THE 

 LONDON, EDINBURGH, and DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE. 



[SIXTH SERIES.] 





FEBRUAR Y 1911. 



XXIII. Hydrodynamical Notes. 

 By Lord Rayleigh, O.M., F.R.S* 



Potential and Kinetic Energies of Wave Motion. — Waves moving 

 into Shallower Water. — Concentrated initial Disturbance with inclusion 

 of Capillarity. — Periodic Waves in Deep Water advancing without 

 change of Type. — Tide Races. — Rotational Fluid Motion in a Corner. — 

 Steady Motion in a Corner of Viscous Fluid. 



IN the problems here considered the fluid is regarded as 

 incompressible, and the motion is supposed to take place 

 in two dimensions. 



Potential and Kinetic Energies of Wave Motion. 



When there is no dispersion, the energy of a progressive 

 wave of any form is half potential and half kinetic. Thus in 

 the case of a long wave in shallow water, " if we suppose 

 that initially the surface is displaced, but that the particles 

 have no velocity, we shall evidently obtain (as in the case of 

 sound) two equal waves travelling in opposite directions, 

 whose total energies are equal, and together make up the 

 potential energy of the original displacement. Now the 

 elevation of the derived waves must be half of that of 

 the original displacement, and accordingly the potential 

 energies less in the ratio of 4:1. Since therefore the 

 potential energy of each derived wave is one quarter, and 

 the total energy one half that of the original displacement, 



* Communicated by the Author. 

 Phil. Mag. S. 6. Vol. 21. No. 122. Feb. 1911. N 



