13_811ps2sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF OCEAN ENGINEERING 13.811 Advanced Structural Dynamics and Acoustics Second Half - Problem Set 2 Solution Problem 1 Using equation (2.52) from text and substituting the expression for p(x,y,0), ( ) y x k k P , = dxdy e e e e e P y ik x ik kx i kx i t i o y x ∫∫ + ) ( 2 2 ω = dxdy e e e e P y ik x k k i x k k i t i o y x x + + ) ( ) 2 ( ) 2 ( Using equation (1.5) from text, ( ) y x k k P , = [ ] ) ( y x x t i o k k k k k e P δ π 2 2 4 2 + + ( 1 ) Using equation (2.50) and substituting equation (1) above,
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2 () z y, x, p = [] y x y ik x ik z ik y x x t i o dk dk e e e k k k k k e P y x z ∫∫ + + ] ) ( ) 2 ( ) 2 ( 4 [ 4 1 2 2 δ π ω = y y k z k i y x x ik x x t i o dk e k dk e k k k k e P y z x + + + ) ( ) ( ) 2 ( ) 2 ( = 1 ) 2 ( ) 2 ( I dk e k k k k e P x x ik x x t i o x + + ( 2 ) Where 1 I = y y k z k i y dk e k y z + ) ( ) ( Using sifting property of the delta function integral (equation (1.37) of text) and recognizing that
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This note was uploaded on 02/24/2012 for the course MECHANICAL 2.067 taught by Professor Davidbattle during the Spring '04 term at MIT.

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13_811ps2sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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