quiz_soln - w ( k x , k y , 0) iL 2 = 4 [sinc(( k x k y / 2...

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13.811 Advanced Structural Dynamics and Acoustics Quiz - Acoustics April 21, 2004 Question 1. 1 ik z h p ( x, y ; 0) = e p ( k x , k y , h ) e ik x x e ik y y dk x dk y (4) 4 2 Solution Problem 1 Fourier transform of field at distance h p ( k x , k y ; h ) = p ( x, y, h ) e ik x x e ik y y dxdy (1) Backpropagate to radiator p ( k x , k y ; 0) = p ( k x , k y ; h ) e ik z h (2) where k z is the vertical wavenumber x k 2 k 2 + k 2 ± k 2 k 2 y , x y k 2 (3) k z = ² i k 2 + k 2 y k 2 , k 2 + k 2 > k 2 x x y Question 2. ik x 0 x θx e ik x 0 x e (5) p ( x, y, h ) = sin( ) = 2 c 2 i where k x 0 = k/ 2 = θ/ 2 c . Insert into eq. 1 and use EGW eqn. 1.5 4 2 p ( k x , k y ; h ) = [ ± ( k x k x 0 ) ± ( k x + k x 0 )] ± ( k y ) (6) 2 i Insert into Eq. (4) 1 [ e ik z 0 h e ik x 0 x e ik z 0 h e ik x 0 x ] p ( x, y ; 0) = 2 i θx = e ik z 0 h sin( ) (7) 2 c with k z 0 = k 2 k 2 = k ± 1 0 . 25 = k ± 3 / 2 x 0 1
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Solution Problem 2 y x L L x’ y’ y k k x k’ k’ x y k’ x0 k’ y0 k x0 k e ik x 0 x + e ik x 0 x θx w ˙( x, y, 0) = cos = (8) 2 c 2 with k x 0 = θ/ 2 c = k/ 2. Wavenumbers in rotated coordinate system k x k y (9) k = x ± 2 k x + k y k = (10) y ± 2 k x 0 θ = = (11) k x 0 ± 2 2 c ± 2 k x 0 θ k = = (12) y 0 ± 2 2 c ± 2 Williams eq. 2.102 leads to L 2 w ˙( k , k y ; 0) = [sinc(( k x k x 0 ) L/ 2)sinc(( k y k y 0 ) L/ 2) x 2 +sinc(( k + k x 0 ) L/ 2)sinc(( k y + k y 0 ) L/ 2)] (13) x Directivity function using transformations in eq. (9)-(12), 2
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iπθ D ( , δ ) = 2 ±
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Unformatted text preview: w ( k x , k y , 0) iL 2 = 4 [sinc(( k x k y / 2 c ) L/ 2)sinc(( k x + k y / 2 c ) L/ 2) +sinc(( k x k y + / 2 c ) L/ 2)sinc(( k x + k y + / 2 c ) L/ 2)] (14) with k x = k sin cos (15) k y = k sin sin (16) Question 3. Surface of radiater has a standing waveeld with crests parallel to the y-axis, generating two plane waves propagating at grazing angle cos 1 (1 / 2) = 60 , interfering at all distances h to produce a standing waveeld in the horizontal plane, but propagating vertically 3...
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quiz_soln - w ( k x , k y , 0) iL 2 = 4 [sinc(( k x k y / 2...

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