and the wave length at the depth of 125 feet would be 769.2 feet. 
Note that no special tables or graphs were used. 
The Itcoth has an additional property which is given as equation 
(12.11). If the hyperbolic cotangent of the product of Hp/p and 
the Itcoth of Hu “/g is formed, it will again equal the Itcoth of Hp-/e. 
This is shown by Table 17. If the Itcoth is treated as the dependent 
variable, I, equation (12.12) follows. The inverse of the equation 
then yields Hp °/¢ as a function of I, and in equation (12.13), Hp ?/e 
is given as a function of I. The function given by eouation (12.13) 
is graphed in figure 31. Other relationships of a useful nature are 
also given in the figure. 
Since the wave length of a wave with a known spectral frequency 
(or period) has now been given as a function of that spectral fre- 
quency and the depth, H, of the water, it is now possible to write 
down the expression for the free surface for one pure sine wave, in 
water of constant depth, H. The free surface is given by equation 
(12.14) in which the constant spectral frequency is given by I and 
the depth is He It is easy to show that this expression reduces to 
the forms given before if H becomes infinite. 
Equation (12,15) then yields the potential function. It is 
again easy to show that the potential function satisfies all required 
properties and that it reduces to the appropriate form in water of 
infinite depth. 
The appropriate Gaussian systems then follow immediately from 
previous considerations. The free surface is given by equation (12.16). 
Boy (CH 98) is the cumulative power distribution function for waves in 
water of depth, H. The function, ¥(,@) and the function, E5,(p ,6) 
a2 
