Computatton of Shallow Water Shtp Mottons 
4 5) 
Boe =: fag = gee :. a dx[S(x)z _ (x) +5B(x)] 
yy 
- (5.12) 
where Z_(x) is the z-co-ordinate of the centroid of the section at x, 
Unfortunately if i= 4, the element n, = yng - zn, cannot be written 
as the normal derivative of a harmonic function, so that the two- 
dimensional Green's theorem cannot be used, as in the above deriva- 
tion, It would appear that we must leave the formula for T,, in the 
form 
2 
T sae ier d At aan (G13) 
al mag! : 4°4 ; 
Lt H 
and evaluate the contour integral explicitly. 
Computation of all quantities (apart from Tye ) in the horizont- 
al equations of motion now proceeds via preliminary computation of 
the potential jumps A o ;(x) . These are related to the inner streaming 
velocity V;(x) by (Tuck 1970, Equation (54) ) 
cee? p 
V.(x) -4|4 + | J seston” (|x - g) ae (5. 14) 
- dx 
which comes from the outer expansion, and 
dah sxe Nsfae vet 1/2 dg. (x) as y—>t o (54215) 
which is the inner boundary condition. Solving the inner flow problem 
leads to a connection between V; and Ad; » which in combination with 
(5.14) gives an integro-differential equation for Ad ;(x) . 
For example, if we solve the canonical problems indicated by 
Figure 5.2, i= 2,4,. weshave 
@, = Voy + (V5-1) ¥, (5. 16) 
Sos 
