HW4_solution - ECE 473/TAM 413 Homework Assignment #4...

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Homework Assignment #4 Solutions - 1 ECE 473/TAM 413 Homework Assignment #4 Solutions Note to graduate students taking the course for 4 hours of graduate credit (NB: only graduate students can receive 4 hours of credit) : For the additional 1 hour of credit, you are required to write a paper (typically about 10 pages, double spaced) that discusses in some detail any topic on acoustics for which the fundamentals of engineering acoustics are explicitly described . The paper needs to be based on 5 peer-reviewed publications. The paper will be due Monday, December 5, 2008. However, I must approve the topic and peer-reviewed publications. For the approval process, prepare a one-page outline (including 5 peer-reviewed citations) for submission Monday, October 6, 2008 . 1. Problem 5.7.5 in Kinsler et al. a) Euler’s equation with gravity is given in Eq. 5.4.9. If density is not approximated by ! o , then the s term does not go to zero and the ! gs term remains. This gives for the linearized Euler’s equation with gravity: ! o " ! u " t = #$ p + ! o ! gs . To obtain the wave equation, first take the divergence of Euler’s equation giving: ! i " o # ! u # t = $! p + " o ! gs % ( ) * => ! i " o # ! u # t $ % ( ) = *! 2 p + ! i " o ! gs [ ] Using Eq. 5.5.2: !" o # 2 s # t 2 = !$ 2 p + $ i " o ! gs [ ] And from s = p ! o c 2 : 1 c 2 ! 2 p ! t 2 = " 2 p # " i $ o ! gs [ ] b) Assume a plane wave solution in the form p = ! Ae j ! t " ! k i ! x ( ) satisfies the wave equation. Insert this solution into the wave equation: !" 2 c 2 p = ! k 2 p ! # i ! gp c 2 $ % ( ) = ! k 2 p ! 1 c 2 * * x g x p + * * y g y p + * * z g z p + , - . / 0 !" 2 c 2 p = ! k 2 p + j c 2 k x g x + k y g y + k z g z # $ % p ! 2 c 2 " j c 2 k x g x + k y g y + k z g z # $ % = k 2 ! 2 c 2 " j c 2 ! k i ! g = k 2 ! 2 c 2 " j c 2 k ˆ n k i ! g = k 2 ! 2 c 2 " j c 2 ! c ˆ n k i ! g = k 2 ! 2 c 2 " j c 2 ˆ n k i ! ! g c # $ % ( = k 2
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Homework Assignment #4 Solutions - 2 If ! ! " g c , then the second term is negligible and the plane wave satisfies the equation. Calculating ! g c for air and water: 9.8 343 = 0.029 /s (air) and 9.8 1481 = 0.006 /s (water) Therefore, for air and water the term is small. 2. Problem 5.9.2 in Kinsler et al. a) P P o = ! ! o T T o is the ideal gas law where P = P o + p , ! = ! o + ! e and T = T o + ! T . P P o = ! ! o " # $ % ( (adiabatic process) Combining to eliminate ! : P P o = ! ! o " # $ % ( = P P o T o T " # $ % ( = P P o " # $ % ( T o T " # $ % ( ) P P o " # $ % ( * 1 = T T o " # $ % ( P o + p P o ! " # $ % ’ ( 1 = T o + ) T T o ! " # $ % * 1 + p P o ! " # $ % ’ ( 1
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This note was uploaded on 09/19/2009 for the course ECE 473 taught by Professor Obrian during the Fall '08 term at University of Illinois at Urbana–Champaign.

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HW4_solution - ECE 473/TAM 413 Homework Assignment #4...

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