THE 

 LONDON, EDINBURGH, and DUBLIN 



PHILOSOPHICAL MAGAZINE 



AND 



JOURNAL OF SCIENCE. 



♦ 



[FIFTH SERIES.] 



APRIL 1876. 



XXXII. On Waves. By Lord Rayleigh, M.A., F.R.S.* 



THE theory of waves in a uniform canal of rectangular sec- 

 tion, in the case when the length of the wave is great in 

 comparison with the depth of the canal, and when the maxi- 

 mum height of the wave is small in comparison with the same 

 quantity, was given long ago by Lagrange, and is now well 

 known. A wave of any form, subject to the above conditions, 

 is propagated unchanged, and with the velocity which would 

 be acquired by a heavy body in falling through half the depth 

 of the canal. The velocity of propagation here referred to is 

 of course relative to the undisturbed water. If we attribute 

 to the water in the canal a velocity equal and opposite to that 

 of the wave, the wave-form, having the same relative velocity 

 as before, is now fixed in space, and the problem becomes one 

 of steady motion. It is under this aspect that I propose at 

 present to consider the question ; and we will therefore suppose 

 that water is flowing along a tube, whose section undergoes a 

 temporary and gradual alteration in consequence of a change 

 in the vertical dimension of the tube. The principal question 

 will be how far the pressure at the upper surface can be made 

 constant by a suitable adjustment of the velocity of flow to the 

 force of gravity. 



That the two causes which tend to produce variation of pres- 

 sure at the upper surface act in opposition to each other is at 

 once evident. If there were no gravity, the pressure would 

 vary on account of the alteration in the velocity of the fluid. 

 Since there must be the same total flow across all sections of 



* Communicated by the Author. 

 Phil. Mag. S. 5. Yol. 1. No. 4. April 1876. T 



