742 Lord Kelvin on 



For our present case, of 8— J, (82) gives 



<% /• i i\f i ! ecosd e 2 cos 20 , ^cos/0 ) /Q „. 



1 y ^ 2 7 2 7 2 2 . J 



With «/ and <^ thus expressed, (83) gives the solution of our 

 problem. 



§46. In all the calculations of §§46-61 I have taken 

 e = '9, as suggested for hydrokinetic illustrations in Lecture X. 

 of my Baltimore Lectures, pp. 113, 114, from which tig. 12, 

 and part of fig. 11 above, are taken. Results calculated from 

 (83), (86), (87), are represented in figs. 13-16, all for the 

 same forcive, (74) with e = "9, and for the four different 

 velocities of its travel, which correspond to the values 20, 9, 

 4, 0, of j. The wave-lengths of free waves having these 

 velocities are [(77) above] 2a/41, 2a/ 19, 2o/9, and 2a. The 

 velocities are inversely proportional to y'41, Vl9, V^, V2. 

 Each diagram shows the forcive by one curve, a repetition 

 of fig. 12 ; and shows by another curve the depression, d, of 

 the water-surface produced by it, when travelling at one or 

 other of the four speeds. 



§ 47. Taking first the last, being the highest, of those 

 speeds, we see by fig. 16 that the forcive travelling at that 

 speed produces maximum displacement upwards where the 

 downward pressure is greatest ; and maximum downward dis- 

 placement where the pressure (everywhere downward) is 

 least. Judging dynamically it is easy to see that greater 

 and greater speeds of the forcive would still give displace- 

 ments above the mean level where the downward pressure of 

 the forcive is greatest, and below the mean level where it is 

 least ; but with diminishing magnitudes down to zero for 

 infinite speed. 



And in (75) we have, for all positive values of J<1, a 

 series always convergent, (^though sluggishly when #==1,) by 

 which the displacement can be exactly calculated for every 

 value of 0. 



§ 48. Take next fig. 15, for which J = 4J, and therefore, 

 by (77), v= s/ga/dir, and X = a/4'5. Remark that the scale 

 of ordinates is, in fig. 15, only 1/2*5 of the scale in fig. 16 ; 

 and see how enormously great is the water-disturbance now 

 in comparison with what we had with the same forcive, but 

 three times greater speed and nine times greater wave-length 

 (u= V 'gajir, X = 2a). Within the space-period of fig. 15 we 

 see four complete waves, very approximately sinusoidal, 

 between M, M, two maximums of depression which are almost 

 exactly (but very slightly less than) quarter wave-lengths 



