Assignment_6_08_revised - Physics 214 Assignment 6...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Physics 214 Assignment 6 Concepts: Fourier Analysis Doppler Effect Beats Energy in Waves Reading: AG Notes on Mechanical Waves, Superposition and Standing Waves (from website); Y&F, Vol. 1, Chapter 16; Y&F Vol. 2, Section 37.6 Assignment: Due in lecture on Thursday, March 6. Please turn in this sheet stapled to the top of your work. Math Warm-Up Problem: 1. Bonus problem (i.e., this problem is optional.) The Fourier series of a periodic function f(x) can be written in the following forms: (A) [] + + = = 1 0 ) sin( ) cos( 2 ) ( n n n n n x k b x k a a x f where = 2 / 2 / ) cos( ) ( 2 L L n n dx x k x f L a and = 2 / 2 / ) sin( ) ( 2 L L n n dx x k x f L b (B) + + = = 1 0 ) cos( 2 ) ( n n n n x k c c x f φ or (C) . = −∞ = n x ik n n e C x f ) ( (a) By expressing the the cosine and sine terms in forms (A) and (B) using complex exponentials, determine expressions for c n and φ n in terms of a n and b n . (b) What is the relationship between C n in form (C) and c n and φ n of the second form? (Hint: C n is in general complex.) Physics Problems: 2. Suppose that you drive the damped oscillator we discussed earlier in the course with a periodic force in the form of a square wave, whose (fundamental) frequency is equal to the resonant frequency ω 0 of your oscillator when the damping is small. What will be the form of the oscillations that result? Physics 214, Spring 2008 1 Cornell University
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
(a) The Fourier series for a square wave drive is F(t) = A [sin ( ω 1 t) + 1/3 sin (3 ω 1 t) + 1/5 sin (5 ω 1 t) +1/7 sin (7 ω 1 t). . .]. Setting ω 1 = ω 0 , the resonant frequency of the oscillator, and using the expressions we derived for the amplitude and phase of the response of the oscillator to a sinusoidal drive, calculate the magnitude and phase of the response of the oscillator to each of the Fourier components of the drive, assuming (i) the system is underdamped with a Q =100 and (ii) that the system is critically damped, and (iii) that the system is overdamped with a Q=0.01. Evaluate the response to the first 10 nonzero terms in the Fourier series, using a spreadsheet or other program. (b) Since the system is linear, its response to a sum of inputs is the sum of the responses to each input applied individually. Evaluate the response to our square wave input by summing the responses found in (a). (c) Plot (i) the sum of the first 10 nonzero Fourier components of the drive and (ii) the response calculated in (b) versus time, for the underdamped, overdamped and critically damped cases in (a). Do any of the responses look more "useful", and if so, why?
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 03/31/2009 for the course CS 101 taught by Professor Gries during the Spring '08 term at Cornell University (Engineering School).

Page1 / 5

Assignment_6_08_revised - Physics 214 Assignment 6...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online