450 Sir William Thomson on Stationary 



D is found by (15 ), which is a cubic equation in D , most 

 easily solved by successive approximations according to the 

 process obviously indicated by the form in which the equation 

 appears in (15). (As a first approximation take D for D in 

 the second member, and so on.) 



To work out the formula (19) for the case of infinitesimal 

 displacement, we may take ty at a great enough distance from 

 inequalities to let the surface in its neighbourhood be sensibly 

 a curve of sines, and the motion simple harmonic. The in- 

 vestigation is facilitated by also taking ^} at a node, as in the 

 diagrams. If we take 



l) = hsinmx (20) 



as the equation of the free surface, the known solution for 

 simple harmonic waves in water of depth D gives, 



C 6 m(D-y) _j_ e -m(T)-y) -\ 



u = U^l + mh — emD _ e _ mJ) sin ma? j, 



v= Um/j- — - * D cos mx, J> • (21). 



where 



V \m e mD + e- mT >) ' 



Hence, where x=§, as in the nodal section ^5 P B, 



6 m(D-y) _ 6 - m(D-y) 



«=U, and v=Vmh ,.„ „ . . (22); 



also 



J D «%=iu 



2»D_ 6 -2mD_ 4mD 



mh (< «j>_ <r «P)i • • • • (23), 



= i^{l- £ » D 4 r^r» } • • • ( 24 )- 



Now going back to (19) we see that when U approaches 

 the critical velocity \/ gj) __ — ° , the first term might 



become important, even though the corrugations at a great 

 distance down-stream from the inequalities were infinitesimal. 

 Reserving consideration of this case, and supposing for the 

 present U to be considerably smaller than the critical value, 

 we may neglect the first term in comparison with the second, 

 remembering that in fact quantities comparable with the first 



