d)(x ,y ,t ) = ilx + my + ft + a, (2.9) 



(b(x + X, y + Y, t + T) = i(x + X) + m(y + Y) + f (t + T) + a 

 ^^o '-^o o o o o 



(2.10) 



This gives a phase difference of 



A()) = (AX + mY + fT). (2.11) 



To obtain a more complicated wave system consisting of many waves 

 of various frequencies and directions, we can linearly superimpose (add 

 up) many waves of the form given above. If we do this, we can write 



M 



For this wave system to be real, the terms must occur in complex con- 

 jugate pairs as indicated above. 



For completeness, consider a model for an infinite but countable 

 number of distinct (discrete) waves and write 



%»\ 



Again the terms must occur in complex conjugate pairs for the wave 

 system to be real. This will be assumed to be the case in future 

 discussions. 



A model for a wave system in the case where energy exists for con- 

 tinuous intervals of frequency and direction should be considered. In 

 particular, consider the general case of continuous direction from 

 to 2Tr radians and continuous frequency in the interval (-fnj + fn^ > 

 or even the interval (-«>, =°) . In theory the above model does not hold 

 for the continuous case. The power spectrum for the infinite but count- 

 able case would be a set of Dirac delta functions of amplitude a'^ 

 standing on the points (Z^, m^, fn) of a three-dimensional frequency 

 space. The continuous case produces a power spectrum, SQ(£,,m,f), 

 which is everywhere nonnegative and in general continuous over the 

 region of three-dimensional frequency space where power is assumed to 

 exist. A reasonable model for ri(x,y,t) in the continuous case must be 

 determined. Consider a single wave element 



an Exp(i27T(ilnX + mnY + fn*^ + ctn)). ' . (2.14) 



