Growth of a Train of Waves. 17 



work done on any vertical plane is only half the total energy 

 per wave-length. This is only half enough to feed a regular 

 procession, advancing to infinity with abruptly ending front, 

 travelling with the wave-velocity v. It is exactly enough to 

 feed an ideal procession of regular periodic waves, coming 

 abruptly to nothing at a front travelling with half the " wave- 

 relocity" r; which is Osborne Reynolds'"* important con- 

 tribution to the ideal doctrine of " group-velocity. 5 '' 



§ 126. The dynamical conclusion of § 125 is very important 

 and. interesting in the theory of two-dimensional ship-waves. 

 It shows that the approximately regular periodic train of 

 waves in the rear of a travelling forcive, investigated in 

 §§ 18-51, and 65-79 above, cannot be as much as half the 

 space travelled by the forcive, from the commencement of its 

 motion ; but that it would be exactly that half- space if some 

 modifying pressure were so applied to the water-surface in 

 the rear as to cause the waves to remain uniformly periodic 

 to the end of the train ; without, on the whole, either doing- 

 work on them, or taking work from them. 



A corresponding statement is applicable to our present 

 subject, as we shall see in §§ 156, 157 below. 



§ 127. Go back to § 118 ; and first, instead of a sinu- 

 soidally varying pressure, imagine applied a series of impulsive 

 pressures, each of which superimposes a certain velocity- 

 potential upon that due to all the previous impulses ; and let 

 it be required to find the resulting velocity-potential at any 

 time t, after some, or after all, of the impulses, Consider 

 first a single impulse at time t — q ; that is to say, at a time 

 preceding the time t by an interval q. Let the velocity- 

 potential at time t, due to that single impulse applied at the 

 earlier time t — q, be denoted by 



GV(*,s,q) (168). 



According to this notation the instantaneously generated 

 A^elocity-potential is OY(jc, z, 0), and the value of this at the 

 bounding surface of the water is OV(x, 1, 0). Hence, by 

 ■elementary hydrokinetics, if I denotes the impulsive surface- 

 pressure, we have 



I=-CT(*,1,0) .... (169). 



§ 128. Considering now successive impulses at times pre- 

 ceding the time t, by amounts q l9 q 2) . . - . q L ; and denoting 

 by S(;r, ~, t) the sum of the resulting velocity-potentials at 

 time t, we find 



SO, *, 0=C,V(* z, fc) + Q*V(*»*, ?*}+-••■ CiV(«, *, gO (170). 



* ' Nature/ August 1877, and Brit. Assoc. Report, 1877. 

 Phil. Mag. S. 6. Vol. 13. No. 73. Jan. 1907. 



