H 

 u = 2 



gT cosh [2ir (z + d)/L] 

 L cosh [2Tid/L] 



cos 



2irt 



(7-23) 



du 



dt 9t 



9u _ gTTH cosh [2tt (z + d)/L] 



cosh [2iTd/L] 



sin 



271 1 



(7-24) 



Introducing these expressions into equation (7-20) gives 



.2 



^i = s^ps — » ^- 



irD" „ I TT_ cosh [2-n (z + d)/L] 

 L cosh [2TTd/L] 



sin I - 



2irt 



(7-25) 



2 2 



1 2 IgT fcosh [2tt (z + d)/L] | f / 2Trt 



f = C - pg D H < { > > cos 



D D 2 ) 2 cosh [2TTd/L] ( \ T 



4L ^ ' 



cos 



t) 



(7-26) 



Equations (7-25) and (7-26) show that the two force components vary with 

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

 maximum for sin (- 2irt/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 

 the wave crest when t = . 



^D 



coincides with passage of 



Variation in magnitude of the maximum inertia force per unit length of 

 pile with elevation along the pile is, from equation (7-25), 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 fr, ; however, the decrease with depth is more rapid 

 since the attenuation factor, cosh [2it(z + d)/L]/cosh[2Trd/L] , is squared. 

 For a quick estimate of the variation of the two force components relative to 

 their respective maxima, the curve labeled K = l/cosh[2Ttd/L] in Figure 7-68 

 can be used. The ratio of the force at the bottom to the force at the surface 

 is equal to K for the inertia forces, and to K for the drag forces. 



The design wave will usually be too high for Airy theory to provide an 

 accurate description of the flow field. Nonlinear theories in Chapter 2 

 showed that wavelength and elevation of wave crest above Stillwater level 

 depend on wave steepness and the uave height-water depth ratio. The influence 

 of steepness on crest elevation 

 in Figures 7-69 and 7-70. The 

 following examples. 



n and wavelength is presented graphically 

 use of these figures is illustrated by the 



*************** EXAMPLE PROBLEM 17 ************** 



GIVEN ; Depth d = 4.5 m (14.8 ft) , wave height H = 3.0 m (9.8 ft ) , and 

 wave period T = 10 s . 



FIND: 



Crest elevation above Stillwater level, wavelength, and relative 



variation of force components along the pile. 



7-106 



