2oO-23l] INEQUALITIES IN THE BED OF A STREAM. 409 



ere Q represents the area incln 

 leral level of the bed. For a d 



The discussion of the integral 



where Q represents the area included by the profile of the inequality above the 

 general level of the bed. For a depression Q will of course be negative. 



can be conducted exactly as in the last Art. The function to be integrated 

 differs in fact only by the factor f /sinh ; the singular points therefore are the 

 same as before, and we can at once write down the results. 



Thus when c 2 > gh we find, for the surface-form, 



the upper or the lower sign being taken according as x is positive or negative. 

 When c 2 <gh, the 'practical' solution is, for x positive, 



h smh a 

 and, for x negative, 



The symbols a, /3,, A, B 8 have here exactly the same meanings as in 

 Art. 230*. 



Waves of Finite Amplitude. 



231. The restriction to ' infinitely small ' motions, in the 

 investigations of Arts. 216,... implies that the ratio (a/\) of 

 the maximum elevation to the wave-length must be small. The 

 determination of the wave- forms which satisfy the conditions of 

 uniform propagation without change of type, when this restric- 

 tion is abandoned, forms the subject of a classical research by 

 Sir G. Stokesf. 



The problem is, of course, most conveniently treated as one 

 of steady motion. If we neglect small quantities of the order 



* A very interesting drawing of the wave-profile produced by an isolated in- 

 equality in the bed is given in Lord Kelvin's paper, Phil. Mag., Dec. 1886. 



t "On the theory of Oscillatory Waves," Camb. Trans., t. viii. (1847); reprinted, 

 with a "Supplement," Math, and Phys. Papers, t. i., pp. 197, 314. 



The outlines of a more general investigation, including the case of permanent 

 waves on the common surface of two horizontal currents, have been given by 

 von Helmholtz, " Zur Theorie von Wind und Wellen," Berl. Monatsber., July 25, 



