INTRODUCTION TO SONAR 7Finally, making the assuming that z << r, we can drop the term z to get r2l ∼= r − z sin θField calculationFor an element of length dz at position z, the amplitude at the field position (r, θ) is:A 1 −i(kl−ωt) dp = e dz L lWe obtain the total pressure at the field point (r, θ) due to the line array by integrating:p = AL L/2−L/21le−i(kl−ωt)dzbut l ∼= r − z sin θ, so we can write:p = ALe−i(kr−ωt) L/2−L/21r − z sin θeikz sin θdzSince we are assuming we are in the far field, r >> z sin θ, so we can replace 1r−z sin θ with1r and move it outside the integral:p = ArLe−i(kr−ωt) L/2−L/2eikz sin θdzNext, we evaluate the integral:p = ArLe−i(kr−ωt)eikz sin θik sin θL/2−L/2⎡ ⎤ 12 ikL sin θ − e− 12 ikL sin θ A −i(kr−ωt) ep = e ⎣ ⎦ rL ik sin θNext move the term 1 into the square brackets: L12 ikL sin θ − e− 12 ikL sin θ A −i(kr−ωt) ep = e ⎣ ⎦ r ikL sin θix −ix e −e and, using the fact that sin(x) = , we can write: 2i A sin( 1 kL sin θ) −i(kr−ωt) 2 p = e 1 r kL sin θ 2which is the pressure at (r, θ) due to the line array. The square of the term in brackets isdefined as the beam pattern b(θ) of the array:sin( 1 kL sin θ)

Detection, Range and Doppler Estimation