634 GKOEN AND GROVES [CHAP. 17 



where the period of free oscillation Tq may be written 



'^ (41) 



We shall return to these formal relations later on. 



{i ) Wind surges 



Formal solutions of equations (36) or (37) and (38), including varying wind- 

 stress, in combination with the boundary conditions of section 3-B-a, have only 

 been obtained for the very simplest models (see e.g. Lauwerier, 1957, 1959a; 

 Hofsommer et al., 1959). 



Numerical techniques of approximating solutions by stepwise integration, 

 starting from given initial conditions and using appropriate difference equations 

 instead of the differential equations, have been developed by various authors 

 and are discussed briefly in section 3-B-d in connection with the forecasting of 

 wind surges. 



A method of successive approximation by an iterative process, proposed by 

 Groen, has been described by Weenink and Groen (1958). It does not reckon 

 with predetermined initial conditions, so that any free oscillation may be super- 

 posed on the solution thus obtained. The method is briefly as follows. In sea 

 areas of the sort we are especially dealing with in this section 3-B, the quasi- 

 equilibrium state corresponding to and varying with the varying wind-stress 

 field may in many cases be looked upon as a first approximation to reality, as 

 we have seen in section 3-B-b. Writing equation (36) (without the term with 

 V^o) as follows: 



i?(U,VO-fp-iT = -^, (42) 



and writing the equilibrium wind effect and current field as ^o, Uo, we have, 

 apparently, 



^(Uo,V^o) + p-iT = 0, (43) 



VUo = 0. (44) 



We can now proceed to a next approximation, ^i, Ui, by substituting ^o and 

 Uo in the left-hand members of (36) and (38) : 



Q{VuV^i) + p-^x = ^, (45) 



VUi = -^' (46) 



the boundary conditions being as before. An advantage of this method of 

 approximation lies in the fact that no time-derivatives of unknown variables 

 occur in the equations to be solved. (It bears some resemblance to the 

 approximation of the ageostrophic wind component by means of the so-called 

 isallobaric wind.) 



