Keulegan (1967), King (1974), Goodwin (1974), Escoffier and Walton (1979), 

 and Walton and Escoffier (1981) have solved the basic equations of motion and 

 continuity for an inlet-bay system (Fig. 4-74) by various techniques including 

 (1) analytical solution and (2) numerical solution via analog and digital 

 computer. 



The latter four references include the effects of inertia and tributary 

 inflow into the bay. 



King's (1974) solution (as presented by Sorensen, 1977) for the case of no 

 tributary inflow will be given here. The solution is in the form of curves 

 for the dimensionless maximum channel velocity during a tidal cycle V^ and 

 the ratio of bay to sea tidal amplitude ^^/^u »as functions of a friction co- 

 efficient K, and a frequency coefficient Ko (see Figs. 4-75 and 4-76). He 

 defines 



A TV 



V = .^-^ (4-64) 



m 2iia a, 

 s b 



I 



a A, F 



K = ^ ^ (4-65) 



1 2LA 

 a 



and 



2 T ^1 gA^ 



(4-66) 



'm 



where V^ is the maximum velocity during a tidal cycle and 



en ex 4R 



F = k + k +1^ (4-67) 



With values of a^ , T , K^^ , k^^ , f , L , R , A, , and A^ , K^ 

 and K2 can be evaluated from equations (4-65) and (4-66;; V^ and a^/a^ 

 determined from Figures 4-75 and 4-76; and V calculated from equation 

 (4-64). Note in Figure 4-76 that for certain Kj^ and K2 values, %l^s ^^ 

 greater than 1; that is, the bay range is amplified. This occurs when the 

 inertia of the vater in the channel exceeds the frictional resistance. 



The major assumptions implicit in King's (1974) solution are 



(a) The sea tide is sinusoidal; i.e., rig = a^ sin 2Trt/T where t 

 denotes the time elapsed and rig is the instantaneous sea level. Since 

 the channel resistance is nonlinear, the channel velocity and bay tide 

 will not be sinusoidal. However, for a first approximation, 



V = F sin 2iTt/T and rii, " sl-u sin 2iTt/T can be assumed (where n^^ is 

 the instantaneous bay level) . Thus, the average velocity over the flood 

 or ebb phase of a tidal cycle is approximately equal to (2/3)V^ . 



(b) The bay water level rises and falls uniformly (i.e., the bay 

 water surface remains horizontal) . This assumption requires that the 

 tidal period be long compared to the time required for a shallow-water 

 wave to propagate from the inlet to the farthest point in the bay; i.e., 



4-162 



