2 WAVE PATTERNS AND WAVE RESISTANCE. 
and on the calculation of wave resistance, and more recently on the comparison of calculated 
results with experiment; but the fundamental principles remain the same, and it is these 
which I wish specially to keep in view in the following notes. We begin by considering 
freely moving wave patterns; that is, not forced waves produced by the motion of a ship, 
but waves moving freely and steadily over the surface of the water under the action of 
gravity alone. We imagine the pattern to be produced by the mutual interference of 
simple plane waves moving freely in all directions, their phases and velocities being suitably 
adjusted; the elementary properties of the pattern are described from this poimt of view. 
Then, considering the waves produced by a ship, we see that these must approximate, 
at a sufficient distance to the rear of the ship, to such a freely moving pattern; this is 
illustrated by calculations made for certain ship models. Finally, it is shown how the 
wave resistance can be obtained from considerations of energy when we know the structure 
of the wave pattern formed at a great distance in the rear of the ship. 
FREE WAVE PATTERNS. 
2. The simplest form of free waves on the surface of water consists of simple harmonic 
waves with straight parallel crests, the procession of waves extending over the whole surface. 
If the velocity of the waves is c, the wave-length is 2 7c?/qg for deep water; so that if 
we take an origin O in the surface and take Ow in the direction of propagation, the waves 
might be represented by 
£=sin 3 (@ — cd) «put Siveugsty, Rae ane ada UD) 
where ¢ is the surface elevation, and we have taken the waves to be of unit amplitude. 
Suppose now that the waves are travelling in a direction making an angle 6 with O2, 
and that the wave velocity is ccos@; then, with Oy in the surface and perpendicular to 
Ox, the waves are now represented by 
f =sin {x sec? 0 (xcos@ + ysin@ —ctcos@)}. . . . (2) 
where we have written « = g/c*. 
An equal procession of waves moving in a direction making a negative angle @ with 
Ox is given by 
¢ = sin {x sec? 0 (w cos 9 —ysin@ —ctcos6)}. . . . (3) 
Superpose these two sets of plane waves, and we have a wave pattern given by the 
sum of (2) and (3), or 
¢ = 2cos (x y sin 8 sec? @) sin {x (ce —ct)secO}. . . . (4) 
These have sometimes been called corrugated waves. . We may get a rough idea of the 
result by drawing parallel straight lines to represent the positions of the crests and troughs 
of the component systems at a given instant; and so we get the picture of a diamond- 
shaped pattern, covering the whole surface and moving steadily in the direction Ox with 
velocity c. 
We now generalize by supposing that we have simple straight-crested waves like (2) 
travelling forward in all directions included within 90° on either side of Ow. Superposing 
these component plane waves will give a surface elevation 
T 
2 
[= fain Ge seet@ (wos @ | ysind — et0s )} a0 eee) 
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2 
378 
