spectrum which does not satisfy the conditions imposed in Chapter 
11, then the sea surface in the region of superposition will break 
up due to non-linear effects. However for swell arriving from a 
distance this effect is usually of no importance and thus most re- 
fraction problems are easily dealt with. 
Consider equation (12.53). It states that given two power 
spectra, [Ana (H 90)1° and ieee (Paseo for two different wave 
systems present at the same point and time of observation, then the 
total effect is obtained by adding them point for point and calling 
the sum ees Ge) le If equation (12.53) is true, then a 
similar equation holds for any number of power spectra, and the state- 
ments made in the paragraphs above are proved. All of the steps are 
valid for both deep water and the transition zone so that H can also 
be infinite in any of the equations which follow. 
Now, [Apu7(# ,0)]° substituted into an equation like (9.47) 
would yield an expression for a sea surface which will be called 
77, and similarly (Aouqtr( #59) 1° would yield 7,,. Consider the 
power contributed to some one net element in the H ,® plane, upon 
passing to the limit inside the one net element, and consider that 
part of the total integral contributed by TAI and "ATI which 
involves these power contributions. Let AE, be the power contri- 
buted by Any # 9) to "AT and let AEty be the power contributed 
by Bout! # 19) to 7 Ary as defined by C12 G54) 
Then points chosen at random from mz, either as a function 
of time at any fixed point or as points chosen from the whole x,y,t 
space, will be distributed according to equation (12.55). Points 
chosen at random from 1 AIT will be distributed according to equation 
65 
