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HW3_2007_Solution - Homework 3_rev Acoustics I due date...

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Homework 3_rev Acoustics I due date: November 19 2007 (1) An expansion chamber muffler is shown below. If, 2 S a π = , a is 2 cm, , temp = 80 1 20 S = S B 0 C, calculate and plot the power transmission coefficients T for the muffler with an anechoic termination shown below for the frequency range 0-3,000 Hz. Π i P r P S S 1 S L 1 A B Fig. 1. Simple expansion chamber L 1 was not given. One can choose 0.3 m that was given in version 1 or 0.55 m, the total length of the muffler given in problem 2. The overall four pole equation of the system with an anechoic termination always becomes 1 A B T T T T B a T T T T A B A B A B Z C D C D = = ⎦ ⎩ Q Q P P P Q . (1) where, the particle velocity B o c ρ = P U at B , B B o S S c ρ = = Q U P ; therefore o B a B c Z S ρ = = P Q in this particular case. There is only one element between the input and output ports; therefore, 1 1 1 1 1 1 sin cos sin cos o T T T T o jS kL kL c A B C D j c kL kL S ρ ρ = . (2) Because P P and A i = + r P ( A i o S c ρ = ) r P P Q , Eq. (1) can be re-written; ( o i r T T a c A B Z S ) B ρ = + P P Q (3) ( ) i r T T a C D Z + = + P P Q B (4) Dividing Eq. (3) by Eq. (4), we obtain 1 1 p i r o T T a i r T T a c A B Z S C D Z ρ p + = = + + R P P P P R + . (5) 1
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