be considered in terms of combinations of sensor separations, or lags. The 

 lag axis ranges from a value of (the separation of a sensor with itself) to 

 the length of the array (the separation of the end sensors) . It is usually 

 labeled in terms of minimum lag, which is the separation of the two closest 

 sensors. In the case of a 1-2-4 array, the lag unit would be YIAG , and the 

 lag axis would range from to 7 -YIAG . 



21. A useful mechanism often employed to understand the potential of a 

 linear array is to graph phase differences between sensors as a function of 

 lag separation for ideal waves having various Ly (longshore component of 

 wavelength, equal to 27r/ky ). To fully appreciate this, some of the details 

 in obtaining phase information from the sensors must be examined. 



22. Phase information is extracted from the coefficients of a Fourier 

 series representation of time series of sea surface elevations at wind wave 

 frequencies. A Fourier transform of the plane wave signal given by 

 Equation A2 yields the complex Fourier coefficient 



F(f) = a(f) [cos(k^x + kyy) + i sin(k^x + kyy) ] (AlO) 



where i = J^ and indicates complex notation. 



23. The phase difference between one sensor located at (x,yi) and a 

 second at (x,y2) is found from the Fourier coefficients, called F]^(f) and 

 F2(f) , respectively, of signals from the two sensors. The product of the 

 Fourier coefficient of one sensor with the complex conjugate of the other is 



Fi(f)-F2*(f) = [a(f)]2.[cos(k^x + kyyi) + i sin(k^x + kyy^) ] 



■[cos(kxX + kyy2) - i sin(k^x + kyy2) ] (All) 



where the asterisk (*) denotes complex conjugation. With some trigonometry 

 this can be written as 



Fi(f)-F2*(f) = [a(f)]2.(cos[ky(yi - y2) ] + i sin[ky(yi - y^)]) (A12) 



24. The terms in braces contain the phase difference information. The 

 real part of Equation A12 is the in-phase or coincident component and the 

 imaginary part is the quadrature component. These are commonly called the Co 

 and Quad terms, respectively. Note that these terms are modulated by sensor 



A9 



