272 Lord Rayleigh on Waves', 



the sum of the squares of the same number of equal parts of a 

 shorter line. 



It follows that when a particle starting from A' has arrived 

 at B', another particle starting at the same moment from A 

 will fall short of B. Thus in a progressive wave the water 

 near the surface has on the whole a motion of translation in 

 the direction in which the waves advance. 



Oscillations in Cylindrical Vessels. 



If liquid contained in a cylindrical vessel of any section, whose 



generating lines are vertical and whose depth is uniform, be 

 isturbed from the position of equilibrium, oscillations will 

 ensue in consequence of the tendency of the fluid to recover 

 its horizontal boundary. 



Let us consider in the first place the small vibrations in two 

 dimensions of a compressible fluid such as air when contained 

 within a cylindrical rigid boundary. If x and y be the rectan- 

 gular coordinates of any point, and (/> the velocity potential, it 

 is known that <p will satisfy over the whole area 



'd(>- a Xd7 + df)> w 



a being the velocity of sound ; while round the contour 



£-* < B > 



where -^- denotes the rate of variation of 6 in a normal direction. 



an T 



Whatever the motion of the air may be, it can be analyzed 

 into components of the harmonic type. Suppose that for one 

 of these (j> varies as cos Kat ; then, from (A), 



S+2+W-o (0) 



is an equation which </> must satisfy for the component vibra- 

 tion in question. The equations (C) and (B) can only be 

 satisfied with certain definite values of k ; and the functions <f> 

 corresponding to these values are proportional to what may be 

 called the normal functions of the air-system. We may denote 

 these functions by u K . Any function arbitrary over the area 

 can be expanded in a series of the functions u *. 



Returning to the liquid-problem, we see that the elevation 

 h of the surface at any point above the undisturbed position 



* See on this subject several papers by the author, especially " General 

 Theorems relating to Vibrations, Math. Society, Proceedings, vol. iv. 

 No. 63, and Phil. Mag. December 1873. 



