separation [the (y^^ - y^) in Equation A12] . A little more trigonometry 



reveals 



-1 r 2uad ] ^ ^^^-1 r sin[k^(yi - y^) ] ^ 

 I Co J \ cos[ky(yi - yz)] J 



= ^r (yi - 72) 



= h4> (A13) 



where l\4> is the same as that expressed in Equation A5 . 



25. Note that if the Fourier coefficient of one sensor is multiplied by 

 its own complex conjugate, Equation A12 reduces to an expression for the 

 squared amplitude (variance) at that frequency, since Yi = Yz ■ The in- 

 phase, cosine term would have a value of one and the quadrature, sine term 

 would be zero, indicating, correctly, that the sensor is completely in-phase 

 with itself. 



26. The phase characteristics of a multi-directional wave field are 

 more easily interpreted if one examines the Co and Quad components of Equation 

 A12 instead of the phase term of Equation A13. The Co and Quad terms can be 

 graphed in terms of the array lag space. To illustrate this. Figure A4a shows 

 a snapshot of a plane wave passing through a 1-2-4 array. The sinusoid is a 

 cross section of the pattern of crests and troughs along the array as a 

 function of absolute (x,y) coordinates. The Co and Quad plots for the same 

 wave are shown in Figure A4b and A4c , respectively. The crests and troughs of 

 the Co plot indicate lag separations where signals from two sensors would be 

 in-phase and 180 deg out of phase, respectively. Similarly, the crests and 

 troughs of the Quad plot indicate sensor separations where signals would be 90 

 and 270 deg out of phase, respectively. 



27. The Co and Quad plots of a single plane wave are sinusoids with 

 wavelengths equal to the longshore component of the incident wavelength; that 

 is, they provide an estimate of ky . Knowledge of ky , f , and d allows 

 an estimate of 6 by way of Equations A6 and A8 . This shows how all the 

 sensor separations may be used together to get an estimate of the longshore 

 wavelength of an incident wave and, from this, its propagation direction. 



AlO 



