HW2_solution

HW2_solution - ECE 473/TAM 413 Homework Assignment #2...

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Homework Assignment #2 Solutions - 1 ECE 473/TAM 413 Homework Assignment #2 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. The tension in a string is provided by hanging a 3-kg mass at one end; the other end is rigidly fixed. The length of the string is 2.5 m and its mass is 50 g. Determine the phase speed of waves on the string. Tension in the string is T = F = mg = (3 kg)(9.81 m/s 2 ) = 29.4 N Mass per unit length is ! L = m L = 0.05 kg 2.5 m = 0.02 kg /m Phase speed is c = T ! L = 29.4 N 0.02 kg / m = 38.3m /s 2. Represent the calculations to the correct number of significant figures: (a) 32.4 x 41.4 = 1,340 (b) 32.1 x 41.43 = 1,340 3. Problem 2.3.1 in Kinsler et al. (show all work). Put your final answers in the form of a wave equation. (a) Assume the linear density varies with position, that is, ! L = ! L x ( ) . The tension force on each element is given by: df y = ! ! x T ! y ! x " # $ % & dx = T ! 2 y ! x 2 dx (see Eq. 2.3.4) Using Newton’s 2nd Law: f y = ma = m ! 2 y ! t 2 , but the mass for an element is: dm = ! L x ( ) dx for some element at x, giving df y = dm ( ) a = ! L x ( ) " 2 y " t 2 dx . Equating forces T ! 2 y ! x 2 dx = " L x ( ) ! 2 y ! t 2 dx and rearranging yields: ! 2 y ! t 2 = T " L x ( ) ! 2 y ! x 2 . (b) Assume the string hangs vertically supported only at the upper end. Therefore, the tension is given by the force of gravity and the mass of the string hanging below the position x for a dx element at x. Thus, the tension at x is given by: T = m x ( ) g = !
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HW2_solution - ECE 473/TAM 413 Homework Assignment #2...

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