516 T. H. Havelock. 
anticipated, to be somewhere about the speed +/(gf). A line doublet is first 
examined, and the surface elevation immediately over the doublet is calcu- 
lated ; it is found to be zero at approximately a speed 0-86 1/(gf). To illus- 
trate the difference for speeds greater or less than this value, curves are shown 
in fig. 1 for the complete surface elevation when gf/c? has the values 4 and 0-5. 
A three-dimensional doublet is then considered and a similar calculation for 
the surface elevation immediately over the doublet gives a critical speed of 
about 0-84 +/(gf). 
The second point is the variation of the wave pattern. We may compare 
it with the pattern due to an ideal point disturbance of the surface of the 
stream. In that case the approximate evaluation of the integrals by the 
method of stationary phase gives the system of transverse and diverging waves 
established in the rear. But in our case there is a variable amplitude factor for 
the constituent harmonic terms of the integrals, and we notice that the velocity 
/(gf) has here also a special significance ; for the amplitude factor itself 
possesses an additional stationary value, a maximum, when the velocity 
exceeds 1/(gf). The difference this makes in the wave pattern is examined ; 
roughly, at lower speeds the pattern consists chiefly of transverse waves, while 
at higher speeds the diverging waves become of increasing relative importance. 
A direct numerical study has been made of the integral for this part of the 
surface elevation for two values of gf/c?, namely, 4 and 0-5; graphs are given 
in figs. 3 and 4 for the surface elevation along various radial lines from the 
origin, including some outside the limits of the ideal wave pattern. 
2. Take Ox in the undisturbed surface of the stream, and Oy vertically 
upwards, and let the velocity of the stream be c in the negative direction of Oz. 
Let there be a two-dimensional horizontal doublet of moment M at the point 
(0, —f). The solution of the problem is familiar as the first approximation for 
the effect of a submerged cylinder of radius a, if we take M = ca?. We quote 
here the complete expression for the surface elevation 1 in the form used in 
previous calculations* 
ny 2Mf 29M. [ m cos mf — ky sin mf ,—-me din, (1) 
eeepc Jo mBE a? 
for z > 0, and 
9 aD 28 . a 
xa aM - + 2a | mM COs mf oe mf = dm 
c (a? + f?) c Jo me + Ko 
+ (4 pM ic) e~% sin xyz, (2) 
for x < 0, with ky = g/c’. 
* «Roy. Soc. Proc.,’ A, vol, 115, p. 271 (1927). 
