closed orbits by linear wave theory; i.e., each particle returns to its ini- 

 tial position after each wave cycle. Morison and Crooke (1953) compared labo- 

 ratory 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 transport phenom- 

 enon, which is discussed in a subsequent section. 



Examination of equations (2-22) and (2-23) shows that for deepwater condi- 

 tions, A and B are equal and particle paths are circular. The equations 

 become 



H 

 A = B = - e 

 2 



jTTz/L 



for 1 > i 

 L 2 



(2-24) 



For shallow-water conditions, the equations become 



A = 



H 



2 2Trd 



H 

 B = - 



2 



z + d 



. d 1 

 for - < — 



L 25 



(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 dis- 

 placement 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 = L /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 break- 

 er zone where wave action and turbulence lift bottom sediments into suspen- 

 sion. 



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 3*************** 



PROVE: 



(a) 



lit' 



^ tanh ^ 



(b) 



irH cosh[2Tr(z + d)/L] 



u = : cos 



T sinh(2ird/L) 



2Trx 



SOLUTION: 



(a) Equation (2-3), 



gT , /2ird 



C = — tanh 



2Tr \ L > 



2-18 



