427 Dr. T. H. Havelock. The Propagation of [Aug. 26, 
defining the lines of cusps within which the wave pattern lies, are given by 
such values of « that 
8 (1—n) 
(3 —n)?" 
Whatever the value of «, can only lie between 1 and 0; hence « can 
only lie between cos~12,/2/3 and 7/2, or between 19° 28’ and 90°. The 
smaller value is the limiting angle for deep water, when 7 is considered zero 
for all values of x. 
We see from (115) and (116) that the equal values of cw occur when 
1—lpm (1 +n) (8—n) = 3 (1—n), 
cos? a = 
(117) 
or when m(3—n) = 2/>p. 
The greatest possible value of m (3—7) is 2; hence we have the limitation 
p>1. Only in this case is there a double wave system with a line of cusps. 
As p decreases to 1, that is as the velocity ¢ approaches the critical value 
s/(gh), mand n at the line of cusps both approach their limiting value 1 ; 
and at the same time the cusp angle widens out, approaching a right angle. 
Further, along the axis we have 
(oi = I, m =1/p =e] gh. 
Hence on the axis the transverse waves are the simple waves travelling 
with velocity c on water of depth 4. As p decreases to 1, the wave-length 
increases indefinitely ; m, and consequently n, approach unity on the axis. 
Now if 2 is 1, the group-velocity U equals the wave-velocity V, and the 
medium is non-dispersive. Thus at the critical velocity c, equal to ,/(gh), 
we have a source emitting disturbances and travelling at the rate of propa- 
gation of the disturbances ; we see that the whole effect is practically concen- 
trated into a line through the source at right angles to the direction of 
motion. This agrees with observations of ship waves when approaching 
shallow water at the critical velocity.* 
(b) c>,/(gh); p<1.—We may now have the greatest value, unity, of m ; 
it is easily seen that for less values of m and n the values of a given by (113) 
become smaller. 
At the outer limit we have 
cos? « = 1—p, si@a = p = gh/c?. (118) 
Consequently the wave pattern is contained within two lines making with 
the axis an angle which diminishes as ¢ increases. 
* ‘Trans. Inst. Nav. Arch.,’ vol. 47, p. 353 (1905). Compare also the motion of an 
electron with the velocity of radiation. 
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