Steered Line ArrayRecall importance of phase:• Spatial phase: kz sin θ = 2λπ z sin θ• Temporal phase: ωt = 2Tπ ; T = f1-L/2L/2rθA/Lzxθ 0k sin θ = vertical wavenumberk sin θ0 = vertical wavenumber referenceTo make a steered line array, we apply a linear phase shift −zk sin θ0 to the excitation ofthe array:dp = A/Leiz(k sin θ−k sin θ0)eiωtdz (1) rWe can writesin θ0 zk sin θ0 = ωzcz sin θ0 zk sin θ0 = ωT0(z) ; T0(z) = cThe phase term is equivalent to a time delay T0(z) that varies with position along the linearray. We can re-write the phase term as follows.iz(k sin θ−k sin θ0)eiωt ikz sin θ −iω(t+T0(z)) e = e eintegrating Equation 1 yields: A sin( kL2 [sin θ − sin θ0]) −i(kr−ωt) p = r ( kL e2 [sin θ − sin θ0])The resulting beam pattern is a shifted version of the beampattern of the unsteered linearray. The center of the main lobe of the response occurs at θ = θ0 instead of θ = 0.⎡01 INTRODUCTION TO SONAR 9steered and unsteered line array (theta_0 = 20 deg)Source level normalized to on−axis response−10−20−30−40−50−60−70−80unsteered beamsteered beam−90 −60 −30 0 30 60 90 120theta (degrees)This plot shows the steered line array beam pattern2 sin( [sin θ − sin θ0]) kL ⎤2b(θ) = ⎣ ⎦ (kL2 [sin θ − sin θ0])for θ0 = 0 and θ0 = 20 degrees

Source level normalized to on−axis response