2:30-231] INEQUALITIES IN THE BED OF A STREAM. 409 



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. 



The discussion of the integral 



...(iv) 



I 



cosh f - cfh/c 2 . sinh 



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

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

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



Thus when c 2 &amp;gt; gh we find, for the surface-form, 



&amp;lt; v &amp;gt;&amp;gt; 



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

 When c 2 &amp;lt; gh, the practical solution is, for x positive, 



*i sm , 

 and, for x negative, 



The symbols a, /3 g , A t B B 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. 



+ &quot;On the theory of Oscillatory Waves,&quot; Cavrib. Trans., t. viii. (1847); reprinted, 

 with a &quot;Supplement,&quot; 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, &quot; Zur Theorie von Wind und Wellen,&quot; Berl. Monatsber., July 25, 

 1889. 



