423 Dr. T. H. Havelock. The Propagation of [Aug. 26, 
We notice first the difference of phase of 4a between this and the 
expressions for the separate systems where they cut the outer boundaries ; 
this is analogous to the change of phase along an optical ray in passing a 
focus. We saw that the separate transverse and diverging crests converged 
towards points of equal phase on the outer boundaries given by 
wo = 2c? (2n+ 4) r/9\/3, 
but with the result given in (96) we see that the actual crests on the line of 
cusps are given by 
w= 2c? (Qn+3) w/q,/3. (97) 
The amplitude of the cusped waves diminishes at a slower rate than the 
transverse waves, so that their size becomes relatively more marked towards 
the rear of the disturbance. The amplitude of successive crests is given by 
(96) and (97) as 
me = 3 I a 98 
s 2°T (2) (24 3)4xr? cp ee) 
The amplitude of successive crests of the transverse waves where they cut 
the axis are given by (82) and (84), and we find 
92 
See er RY 99 
oma (2n+4)*0r ctp (99) 
Taking the ratio of these two quantities we have an expression for the 
magnitude of the crests at the cusps compared with the transverse crests on 
the axis ; approximately 
Gre (2n+4) 
sae — 199) SS 100 
Ena Ont) ae 
The following table and curve show how the successive crests at the axis 
and outer line diminish, and exhibit their relative magnitudes for different 
values of n.* 
* On August 3, 1887, Lord Kelvin delivered a lecture “On Ship Waves” before the 
Institution of Mechanical Engineers at Edinburgh, in which he appears to have shown a 
model to scale of the theoretical wave pattern produced by a ship. Only a diagram of 
the crest curves has been published (‘Popular Lectures,’ vol. 3, p. 482); the form of the 
crests agrees with that deduced above, except of course near the disturbance or the 
radial boundaries. It has, in fact, been verified that on substituting his expressions for 
x, y in terms of a parameter w in the present equations, the latter are satisfied identically. 
The law of amplitude along the waves is not stated by Lord Kelvin: as Prof. Lamb 
conjectures, his result seems to have been obtained by an application of the idea of group- 
velocity (H. Lamb, ‘Hydrodynamics,’ 1932 edn. pp. 406/7.) 
26 
