The cumulative power density of the u component of the velo- 
city must be bounded as stated by equation (11.15). Eauation (11.14) 
shows that for z not equal to zero, the term, exp[2(y *)*2/g], can 
cause F(y ) to be bounded for all » even if [a,(p ,e)]° is of a 
form in which E,, is unbounded. Thus any admissible [a(n ,e)]° 
which has a bounded Boat must also result in reasonable velocities 
below the surface. 
For z equal to zero in the equations for u, the equations give 
values for the surface water velocities due to the waves. It is true 
that the crest particle velocities occur at values of z greater than 
zero and the trough particle velocities occur at values of z less 
than zero, but such refinements are not justified in a linearized 
theory. 
In equation (11.14), consider the integration for the case where 
z is zero. Suppose that the integration overyp and 96 for yu less 
than pw K is bounded. Also suppose that [an(u ,0)1° can be expressed 
in a series form for » greater than # K such that 
[a(n O)]2 =f, +2,00)K 2+ 2,(e)/u 2+2,(0)/p * + 
are? ae aed oe ene: head 0 a os 
Then the integration over 6 of this series times (cose)* must yield 
constant or zero values such that 
er 2 4 
[D(H] = Cy + Co © + Cop 3+ of *+.... 
2 2 2 
It then follows that »“[D(yp )]° = CjH + Cy + C3/u + C,/ : eee 
Now, integration of yu 2rp¢z 7° from , to infinity in the above 
form would yield infinitely large values of Ee unless Ci» Coy and 
C3 were zero. Therefore, they must be zero or else the power integral 
will break down and predict infinitely strong u velocity components 
