180 



Lord Rayleigh : Hydrodynamical Notes. 



Equations (2), (5), (6), (9) may be regarded as giving the 

 solution of the problem in terms of a known <x. It is perhaps 

 more practical to replace a in (5) by \ , the corresponding 

 wave-length in a great depth. The relation between <r and 

 X being a 2 -— 27n//A, , we find in place of (5) 



l' = 27rl/\ = k l (10) 



Starting in (10) from A aQ d I we may obtain l\ whence 

 (6) gives kl, and (9) gives U/U . But. in calculating 

 results by means of tables of the hyperbolic functions it is 

 more convenient to start from kl. We find 



kl. 



V. 



U/U . 



1-000 



kl. 



V. 



u/u . 



00 



kl 



•6 



•322 



•964 



10 



kl 



l'OOO 



•5 



•231 



•855 



5 



4-999 



1001 



•4 



•152 



•722 



2 



1-928 



1-105 



•3 



•087 



•566 



1-5 



1-358 



1-176 



•2 



•039 



•390 



1-0 



•762 



1-182 



•1 



•010 



•200 



•8 



•531 



1-110 



kl 



(kl? 



2kl 



•7 



•423 



1048 



__ 



— 



— 



It appears that U/Uo does not differ much from unity 

 between I' — *23 and £' = 00, so that the shallowing of the 

 water does not at first produce much effect upon the height 

 of the waves. It must be remembered, however, that the 

 wave-length is diminishing, so that waves, even though they 

 do no more than maintain their height, grow steeper. 



Concentrated Initial Disturbance with inclusion of 

 Capillarity. 



A simple approximate treatment of the general problem 

 of initial linear disturbance is due to Kelvin *. We have 

 for the elevation y at any point x and at any time t 



lf°° 

 r) = — 1 cos k,v cos at dk 



^ /*co i r°° 



= -— 1 cos (kx — at)dk + ^— 1 cos (kx + at)dk, . (1) 

 27r Jo j7r Jo 



in which a is a function of &, determined by the character 



* Proc. Roy. Soc. vol. xlii. p. 80 (1887) j ' Math, and Phys. Papers/ 

 iv. p. 303. 



