184-185] TIDES OF SECOND ORDER. 301 



The occurrence of the factor x outside trigonometrical terms in (iv) shews 

 that there is a limit beyond which the approximation breaks down. The 

 condition for the success of the approximation is evidently that go-axfc? 

 should be small. Putting c z =gk, \ = / 2irc/a; this fraction becomes equal to 

 2;r (a/h) . (#/A). Hence however small the ratio of the original elevation (a) 

 to the depth, the fraction ceases to be small when x is a sufficient multiple of 

 the wave-length (X). 



It is to be noticed that the limit here indicated is already being over- 

 stepped in the right-hand portions of the figure above given ; and that 

 the peculiar features which are beginning to shew themselves on the rear 

 slope are an indication rather of the imperfections of the analysis than 

 of any actual property of the waves. If we were to trace the curve further, 

 we should find a secondary maximum and minimum of elevation developing 

 themselves on the rear slope. In this way Airy attempted to explain the 

 phenomenon of a double high-water which is observed -in some rivers ; but, 

 for the reason given, the argument cannot be sustained*. 



The same difficulty does not necessarily present itself in the case of a canal 

 closed by a fixed barrier at a distance from the mouth, or, again, in the case 

 of the forced waves due to a periodic horizontal force in a canal closed at both 

 ends (Art. 176). Enough has, however, been given to shew the general 

 character of the results to be expected in such cases. For further details 

 we must refer to Airy's treatise f. 



Propagation in Two Dimensions. 



185. Let us suppose, in the first instance, that we have a 

 plane sheet of water of uniform depth h. If the vertical accelera- 

 tion be neglected, the horizontal motion will as before be the same 

 for all particles in the same vertical line. The axes of x, y being 

 horizontal, let u, v be the component horizontal velocities at the 

 point (x, y), and let ? be the corresponding elevation of the free 

 surface above the undisturbed level. The equation of continuity 

 may be obtained by calculating the flux of matter into the 

 columnar space which stands on the elementary rectangle 

 viz. we have, neglecting terms of the second order, 



\M / T CN \ (S Cv / -f rv \ CX w 



whence 



* Cf. McCowan, 1. c. ante p. 277. 



t " Tides and Waves," Arts. 198, ...and 308. See also G. H. Darwin, "Tides, 

 Encyc. Britann. (9th ed.) t. xxiii., pp. 362, 363 (1888). 



