These equations express the local fluid velocity components any distance (z + 

 d) above the bottom. The velocities are harmonic in both x and t. For a 

 given value of the phase angle 9 = (2itx/L - Zirt/T), the hyperbolic functions 

 aosh and sink, as functions of z result in an approximate exponential decay 

 of the magnitude of velocity components vd.th increasing distance below the 

 free surface. The maximum positive horizontal velocity occurs when 6=0, 

 2ir , etc., while the maximum horizontal velocity in the negative direction 

 occurs when 6 = tt, Sit, etc. On the other hand, the maximum positive vertical 

 velocity occurs when 9 = ir/Z, 5tt/2, etc., and the maximum vertical velocity in 

 the negative direction occurs when 9 = 3Tr/2, 7Tr/2, etc. (see Fig. 2-3). 



The local fluid particle accelerations are obtained from equations (2-13) 

 and (2-14) by differentiating each equation with respect to t. Thus, 



girH cosh[2Ti(z + d)/L] _, /2irx 2Trt , ,. ,cn 



a = + — — sin (2-15) 



x L cosh(2TTd/L) \ L T ' 



gTTH sinh[2Ti(z + d)/L] ... ... 



a = — -— cos (2-16) 



z L cosh(2Trd/L) ' ' 



/2inc 2irt\ 



Positive and negative values of the horizontal and vertical fluid acceler- 

 ations for various values of 6 = 2Trx/L - 2irt/T are shown in Figure 2-3. 



The following problem illustrates the computations required to determine 

 local fluid velocities and accelerations resulting from wave motions. 



*************** EXAMPLE PROBLEM 2*************** 



GIVEN ; A wave with a period T = 8 seconds, in a water depth d = 15 meters (49 

 feet), and a height H = 5.5 meters (18.0 feet). 



ations a and a at an elevation z = -5 meters (-16.4 feet) below the 



FIND : The local horizontal and vertical velocities u and w, and acceler- 

 ations a and a at an elevation z 

 SWL when 9 = 2irx/L - 2irt/T = tt/3 (60°). 



SOLUTION: Calculate 



L = 1.56t2 = 1.56(8)2 = 99,3 „ (327 ft) 

 o 



d 15 



= 0.1503 



L 99.8 

 o 



From Table C-1 in Appendix C for a value of 



d 



— = 0.1503 

 L 

 o 



d 2TTd 



-as 0.1835; cosh = 1.742 



L L 



2-13 



