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13_811ps1sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF OCEAN ENGINEERING 13.811 Advanced Structural Dynamics and Acoustics Second Half - Problem Set 1 Solution Problem 1 . (a) Using equation (2.99) of text and substituting w for surface velocity, o . . L k L k y y ) x x k k W 0 , ) = w L L y sin c ( )sin c ( (1) ( , y x o x 2 2 Substituting L x =3L y . . L 2 L k L k y x ) x x x k k W 0 , ) = w sin c ( ) sin c ( (2) ( , y x o 3 2 6 (b) Using equation (2.86), directivity function, . L 2 kL kL x x x , D ( φ θ ) = i ρ ck w sin c ( sin θ cos φ )sin c ( sin θ sin φ ) (3) o o 6 π 2 6 (c) Equation (2) can be rearranged in the normalized form, . L k L k y x ) x x W sin c ( )sin c ( ] = 2 6 log 20 [ (4) 10 . w L 2 3 o x 1
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. W ], is plotted against k x L x and k y L x values The normalized expression, log 20 [ 10 . w L 2 o x for the range from -40 to 40 on the plot as shown in Fig. 1. A cutoff level of -40dB was applied to the plot. . W Fig 1 . Plot of log 20 [ ] against k x L x and k y L x 10 . w L 2 o x For a normalized frequency of kL x = 30, a circle with radius of 30 is drawn on the plot. The directivity function can be sketched out using Ewald sphere construction 2
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with a hemisphere on a 3-D plot of the contour plot as shown on Fig. 2 and the circle
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