410 T. H. Havelock 
second order. One purpose of the present work is to examine that assump- 
tion; the approximate method is extended in a certain case to give the 
variation in amplitude of the surface oscillation along the side of the ship. 
The view has been put forward recently that the mean extra resistance 
to a ship advancing through waves is due to the reflexion of the waves by 
the sides of the ship, being in fact analogous to the pressure of radiation: 
it has been stated, for instance, that the resultant amplitude at the bow is 
about one-third greater, and that at the stern one-third less, than the 
amplitude of the incident waves, and empirical formulae for the pressure 
have been constructed on that basis. The problem requires, however, 
a consideration of second order terms which does not appear to have been 
made for water waves even in simple cases. We consider total reflexion, 
normal or oblique, by a plane wall, and diffraction by a cylinder of circular 
or parabolic section, together with approximations for a section of ship 
form: the results are discussed in relation to the ship problem. 
DIFFRACTION OF WATER WAVES 
2. Consider a fixed cylindrical obstacle in the water, the sides vertical 
and extending down to an infinite depth; let C be the contour of any hori- 
zontal cross-section. We suppose plane waves of amplitude A to be travelling 
in the negative direction of Ox; the origin O is in the free surface and Oz 
is vertically upwards. The velocity potential of the fluid motion is of the 
form 
g ze DE piotsraines + plots $! (a, 4). (1) 
The pressure condition at the free surface is satisfied, to the usual first order 
terms, by o? = gx. Further, we have 
CP OP ay 2 
ie oye =O (2) 
and 0¢/dv = 0 on the contour C. The potential may be expressed in terms 
of a source distribution over the surface of the cylinder, but that is, in 
general, merely a restatement of the problem. We are concerned meantime 
with an approximate solution when the contour C is of ship form; that is, 
we assume C to be a contour of small breadth compared with its length. 
We take Ox in the direction of the length and to be an axis of symmetry 
of the contour. The approximation is the same as that used in determining 
the waves produced by a moving ship. We take the source strength at any 
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