630 UROEN AND GROVES [CHAP. 17 



From (20), (21) or (22), (23) one can calculate height differences between 

 any two points once the current field V{x, y) is known. In order to find absolute 

 values of ^ one more condition is necessary, viz. either, in the case of a wholly 

 enclosed body of water, a normalization condition stating that the integral of 

 t, over the whole area is zero, or a boundary condition giving an absolute 

 reference level somewhere at an open boundary. If, according to equation (12), 

 we assume that ^ = at a point of the boundary (see Fig. 9), 



UO) = 0, (24) 



we can find the height ^(P) at any place P by integrating (20), (21) or (22), (23) 

 from O to P, supposing the current field is known. This integral can be split up 

 into two parts, 



^P) = UP) + ^t/(P), (25) 



where Z,r means the part of the integral found by integrating only the terms 

 containing Ta, and l,u the part found by integrating only the terms containing 

 U. We shall call ^t the "static wind effect" and l,u the "current effect". In 

 order to have these two parts unambiguously defined, the path of integration 

 from to P should be fixed by some suitable agreement. 



For the following we shall confine ourselves to the linear equations (22) and 

 (23). Introducing the stream function ^{x, y) according to (17) and (19), and 

 introducing vector notation, we have 



U = [j X V(^], (26) 



^.(P) = p-i^-iJJ/.-iT(^s, (27) 



^f/(P) = g-' ^l ifh-^ V<?^-r/.-2[j X V<^]) ds, (28) 



where j is the unit vector pointing vertically upwards, x means the vectorial 

 product, ds is an infinitesimal vectorial line element, t= (1 +m)Ta, and where, 

 as has already been said above, the path of integration from O to P is supposed 

 to have been fixed by agreement for all P, since the integrands of (27) and (28) 

 are, in general, not irrotational. 



By differentiating (22) with respect to y and (23) with respect to x and sub- 

 tracting, we find the equation which ^ must satisfy (/ is treated as a constant 

 within the area concerned) : 



If the sea area is wholly enclosed, the only boimdary condition which ^ has 

 to satisfy follows from (9) : 



(j) = constant along the boundary. (30) 



If, moreover, the depth is constant, equation (29) becomes very simple : 



V2<^ = p-ir-iA curl T. (31) 



