where n = 0, -1, +1, -2, +2, ... . Each value of the index n relates to 

 a particular crest. The Intersections of the crests with the x and y 

 axes move along the respective axes with time velocities V^ = -fit 

 and Vy = -f/m. This follows from the differential expressions 



obtained from the wave crest relationship given above. 



From Euler's equation we know that cos y = (ExpCiy) + Exp(-iY))/2. 

 If we consider y as 2Tr(Jlx + my + ft + a) we can write 



n(x,y,t) = 1/2A Exp(i2Ti(ilx + my + ft + a) + 



1/2A Exp (i27r(-J!,x - my - ft - a)) (2.6) 



where -°= < % < jt_°°» -~ < m < + "^ and -<» < f < + oo. 



In the above we have introduced the notion of negative time frequencies . 

 This makes it possible to express an elementary wave in the mathemati- 

 cally convenient form 



n(x,y,t) = a Exp (i2Tr(£x + my + ft + a)) (2.7) 



where a = 1/2A. In the real world a complex wave of this type implies 

 the existence of another wave n*(x,y,t) which is the complex conjugate 

 of ri(x,y,t) above. This complex conjugate is given by 



n*(x,y,t) = a Exp (-i2TT(£x + my + ft + a)) 



= a Exp (i2TT[(-Jl)x + (-m)y + (-f)t + (-a)]). (2.8) 



The fact that negative frequencies are considered is explicit in the 

 above relation. 



A property of the above model, which will be used later in connec- 

 tion with the directional analysis of waves from measurements obtained 

 from an array of detectors, is expressed by the equatio.n for the phase 

 difference of two measurements made at two different points in space 

 and time. Assume we know the value of n(x,y, t) at the three-dimensional 

 coordinates (xQ.yo.to) and (xq+X, yo+Y, t^+T) , where X, Y, and T 

 are constants. The phases at the two points are given by- 



