SECT. 5] SURGES 633 



the North Sea, used inhomogeneous wind field models composed of a number 

 of homogeneous subfields. Workers of the Mathematisch Centrum, Amsterdam, 

 on the other hand have computed wind effects in geometrically simple basins 

 under linear wind fields (Veltkamp, 1954; Lauwerier, 1959). 



The effect of the current field upon surge heights is twofold, according to 

 equation (28) : a Coriolis-force effect which gives a raising of the sea-levels to 

 the right-hand side, a lowering to the left-hand side of the current, and a 

 friction effect which adds a sloping down in the downstream direction of the 

 path of integration used in defining the current effect according to (25), (27) 

 and (28). (Cf. Fedorov, 1956.) 



c. General case 



The linearized, time -dependent equations of motion are written as follows : 



-^ = -/[ j X U] - rh-^V -ghV^ + p-^x - hp-^ Wpo, (36) 



where, as before, t = (1 + ni)'Za. If the bottom stress term is kept in the quadratic 

 form, we have 



^= -fUxV]-sh-WV-ghWUp-''T-hp-^Vpo. (37) 



The continuity equation is 



I = -VU. (38) 



The boundary conditions are those of section 3-B-a. 



The most simple case of non-equilibrium is the case of a free oscillation, 

 without pressure differences or wind stresses acting and without non-stationary 

 boundary actions (external surges). The result is a damped wave or "seiche", 

 the possible periods of which depend on the geometry of the basin, its bound- 

 aries (whether closed or open), the effectiveness of friction and the direction in 

 which the body of water has been brought out of equilibrium. The mathematical 

 theory of such oscillations will not be dealt with here. Undamped seiches with 

 geostrophic (Coriolis) effects and damped seiches without geostrophic effects 

 have for a long time been studied theoretically (cf. Proudman, 1953, ch. XI and 

 XIV; Saito, 1949; Harris, 1954; Reid, 1957; Hofsommer, 1958). Inclusion 

 of both friction and Coriolis force makes the problem much more difficult, 

 mathematically. This problem has been treated by van Dantzig and co-workers 

 (1958, 1959). 



Kinematically, the local variation of elevation at a certain point under a 

 simple free oscillation may be described as a damped sinusoid 



^ = ^(0)e-«< cos 27TtlTo for t > h, (39) 



satisfying the equation 



i + 2a§ + 6=? = 0. (40) 



