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WAVE PATTERNS AND WAVE RESISTANCE. 
We could also have used a form 
C= |GOsG@mede ss 6 56 5 go 5 (). 
which may be called a simple cosine pattern. The general form of the pattern is the same 
in both cases, with the necessary changes in wave heights due to the interchange of crests 
and troughs. It would be of interest to have a more detailed mathematical and numerical 
analysis of these two simple forms. 
In (7) and (8) the amplitudes of the component plane waves are taken to be the same 
for all directions. We may now proceed to a final generalization by supposing that in each 
case there is an amplitude factor depending upon the direction of each component; adding 
the two forms, we arrive at a general expression for a freely moving wave pattern, namely 
t -{i (8) sin (x, y) a0 +| 
It is true that the amplitude factors may alter considerably our picture of the pattern, 
especially if they have pronounced maxima or minima; however, we shall see that most 
cases which have been calculated for ship models can be reduced to terms like (9) with 
simple amplitude factors. 
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SHip WAVES. 
5. We have been dealing so far with a free wave pattern; that is, we have supposed 
the system to be completely in existence at some instant, and then afterwards it moves 
freely and steadily forward. 
Consider now the disturbance produced, in a frictionless liquid, by a moving ship or 
by a disturbing pressure system moving steadily forward. At some distance in advance 
of the ship there can be no appreciable disturbance, as we suppose it moving forward into 
still water. In the immediate neighbourhood of the ship the disturbance will be of a 
complicated character. But as we go further and further to the rear, the surface disturbance 
must approximate more and more to some freely moving wave pattern following on with the 
same speed as the ship. 
For instance, if a long cylindrical log is moved with steady velocity ¢ at right angles 
to its length, the disturbance at a great distance in the rear must approximate to a simple 
plane wave of velocity c, whose wave-length is therefore 27c?/g. It could be expressed 
by (1), taking some suitable point as the origin O, and including some definite amplitude 
factor; this amplitude factor would be the important thing left to be determined from the 
form of the cross-section of the cylinder and its velocity. Similarly for an ordinary ship 
form, the waves at a great distance in the rear must approximate to some freely moving 
wave pattern such as we have been considering; and for some suitable origin O, in or near 
the ship, they must therefore be expressible in the form (9), with amplitude factors f (@) 
and (8) depending upon the form of the ship and the speed. Without going into the 
details of calculating these expressions we shall now examine a few cases in order to illustrate 
the types of wave pattern which occur in such problems. 
Point DistURBANCE AND SPHERE. 
6. On account of its historical interest we may mention first the travelling point 
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