4 WAVE PATTERNS AND WAVE RESISTANCE. 
by (5) is only a first approximation; detailed mathematical analysis is necessary for a more 
correct and intimate knowledge of the surface elevation. 
Examine more closely one of the curves of Fig. 2, say the portion OAB which is 
shown in Fig. 3 along with the crest lines of the component plane waves. 
We find that the transverse part AB is made up from those plane waves whose 
direction angles range from zero up to an angle 6@,, which is such that cos?@,—= 3, or 
0, = 35° 16’ approximately; the diverging part OA comes from the plane waves whose 
directions range from 6, to 90°. The angle between the crest line OA and the central 
line OB is 19° 28’, nearly. To complete our picture we require some information about 
the height of the waves in the pattern defined by the expression (5). All that need be 
said here is that, following a curve such as BAO, the height is fairly constant over the 
central portion of the transverse wave, increases in the neighbourhood of a crest point A 
and then decreases along the diverging wave to zero at the point O. 
It may also be noted that the wave-length A of a component plane wave being 
(2 7 c?/g) cos? 0, these wave-lengths range from 2 7 c?/g to zero. 
3. Consider for a moment the difference in these general results if the water, instead 
of being very deep, is of given finite depth 4. The relation between velocity and wave- 
length for a simple plane wave is different, and, moreover, there can be no plane wave 
A 
Rie. 3: 
whose velocity is greater than \/ (gh). Suppose we build up a pattern like (5) when the 
velocity c of the pattern is less than this critical value \/ (gh). We could trace the envelope 
curves in the same way and obtain a wave pattern similar to Fig. 2. The chief difference 
is that the wave pattern widens out; the angle of the cusp line is greater than the value 
19° 28’ for deep water and it increases with the velocity c. In addition, the transverse 
waves become less curved, the angle 0, of Fig. 3 being less than the value 35° for deep 
water and becoming less as the velocity c¢ is increased. 
If the velocity c is made greater than the critical value 1/ (gh), we see at once that 
we must omit a central portion of the integration in (5), because the component plane waves 
can only begin to exist at such an inclination 6 that their wave velocity ccos@ is equal 
to 4/ (gh). On working out the wave pattern in more detail, it is found that it consists 
then of only diverging waves. 
4. We return to the expression (5) for deep water. The origin O was taken at a fixed 
point, but it is more convenient to take a moving origin for the co-ordinates at the focus 
point of the wave pattern; so in what follows we shall write x instead of x —ct. Further, 
for brevity we shall write 
(x, y) = « sec? 6 (~ cos 6 + y sin @) Alpi oe tite (6) 
We may call the surface elevation given by 
7 
= |sneenai rem OM ok lero Heo 1 (C)) 
a simple sine pattern. 
380 
