approaching the array lying along the x-axis, and let the 3 sensors 

 be at the points (x., y.), i = 1, 2, 3. Then at each sensor we 

 measure: ' ' 1{K ^l ^I ^\ - ^■^) 



At each sensor we can get a Fourier transform: 



w 



{.^^. ^i . c.) '- ^^ ' ""''^ (c^-o 



From a coherence computation we obtain a phase between each 

 pair (ambiguous up to multiples of 21^" ): 



only 2 of which are independent. If the sensors are restricted to 

 the y = line, and if the mean x positions are known, then we can 

 unambiguously solve for k. In practice neither the y. nor the x. 

 and precisely known, and the phases are 



where Cjj and^;; represent positioning errors. There will then be 

 a relative wave-number error of: 



Ignoring the y- variation for simplicity, this is: 



and is directly proportional to the uncertainty in the mean position 

 values. To bring the wave-number uncertainty down to the 1% level, 

 the position differences cannot exceed 10 m. out of 10"^ m. 



If the sensors oscillate at the frequency 0" and are decoupled, then 

 the phase estimates will be uncertain (the coherence will be reduced 

 due to the lack of phase lock) so that we have constraints upon both 

 the absolute uncertainty of sensor position and the frequency of 

 change. The out-of-line movement (y-displacement) is less severely 

 constrained than the x, since the determination of the x wave-numbers 

 is relatively insensitive to this motion, assuming the waves come in 

 dominantly along the array ( a topographic consideration we will not 

 go into here). 



198 



