nU sinh [27r(z + d)/L] 2itx 



w = — C r~r-r: — 7r~^ — sin 1 



L sinh(27rd/L) \ L 



+ 1 i'^\ r si"h [4n(z + d)/L] . Mttx 

 4 \l/ sinh'»(27rd/L) ''" \l 



(2-52) 



Second-order equations for water-particle displacements from their 

 mean position for a finite amplitude wave are: 



? = - 



1 - 



HgT^ cosh [27r(z + d)/L] . Ilrrx _ 2-nt 

 471 L cosh (27rd/L) ^"^ \ L T 



+ 



7:H^ 



8L sinh^ (2;rd/L) 



(2-53) 



3 cosh [47r(z + d)/L] 

 2 sinh^ (27:d/L) 



Mtt: 



sin 



+ 



/ttHV Ct cosh [47r(z + d)/L] 

 It) T sinhM27rd/L) 



and 



HgT^ sinh [27r(z + d)/L] 2nx l-nt 



t = —2 cos — 



^ 47rL cosh (27rd/L) \ L T 



3 ttH^ sinh [47r(z + d)/L] Mttx 



+ ~ — n. ; cos 



16 L sinh''(27rd/L) \L 



(2-54) 



2.253 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, 



U(z) = 



/ttHV C cosh [47r(z + d)/L] 

 T) 2 sinhM27rd/L) 



(2-55) 



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

 waves in the direction of wave propagation. If the mass transport, indi- 

 cated by Equation 2-55 leads to an accumulation of mass in any region, the 

 free surface must rise, thus generating a pressure gradient. A current, 

 formed in response to this pressure gradient, will reestablish the distri- 

 bution of mass. Studies of mass transport, theoretical and experimental, 

 have been conducted by Longuet-Higgins (1953, 1960), Mitchim (1940), Miche 

 (1944), Ursell (1953), and Russell and Osorio (1958). Their findings 

 indicate that the vertical distribution of the mass transport velocity is 

 modified so that the net transport of water across a vertical plane is 

 zero. 



2-38 



