6 WAVE PATTERNS AND WAVE RESISTANCE. 
disturbance examined by Kelvin in the paper to which reference has already been made. 
In this case the waves in the rear approximate to the form 
Nols 
C= A [see @ sin (x, y) dé Mm ORD el ome se ((IO) 
7 
where A is a constant, and we use the notation specified in section 4. 
We may describe this as a sine pattern with an amplitude factor sec? @, which varies 
from unity at #0 = 0° to infinity at 6 = 90°. We have seen, in section 2, that the transverse 
waves of the pattern come from the range 0° to 35° approximately, while the diverging 
waves come from the rest of the range 35° to 90°, taking one side of the central line O x. 
Thus we should expect the diverging waves in this case (10) to be increased in magnitude 
5 SS SS S58 
a 
2 15° 30° 45° 60° 1° g0° 
Fig. 4.—Grapus or Sect e~*/*°* por DIrFERENT VALUES OF xf. 
compared with those for the simple sine pattern (7); these features are unduly prominent 
in the Kelvin pattern in comparison with those made by an ordinary ship model. 
Incidentally we may note that the factor sec? 6 causes the expression (10) to have an infinite 
value at the focus point. 
An interesting contrast is for a small sphere, of radius a, submerged in the water with 
its centre at a depth f and moving with velocity c. The expression is now 
z= Beat | sect etn sin Go 4 6 6 0 (ii) 
2 
the focus point O being vertically above the centre of the sphere. 
In Fig. 4 are shown curves of the amplitude factor sect @ exp. (— kf sec? 6) for different 
values of «xf, that is, of g f/c?. 
From these curves we get at once some idea of the relative importance of the transverse 
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