Two typical problems arise in the use of Equation 7-13. 



(1) Given the water depth d the wave height H and period 

 T, which wave theory should be used to predict the flow field? 



(2) For a particular wave condition, what are appropriate 

 values of the coefficients C^, and C^ ? 



7.313 Calculation of Forces and Moments . It is assumed in this seotion 

 that the aoeffiaients C^ and C^^ are known and are constants. (For 

 the selection of Cp and Cyi^ see Section 7.315, Selection of Hydro- 

 dynamic Force Coefficients C^ and Cj^.) To use Equation 7-13, assume 

 that the velocity and acceleration fields associated with the design wave 

 can be described by Airy wave theory. With the pile at x = 0, as shown 

 in Figure 7-39, the equations from Chapter 2 for surface elevation, 

 (Equation 2-10), horizontal velocity (Equation 2-13), and acceleration 

 (Equation 2-15) , are 



H 



7? = — COS 



2 



T 



H gT cosh [27r (z + d)/L] / lirt 



u = — — : cos — 



2 L cosh [27rd/L] V T 



du 3u sttH cosh [27r (z + d)/L] / Int 

 — as — = 2 5jj^ 



dt at L cosh [27rd/L] \ T 



Introducing these expressions in Equation 7-13 gives 





IT cosh [In (z + d)/L] 

 L 



cosh [27rd/L] 



27rt 



^D = Co r Pg D H^ 





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

 cosh [27rd/L] I 



T 



27rt 



(7-15) 

 (7-16) 

 (7-17) 



(7-18) 

 (7-19) 



Equations 7-18 and 7-19 show that the two force components vary with 

 elevation on the pile z and with time t. The inertia force fi is 

 maximum for sin (- 2TTt/T) = 1, or for t = - T/4 for Airy wave theory. 

 Since t = corresponds to the wave crest passing the pile, the inertia 

 force attains its maximum value T/4 sec. before passage of the wave 

 crest. The maximum value of the drag force component f^j coincides 

 with passage of the wave crest when t = 0. 



Variation in magnitude of the maximum inertia force per unit length 

 of pile with elevation along the pile is, from Equation 7-18, identical 

 to the variation of particle acceleration with depth. The maximum value 

 is largest at the surface z = and decreases with depth. The same is 

 true for the drag force component fj^ ; however, the decrease with depth 

 is more rapid since the attenuation factor, cosh[2Tr(z+d)/L]/cosh[2ird/L], 



7-69 



