Tides and Tidal Currents in the Proximity of Land 345 



the superposition of the two gives a progressive wave, which differs the more 

 from a standing wave, the greater the amount of tidal energy is lost inside 

 the bay. Many bays and adjacent seas become very shallow in their inner 

 regions, where a considerable portion of the tidal energy is absorbed and 

 thus lost for reflection (see Defant 1928, p. 274). In such adjacent seas the 

 co-oscillating tide loses its character of a standing wave, even when the depth 

 is very large elsewhere. The case of a rectangular bay of constant depth, 

 at whose inner end tidal energy gets lost, can be computed mathematically. 

 Let the bay at y = 1 open into a sea where the tide is 



r\ = Z cos at . 



The time is counted from the time of high water at the open end. The am- 

 plitude of the incoming wave (in the negative A-direction) is a. The reflected 

 wave, however, has the amplitude b, so that the quantity of energy (a 2 —b 2 )/a 2 

 has been lost by the reflection at the inner end of the canal, b = a, means 

 total reflection. The horizontal and vertical displacements of the water 

 particles along the bay (yn = al/c) can the be written 



£ = a sin (at— e-\- vny) — bs\n(at—e— vny) , 



ha 

 tj = [acos(at— e-\- vny) -\-b cos (at— e— vny)]. 



(XI. 20) 



ajb is given by the loss of energy; the amplitude a and the phase e must 

 be given by the only boundary condition at the opening. We obtain 



cost' sine 



cos vny cos(at — s) : sinvnysm(at— e) 



>1 



cos jot smv7i 



in which [(a— b)/(a + b)] tan vn = tane 



cZ sin(v7r + e) 



(XI.21) 



and a = — 



ha sin 2vn 



For a = b the equation (XI.21) changes into a standing wave as given by 

 (XI. 16) for a total reflection. In the case alb the co-oscillating tide is the 

 superposition of two orthogonal waves (phase difference = one-quarter 

 period). Along the bay the phase retards gradually against the phase at 

 the opening by increasing amounts, until it reaches the full amount e at the 

 inner end. For one nodal line the phase difference then is no longer 180° 

 but, according to the value of V, 180°±£. 



When an adjacent sea has in itself an orographical configuration such that 

 its tides are only little disturbed by the friction, its co-oscillating tide with 

 the open sea will no longer have the form of a simple standing wave, when 

 in its inner end a portion of the wave energy is lost. This is also the case, 

 when for instance the inner end of a more or less closed sea becomes a very 

 shallow one, where the dissipation of tidal energy is large. The co-oscillating 



