Assignment_4_09

# Assignment_4_09 - TA's Name:_ Section: _ Your Name: _...

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TA's Name:____________________ Section: ____ Your Name: _________________________________ Physics 2214 Assignment 4 Concepts: Sound waves Fourier analysis Standing waves Response to arbitrary inputs Standing waves on strings Dispersion Standing sound waves in pipes Beats Reading: AG Notes on Mechanical Waves and Superposition (from website); Y&F, Vol. 1, Chapters 15 and 16. Assignment: Due in lecture on Tuesday, February 18. Please turn in this sheet stapled to the top of your work. Math Warm-Up Problem: 1. This problem is for review. Please don't turn it in. 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 214, Spring 2009 1 Cornell University

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Physics 214, Spring 2009 2 Cornell University Physics Problems: 1. Threshold of hearing. A standing sound wave in air at 20ºC and 1 atm has a frequency of 1 kHz and a pressure amplitude of 3 × 10 –5 Pa. (This value is called the threshold of hearing—the smallest pressure amplitude that a person with excellent hearing can hear.) What is the displacement amplitude? Compare it to the size of an atom (0.1 nm to 0.5 nm). 2. Standing waves on a guitar. The D-string of a guitar—a string fixed at both ends— has a fundamental frequency of 150 Hz. [Information given in square brackets may be interesting to those who know something about music or the guitar, but is not needed to work the problem.] (a) The oscillating length of a guitar string [between the nut and the saddle] is 25 inches. What is the wave speed on the D-string? (b What are the next two lowest standing wave frequencies on the D-string? (c) Suppose the string extends from x = 0 to x = L. If you press the string firmly against the neck at x = L/5 [at the 4th fret], only the side of the string with length 4L/5 can vibrate. Assume the tension is the same as before. What are the lowest three standing wave frequencies now? [The note you’re playing is F#.] (d) If instead you touch a finger lightly to the string at x = L/5, the entire length L of string can vibrate but you force a node at the point x = L/5. What are the lowest three standing wave frequencies now? [This is how you play a natural or open-string harmonic. The note sounded is the F# two octaves higher than the F# in part (c).] 3. Optical standing waves in a laser. A He-Ne laser can be made by placing a plasma
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## This note was uploaded on 04/01/2009 for the course PHYS 2214 taught by Professor Giambattista,a during the Spring '07 term at Cornell University (Engineering School).

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Assignment_4_09 - TA's Name:_ Section: _ Your Name: _...

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