5. Shallow-Water Tides. 



The astronomical tides generated in the deep ocean act (in general) like progressive waves 

 as they travel across shallow parts of the continental shelves and into estuaries, where the 

 tides are most important to man. Whenever the amplitude of the tide wave is of the same 

 order of magnitude as the water depth, the crest of the wave travels more rapidly than the 

 trough. As a result, the time interval from low water to high water in the upper reaches of 

 an estuary is generally shorter than the time interval from high water to low water. The 

 complete cycle from low water to liigh water to low water, however, remains unchanged. 

 The asymmetry in wave speed results in a nonsymmetric wave shape which can be described 

 in terms of trignometric functions by adding harmonics of the fundamental waves. If the 

 original wave shape were sinusoidal with frequency co, the additional terms due to shallow 

 water would have the form nco, where n = 2, 3, 4, . . . . The tide wave, however, is a 

 composite of several trignometric terms of nearly the same frequency, and as a result, the 

 structure of tlie shallow-water terms is complex. The principle can be illustrated, but not 

 rigorously established, by considering the estuarine tide to be expressed as a power series of 

 trignometric functions. Thus, if the open-sea tide could be expressed as 



y^ = A cos cjt + B cos 0t (18) 



where a; and are arbitrary frequencies, the estuarine tide might be expressed in the 

 form 



ye = ^iJo + ^270 + «3yo +••• <^^) 



evaluated at a slightly later value of t. Note that y2 has the form 



yj = A2 cos^ cot + B^ cos^ 0t + 2AB cos cjt cos 0t (20) 



Introduction of suitable trignometric identities yields: 



y2 = -^ (A2 -I- b2) + ^ (A2 cos 2wt + B^ cos 20t) 



+ A B [cos (co - 0) t + cos (oj -I- 0) t] (21) 



and y^ can be expanded in a similar way. This procedure can be extended to consider more 

 trignometric terms in y . 



The important principles to note are that tlie tide wave is distorted as the tide is 

 propagated in shallow water. The resulting hydrograph can be described by introducing new 

 trignometric functions who^e frequencies are sums and differences of the frequencies used 

 to describe the tide in the open sea. The fundamental frequencies to be considered in this 

 expansion are those tliat correspond approximately to one or two cycles per day. 



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