HW11_solution473

# HW11_solution473 - ECE 473/TAM 413 Homework Assignment#11...

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Homework Assignment #11 Solutions - 1 ECE 473/TAM 413 Homework Assignment #11 Solutions For your information Hour Exam 2: Wednesday, April 23, 2008, 7:00-8:30 pm in 165 EL Hour exams are open book (the text for the course) with one sheet of notes (you can use both sides). Your class notes and lecture notes are not permitted. You can use calculators. Final Exam: Monday, May 5, 2008; 1:30 - 4:30 pm, 163 EL 1. Calculate the lowest 10 normal modes (eigenfrequencies in Hz) for a rigid-walled rectangular room of dimensions L x = 3.12 m , L y = 4.69 m and L z = 6.24 m . Use air at 20˚C. c = 345 m/s L x = 3.12 m L y = 4.69 m = 1.5L x L z = 6.24 m = 2.0L x From Eq. 9.2.7, f lmn = c 2L x 2 + m 1.5 ! " # \$ % & 2 + n 2.0 ! " # \$ % & 2 = 2 + m 1.5 ! " # \$ % & 2 + n 2.0 ! " # \$ % & 2 Hz 0 0 1 = 27.6 Hz 0 1 0 = 36.8 Hz 0 1 1 = 46.0 Hz 1 0 0 = 55.3 Hz 0 0 2 = 55.3 Hz 1 0 1 = 61.8 Hz 0 1 2 = 66.4 Hz 1 1 0 = 66.4 Hz 1 1 1 = 71.9 Hz 0 2 0 = 73.6 Hz 2. Problem 9.2.3 in Kinsler et al. Note: In part (a), the waves are traveling perpendicular to the z axis. From Eq. 9.2.5, p lm0 = A Cos k xl x ( ) Cos k ym y ( ) e j ! lm0 t because k z0 = 0 a) There is no z dependence. Components of the propagation constant exist in only the x and y directions. b) ! k = k x " x + k y " y . Thus, the speed of the wave is ! k , and obviously the propagation vector lies in the x-y plane, and makes an angle Tan ! 1 k y k x " # \$ % & with respect to the x axis. c) The component of the propagation constant in the y direction is k y so the speed in that direction is ! k y .

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Homework Assignment #11 Solutions - 2 3. Problem 9.2.5 in Kinsler et al. From Eq. 9.2.5, p lmn = A Cos k xl x ( ) Cos k ym y ( ) Cos k zn z ( ) e j ! lmn t where k = l ! L x = l ! L k = m ! L y = m ! L k = n ! L z = n ! L ! = c l " L x # \$ % & ( 2 + m " L y # \$ % & ( 2 + n " L z # \$ % & ( 2 = c " L l 2 + m 2 + n 2 For the lowest 10 normal modes, ! = c " L 2 + m 2 + n 2 are: 1 0 0 = c !
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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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HW11_solution473 - ECE 473/TAM 413 Homework Assignment#11...

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