425 Dr. T. H. Havelock. The Propagation of [Aug. 26, 
with wave front through P at right angles to AP, and each system can be 
expressed in the form 
£ = F(o,2) cos {x (w?— 2cum cos a+ c?u*)? + e}, 
with cw and « as functions of o and a. 
Suppose the medium is such that the group-velocity bears a constant ratio 
to the wave-velocity, that is, suppose 
U=3("4+1)V, (104) 
where 7 is independent of x. 
Then the equations (102) and (103) lead to a quadratic for cw, namely, 
(1—n) @v?+(n—3) cum cos a+ 257 = 0. (105) 
Hence we have the roots 
uh = Taw —n) cosat,/{(3—n)? cos?a—8(1—x)}]. (106) 
We shall examine some special cases. 
(a) O<n<1.—There are two positive values of cw which are real, provided 
cos? a >8 (1l—n)/(8—n)?. 
Thus there are two wave systems, transverse and diverging, with a line of 
cusps corresponding to the double roots, and the whole wave pattern is 
included within an angle 
2 cos! {8 (1—n)}?/(8—n), (107) 
which increases with n. 
The previous section on deep-water waves is the case of n zero, 
(b)x=1. This is a critical case, implying coincidence of wave-velocity 
with group-velocity, and consequently no dispersion. 
(c) n= 2. This is the case of capillary surface waves. We see that there 
is only one positive root of the quadratic, and it is real for all values of a; 
the root is 
cu = 30 {(cos?«+8)?—cos a}. (108) 
There is only one wave system, but it extends over the whole surface ; 
along the line of motion « is zero in the rear, while in advance of the 
impulse it is of value suitable to simple waves moving with velocity c. 
(d) n = 3. This holds for flexural waves on a plate; there is one system 
of waves extending over the surface, corresponding to the root cw = a. 
The crests, and other lines of equal phase, are given by the curves 
o sin? se = constant. 
(ce) Gravity and capillarity combined—tThe relation between U and V is 
28 
