From a formal point of view, we may 
regard the modification as an empirical introduction of a reducing factor 
to allow for decrease in efficiency of the elements of the ship’s surface 
as we pass from bow to stern. 
There are now available experimental results, for wave profiles as 
well as for wave resistance, which make it possible to attempt such a 
comparison. The following work is limited to a few simple cases, and 
the assumptions are made in as simple a form as possible for the purpose 
of the calculations; these deal with the wave profile and wave resistance 
of a model of symmetrical form, and also with the difference between 
motion bow first and motion stern first for a simple asymmetrical model. 
2. Take the origin O in the undisturbed free surface of the water, with 
Ox horizontal and Oz vertically upwards; and let the origin O be moving 
with uniform velocity c in the direction Ox. We suppose that there is a 
given distribution of sources and sinks over the zx-plane, or, alternatively, 
that the normal fluid velocity is given over this plane; let it be F (H, f) at 
**Proc. Roy. Soc.,’ A, vol. 110, p. 238 (1926). 
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