V = water velocity in the inlet 



y = distance along the cross section 



(t ) = component of the stress tensor in the direction of flow 



g = acceleration due to gravity 



t = time 



The inlet has a width, B, depth, d, length, L, and cross-sectional area, 

 A ; the bay has a surface area, A^^y The water levels in the sea and 

 bay are hg and h^j, respectively (Fig. 1). 



Equation (1) equates the horizontal driving force due to the water 

 surface slope with three terms on the right which are the channel fric- 

 tional resistance, the convective acceleration caused by velocity vari- 

 ation along the channel axis, and the temporal acceleration (or inertia) 

 resulting from velocity variation at a point with time. In nearly 

 prismatic channels, such as many inlets on the Great Lakes, the convec- 

 tive acceleration is often negligible. 



The continuity equation, which relates rate of bay water level change, 

 dhjy/dt, to inlet discharge, Q, is: 



3h2, 

 Q = VA^ = ^bay W • ^2) 



A simultaneous solution of equations (1) and (2) for a sinusoidal 

 sea level fluctuation reveals the important response characteristics of 

 an inlet-bay system (Fig. 2). In this figure, the phase lag between the 

 sea and bay water level fluctuations and the amplification of the forc- 

 ing wave in the bay by the inlet-bay system are plotted as functions of 

 dimensionless period. Dimensionless period is defined as the friction- 

 less inlet-bay system Helmholtz period, T^', divided by the forcing wave 

 period, T. The Helmholtz period is that period of the forcing wave which 

 through resonance will cause the largest water level fluctuation in the 

 bay. The bay water level remains approximately horizontal throughout 

 this fluctuation. 



The inlet-harbor system response is analogous to the response of a 

 slightly damped spring-mass system or its acoustic counterpart, the 

 Helmholtz resonator. The motion of the mass of water in the inlet 

 channel corresponds to the motion of the mass of the spring-mass system, 

 and the action of gravity on the rising and falling harbor water surface 

 corresponds to the restraining force of the spring. 



At values of T^'/T approaching zero (long wave periods) the water 

 level fluctuations in the bay are the same as those in the sea with no 



13 



