415 Dr. T. H. Havelock. The Propagation of [Aug. 26, 
Then we choose u so that 
cS (eto =0, or u(cu—20)(s+eu)? = 0. 
The value giving a regular wave pattern is the positive root 
cu = 20, for w positive. 
Hence in this case we have a regular train of waves of length suitable to 
the velocity ¢ in advance of the moving pressure system, with no regular 
pattern in the rear. 
§ 11. Water Waves due to Gravity and Capillarity. 
If we take account of gravity and the surface tension together, we have the 
velocity function 
V = (Te+g/«)}. (59) 
= Gh _ _3Te+9 
Hence Us ae (KV) = Sraisneaar (Tet gk (60) 
We have not here a simple ratio U/V, independent of «. The velocity V 
has a minimum ¢m for a certain value Km, equal to (g/T), and for this value 
U is equal to V—as in fact follows from the definition of U. For «<#m, U 
is less than V, tending ultimately to }V; while for «>«m, U is greater than 
V and approaches as a limit $V. 
If we consider a travelling line impulse, the whole problem of finding the 
principal groups is contained in the equations 
wre _pyH 3TK?+9 
Uw > 2 (Tx? + x)3 i (61) 
c= V= (Te+9/«)t 
Actas? ae 2 +(ct—cmt)t 
c8—Cn!t 2T 
where the positive sign is taken for o positive (in advance of the impulse), 
and the negative sign for a negative (in the rear). Thus there is no wave 
Hence Cu = 
pattern unless ¢ is greater than the minimum wave-velocity c,; and if so 
there are regular trains both in advance and in the rear, the smaller wave- 
lengths being in advance. With the ratio c/em large, the results approximate 
to very small waves in front and waves in the rear with « equal to g/c’. 
§ 12. Surface Waves in two Dimensions. 
Suppose that the initial data instead of being symmetrical about a 
transverse straight line are symmetrical around the origin. Let the axes of 
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