30 Lord Kelvin on tJtc j 



the motion — A cos 27ra?/\sino>£, which needs no surface- 

 pressure. In the motion thus compounded, we have equal 

 sinusoidal waves travelling outwards in the two directions 

 beyond MN (semi-amplitude A) : and, in the space MN, 

 we have a varying water-surface found by superimposing on 

 the motion P sin cot an arbitrary shape of surface, varying 

 sinusoidally according to the formula — Qcoscof. 



§ 148. A curiously interesting dynamical consideration is 

 now forced upon us. The P-component of motion needs, as 

 we have seen, no surface-pressure. The Q-component of 

 motion is kept correct by the surface-pressure FU')coso>£, 

 which, in a period, does no total of work on the Q-motion; 

 but work must be done to supply energy for the two trains 

 of waves travelling outwards in the two directions. Hence 

 this work is done by the activity of the surface-pressure upon 

 the P-component of the motion. 



§ 149. Another curious question is forced upon us. Our 

 solution of §§ 135-115 has given us determinately and 

 unambiguously, in every variety of the cases considered, the 

 motion of every particle of the water throughout the space 

 occupied. The synthetic method of quadratures which we 

 have used could lead to no other motion at any instant due 

 to the applied surface-pressure ; but now, in § 117, we have 

 considered a Q-motion alone, kept correct by the applied 

 surface-pressure. Would this motion be unstable ? and, if 

 unstable, would it in a sufficiently long time subside into the 

 motion expressed in the determinate solution of §§ 135-145 ? 

 The answer is Yes and No. At any instant, say at £ = 0, let 

 the whole motion be the Q-component alone of § 148. Let 

 now the surface-pressure, F(#) cos cot, be suddenly commenced 

 and continued for ever after. It will, according to §§ 135- 

 145, produce determinately a certain compound motion (P, Q) 

 which will be superimposed upon the motion existing at 

 time t = : and this last-mentioned motion, given with its 

 infinite amount of energy distributed from a?= — go to 

 #=:-|-co, and left with no surface-pressure, would clearly 

 never come approximately to quiescence, through any range 

 of distance from qn the two sides. Thus we see that, 

 though the Q-motion alone of § 148 is essentially unstable, 

 the condition of the fluid does not subside into the deter- 

 minate solution of §§ 135-145. It would so subside, if it 

 were given initially only through any finite space however 

 great, on each side of 0. In fact, any given distribution of 

 disturbance through any finite space however great on each 

 side of 0, left to itself without any application of surface- 

 becomes dissipated away to infinity on the two 



