182-183] SPECIAL PROBLEMS. 297 



Waves of Finite Amplitude. 



183. When the elevation rj is not small compared with the 

 mean depth h, waves, even in an uniform canal of rectangular 

 section, are no longer propagated without change of type. This 

 question was first investigated by Airy *, by methods of successive 

 approximation. He found that in a progressive wave different 

 parts will travel with different velocities, the wave- velocity corre- 

 sponding to an elevation rj being given approximately by 



where c is the velocity corresponding to an infinitely small 

 amplitude. 



A more complete view of the matter can be obtained by the 

 method employed by Riemann in treating the analogous problem 

 in Acoustics, to which reference will be made in Chapter x. 



The sole assumption on which we are now proceeding is that 

 the vertical acceleration may be neglected. It follows, as ex- 

 plained in Art. 166, that the horizontal velocity may be taken to 

 be uniform over any section of the canal. The dynamical equation 

 is 



du du drj /1X 



di + w d X = -9d x ....................... 



as before, and the equation of continuity, in the case of a rect- 

 angular section, is easily seen to be 



where h is the depth. This may be written 



du 



Let us now write 



2P=f(r,) + u, 2Q=f(rj)-u ............... (4), 



where the function f(rj) is as yet at our disposal. If we multiply 

 (3) by/ (77), and add to (1), we get 



dP dP 



* " Tides and Waves," Art. 198. 



