Equation 2-21 is the equation of an ellipse with a major (horizontal) 

 semiaxis equal to A, and a minor (vertical) semiaxis equal to B. The 

 lengths of A and B are measures of the horizontal and vertical dis- 

 placements of the water particles. Thus, the water particles are predicted 

 to move in closed orbits by linear wave theory: i.e., each particle returns 

 to its initial position after each wave cycle. Morison and Crooke (1953), 

 compared laboratory measurements of particle orbits with wave theory and 

 found, as had others, that particle orbits were not completely closed. This 

 difference between linear theory and observations is due to the mass trans- 

 port phenomenon which is discussed in a subsequent section. 



Examination of Equations 2-22 and 2-23 shows that for deepwater 

 conditions A and B are equal and particle paths are circular. The 

 equations become 



A=B = -e'^^/L for->-. (2-24) 



2 L 2 



For shallow-water conditions, the equations become 



B 



(2-25) 



Thus, in deep water, the water particle orbits are circular. The more 

 shallow the water, the flatter the ellipse. The amplitude of the water 

 particle displacement decreases exponentially with depth and in deepwater 

 regions becomes small relative to the wave height at a depth equal to 

 one-half the wavelength below the free surface, i.e., when z = - Lo/2. 

 This is illustrated in Figure 2-4. For shallow regions, horizontal 

 particle displacement near the bottom can be large. In fact, this is 

 apparent in offshore regions seaward of the breaker zone where wave action 

 and turbulence lift bottom sediments into suspension. 



The vertical displacement of water particles varies from a minimum of 

 zero at the bottom to a maximum equal to one-half the wave height at the 

 surface. 



************** EXAMPLE PROBLEM ************** 



PROVE : 



/-„^ /27rV 27rg 



(a) = _S tanh 



(b) 



T / L 



jtH cosh[27r(z+d)/L] /27r: 



u = — ; ; cos 



T sinh(27rd/L) \ L 



2-18 



