37 



A special class of tidal hydraulics pertains to the hydraulics of 

 tidal inlets or entrances connecting the sea to relatively small and deep 

 bays. A simple, coherent theory for predicting water level variation in 

 the bay for a given, sinusoidally forced, sea tide has been presented by 

 Keulegan (1967). Mehta and Ozsoy (1978) have reviewed various approaches 

 including developments previous and subsequent to Keulegan' s contribution. 



4.3 PHYSICAL PRINCIPLES 

 4.3.1 Tidal Propagation 



According to the equilibrium theory of tides, the tidal amplitude can 

 be shown to be proportional (to leading order) to the fourth power of 

 earth's radius, considering the moon-earth system. Since this number 

 (6,378 km) is so large compared to any expected effect of sea level rise 

 (i.e., increase in earth's radius), the corresponding change in the tidal 

 range on this account would be negligible. In order to evaluate the effect 

 of sea level rise on the tidal range, the nature of propagation of tide in 

 very shallow waters must be considered. 



The simplest description of tide in the dynamic sense is that of a 

 shallow water wave moving along the x- direction with a speed or celerity, 

 Cq. If a frictionless bottom is assumed, the wave equation is 



2 2 



3_a = c 2 5_a_ (4.1) 



at ax 



where r?(x,t) is the instantaneous water surface elevation. The celerity, 

 ^o ^ (gti) ' where g is acceleration due to gravity and h is water depth. 



The effect of friction can be accounted for by including an additional 

 term on the right hand side of Eq. 4.1. Thus, for example, this term under 

 the assumption of linearized friction is -gMar;/at, where M is an empirical 

 coefficient accounting for the magnitude of bottom friction. Friction 

 slows down the speed of propagation (celerity) , decreases the current speed 

 and reduces the tidal range compared with frictionless tide. The effect is 

 depth- dependent, and it can be shown that in fact it varies with h'^/ 

 which means that increasing the water depth would decrease frictional 

 damping, thereby increasing the tidal range. Observations in the German 



