﻿Wave of Permanent Type. 357 



5. The Final Equations of the Motion. 

 The fundamental equation (1) may now be written 



b+M0= — 1-25 \/gh{l-'llsec 2 ^(z + i,x)/h} Vl — -40sec 2 J(z + *#)//*, 



which completely determines the motion of the fluid. 



It is convenient, however, to consider the formulae specially 

 suitable to the regions near to, and fairly distant from, the 

 crest. Thus for regions fairly distant from the crest (where 

 exp( — x/h) is small) (5) and (6) give 



-(«+ii»)/U=l-l-24r^— W»+Ao., . . (38) 

 and 



-(f + t<j>)l 1 Gh=(z + M)/h + il-24;€-( x -^/ h -&c., . (39) 



in which U has the value given by (35). 



The equation to the free surface, as given by (7), becomes 



*7/A = l-04e-*/*--44e- to /*&c., . . . .(40) 

 and for the pressure-error given by (14) we have 



8p=-'$9e- 2 *l h gph (41) 



Again, for the neighbourhood of the crest (16) and (17) 

 give 



-(u + mi>)/U=-80 V(?-mO//*{1 + -084(?-mO/A} • ( 42 ) 

 and 

 (^ + ^)/UA=l + -53{(?-^)/A} 3 / 2 {l + -50(?-^)A}. . (43) 



We have already seen that the crest is formed by two 

 surfaces, equally inclined to the bottom, meeting at an angle 

 of 120°, so that the summit of the wave has the form of a 

 blunt wedge. We have also seen that in the free wave these 

 surfaces must be plane or have an infinite radius of curvature 

 at the crest ; we have, however, made no effort to satisfy this 

 condition, but on substituting the values of the constants in 

 (20) we find that it gives the radius of curvature at the crest 

 about equal to thirty times the depth of the water, a result 

 sufficiently indicating the closeness of our approximation. 



The pressure-error near the crest, as given by (27), is 



8p='03gpr 2 /h. ...... (44) 



By (41) and (44) we see that the deviation from constant 

 surface-pressure is everywhere very small ; there is a very 

 slight excess near the crest but vanishing at the crest, and a 

 slight defect near the mean level. The deviation has in fact 

 only an appreciable value over a very limited region, say 

 from x='bh to x — l'bh ; (41) and (44) are hardly applicable 



