sand in saltwater) , Figure 3 provides a graphical solution of equation (5) . The 

 dimensionless value of (Zfrd^/L) is converted to dj^^ by multiplying by { (Lo/2ir) 

 tanh (2Trdj^/L)} (see eq. 3); Figure 4 provides the needed graph of (tanh x) ver- 

 sus X. Another alternative for y' - 1*6 is the first-order analytical approxi- 

 mation, 



dj = 2Hg + 11 o, (6) 



which is convenient and usually within ±5 percent of an exact calculation. 

 The seaward bound to the shoal zone is determined from the relationship 



"max(-d) 

 Y'gD 



(7) 



where D is the median sand diameter characteristic of the shoal zone bed, e.g., 

 at mean water depth of (1.5 d£) . Equation (7) gives the threshold of sand motion 

 by wave action in field conditions, and uniax(-d) will be taken as that due to 

 the median (50 percent) wave condition. For this wave condition, equation (7) 

 along with the depth dependence in equation (2) defines a maximum water depth 

 for sand motion, d-^, which corresponds to the seaward limit of the usual wave- 

 constructed profile. 



The modified exponential distribution gives the median significant wave 

 height as 



^s50 = Hs - 0.3 o (8) 



The period associated_with this median wave height is also taken to be the 

 average wave period T . With these relationships, it follows that 



(^)=r 



(H. - 0.3 a)' 

 sinh 



8Y'gDT^ 



0.5 (9) 



The value of d-^ can be accurately determined using the calculator program pre- 

 sented in the Appendix. Alternatively, after calculating the value of the right 

 side of equation (9) , the value of (2TTd-j^/L) can be determined from the graph of 

 X versus (sinh x) in Figure 4, and converted to d-j^ by multiplying by { (Lo/2Tr) 

 tanh (2TTd-j^/L) } , using the Figure 4 graph of (tanh x) versus x. Another alter- 

 native, for y' =1.6, is to use the empirical approximation 



di = (Hg - 0.3 a) Tg (g/5,000 D) ° • ^ (10) 



which is convenient and usually within ±5 percent of an exact calculation. 



Table 1 summarizes the basic procedure for calculating the water depths at 

 the two bounds of the shoal zone, for specified values of g, y'» D, Hg , a, and 

 Tg. The procedure can be executed using either the calculator program provided 

 in the Appendix or the graphs provided in Figures 3 and 4. A convenient alter- 

 native is to use the simple approximations in equations (6) and (10) , which 



10 



