748 Lord Kelvin on 



surface, having the wave-length o60°/(j + ^) due to the 

 translational speed o£ the forcive. 



§ 52. I confess that I did not expect so small a difference 

 from sinnsoidality through the loliole 240°, as calculation 

 by {(83), (86), (87) J- has proved ; and as is shown in figs. 18, 

 19, 20, by the l)-curve on the right-hand side of G, which 

 represents in each case the value of 



D(0) = d(0)-(-iyd(18O°). sin(j + ±)0 • (89), 



being the difference of d(0) from one continuous sinusoidal 

 curve. The exceeding smallness of this difference for dis- 

 tances from C exceeding 20° or 30°, and therefore through a 

 range between C C of 320°, or 300°, is very remarkable in 

 each case. 



§ 53. The dynamical interpretation of (88), and figs. 18, 

 19,20, is this: — Superimpose on the solution -}(83), (8G), 

 (87) \ a " free wave" solution according to (73), taken as 



_(-iyd(i80°). s m (j+i)d . . . (90). 



This approximately annuls the approximately sinusoidal portion 

 between C and C shown in figs. (13), (14), (15) ; and approx- 

 imately doubles the approximately sinusoidal displacement 

 in the corresponding portions of the spaces C C, and C C on 

 the two sides of C 0. This is a very interesting solution of 

 our problem § 41 ; and, though it is curiously artificial, it 

 leads direct and short to the determinate solution of the 

 following general problem of canal ship-waves : — 



§ 54. Given, as forcive, the isolated distribution of pressure 

 defined in fig. 12, travelling at a given constant speed; re- 

 quired the steady distribution of displacement of the water in 

 the place of the forcive, and before it and behind it ; which 

 becomes established after the motion of the forcive has been 

 kept steady for a sufficiently long time. Pure synthesis of 

 the special solution given in §§ 1-10 above, solves not only 

 the problem now proposed, but gives the whole motion from 

 the instant of the application of the moving forcive. This 

 synthesis, though easily put into formula, is not easily worked 

 out to any practical conclusion. On the other hand, here is 

 my present short but complete solution of the problem of 

 steady motion for which w r e have been preparing, and working- 

 out illustrations in §§ 32-53. 



Continue leftward, indefinitely, as a curve of sines, the D 

 curve of each of figs. 18, 19, 20 ; leaving the forcive curve, 

 F, isolated, as shown already in these diagrams. Or, analy- 

 tically stated: — in (89) calculate the equal values of d(#) for 



