Wave Pattern of a Doublet in a Stream. 521 
axis of a, or lines for which ¢’ = + 2\/2. Within the range © > t > 21/2, 
there are two values of ¢ for which the phase is stationary, namely, the roots of 
2 —tt+1=0, (20) 
the smaller root corresponding to the transverse wave and the larger to the 
diverging wave at each point. In the elementary ideal case the constituent 
harmonic waves have equal amplitudes, but in (19) we have the amplitude 
factor (1 + #)e-**. If 8 > 1, this function has a maximum at t = 0, and 
diminishes steadily to zero as ¢ increases. But if 8 <1, there is a minimum 
att = 0 anda maximum at t = {(1 — @)/@}!. We may expect then a difference 
in the wave pattern according as c? is greater or less than gf. When 8> 1 
and « has moderate values, the main part of (19) comes from small values of ¢ ; 
further, when « is large and the typical wave pattern should be developed, 
we see that the diverging waves will be relatively small. On the other hand, 
when 8 <1, there is increasing importance of the diverging waves; and in 
particular, there will be a value of ¢’, that is a certain radial line, for which 
the maximum of the amplitude factor coincides with the greater root of (20) 
for which the phase is stationary. As we are not calculating the wave pattern 
at large distances we need not put down the general first approximations to 
(19) by the stationary phase method ; we may, however, note the particular 
cases for t’ = oo and t’ = 24/2, that is for radial lines along the rearward axis 
and along the line of cusps of the so-called isophasal lines. For these cases 
(19) gives, by the usual methods, 
24} 4.°M 
¢-G=-— Gaye te e tof cos ( Kol" —5), 
for t’ = o, and 
9393 2 
C=G== Eco AN eet sin (heygr/3), (21) 
0 
for t’ = 24/2. We note here the additional factor e~*-/ in the second case, 
so far as variation of the amplitude with the depth is concerned ; we see that 
the relative prominence of the so-called cusp waves is only a feature of the 
limiting case of a point surface disturbance. 
6. Returning to the exact integral (19) for this part of the surface elevation, 
it seemed of interest to make some numerical calculations directly from the 
integral for points near to the origin, or for moderate values of «. Instead of 
following the isophasal lines, which are not significant in this region, we have 
calculated the surface elevation from (19) along certain radial lines. We take 
in turn the values ¢’ = 00, 3, 2\/2, 2, 1 and zero; these are shown in fig. 2 
as A, B, C, D, E and F respectively. 
294 
