98 Prof. A. R. Richardson on 



Part I. 

 I. Stream-line flow, under gravity, with a free surface. 

 Adopt the usual notation : 



z = x+iy % y being measured vertically upwards, 



q, 6 refer to the fluid velocity. 

 Consider the equation 



-£ = M^*[ {1 - G ' >)}i+2 ' GV) ] = -/ !S - • (1) 



Ifc' along part of any stream-line, say i|r = ^ , Q(iv)i, G'(w), 

 {1 — G /2 (iv)}* are real, and finite, over that part 



y=-| t {c+{G( W )pj (2) 



9»=y{&(«)}i 



i.e. q* + 2 ffl/ =-^0 (3) 



i£ V?=1g (4) 



Hence there will be a free surface, and (1) will give the 



solution of the problem provided G(iv) is chosen so as to 



satisfy the conditions over the rigid boundary. The presence 



of the term {G-{w)}* makes this choice difficult, and in 



dz 

 addition to singularities of -7— which arise where a rigid 



" aw & 



boundary bends, or meets a free surface, others may occur 

 at places such as the crest of a wave of maximum elevation. 

 If z# = is such a place, near w = 



x / V3 i\ ,. 



(4> \h\T + 2) a PP roximatel J' 



^ -£ = - ^* 



acp 



i.e.,i£ <£>G 0=tt/6 



4><0 <9= — 7J-/6, and at the crest ^ = 0. 



