HgT^ cosh[2ir(z + d)/L] /2irx 2Trt\ ttH^ 

 5 = - . .. .. . sin — — + 



4TrL co8h(2ird/L) \ L T / 8L sinh2(2iid/L) 



(2-53) 



3 cosh[4ii(z + d)/L] / 



1 sin - 



I 2 sinh2(2Trd/L) \ 



Attx 4Trt\ /tthV Ct cosh[4TT(z + d)/L] 



+ 



L T / \L / 2 sinh2(2Tvd/L) 



and 



HgT^ sinh[2ir(z + d)/L] /2Trx 2Tit 



c = : cos 



4itL cosh(2iTd/L) 



3 ttH^ sinh[4Tr(z + d)/L] 



+ ; COS 



16 L sinh'*(2TTd/L) 



(2-54) 



c. Mass Transport Velocity . The last term in equation (2-53) is of 

 particular interest; it is not periodic, but is the product of time and a 

 constant depending on the given wave period and depth. The term predicts a 

 continuously increasing net particle displacement in the direction of wave 

 propagation. The distance a particle is displaced during one_wave period when 

 divided by the wave period gives a mean drift velocity, U(z), called the 

 mass transport velocity. Thus, 



/tthV C cosh[4iT(z + d)/L] 



U(z) = — — (2-55) 



y L/ 2 sinh2(2TTd/L) 



Equation (2-53) indicates that there is a net transport of fluid by waves 

 in the direction of wave propagation. If the mass transport, indicated by 

 equation (2-55) leads to an accumulation of mass in any region, the free sur- 

 face must rise, thus generating a pressure gradient. A current, formed in 

 response to this pressure gradient, will reestablish the distribution of mass. 

 Theoretical and experimental studies of mass transport have been conducted by 

 Mitchim (1940), Miche (1944), Ursell (1953), Languet-Higgins (1953, 1960), and 

 Russell and Osorio (1958). Their findings indicate that the vertical distri- 

 bution of the mass transport velocity is modified so that the net transport of 

 water across a vertical plane is zero. 



d. Subsurface Pressure . The pressure at any distance below the fluid 

 surface is given by 



H cosh[2iT(z + d)/L] /2Trx 2Trt\ 

 P = ^^ I cosh(2.d/L) ^°^ \^-—)- '^^ 



3 ttH^ tanh(2Trd/L) (cosh[4Tr(z + d)/L] l\ Mux 



H pg { } cos - 



8 L sinh2(2Trd/L) [ sinh2(2ird/L) 3) \ L 



1 irH^ tanh(2TTd/L) f 4ir(z + d) ) 



pg ; — (cosh 1} 



8 L sinh2(2ird/L) L 



2-36 



