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Unformatted text preview: 1B Midterm 1 Spring 2010 Solutions Problem 1 Consider a pipe open at one end and closed at the other as shown below. The end (at x=0) begins to vibrate and a travelling sound wave is launched which may be described by: χ = Re Ae i ω t − kx + φ ( ) ( ) Here χ is the horizontal displacement of the air. The wave is reflected from the end (x=L) and a standing wave is set up. Assume there is a node at the speaker and the wave is perfectly reflected at the open end. a) Find the expression for the displacement of the air inside the tube once the standing wave is set up, χ x , t ( ) . b) Sketch the magnitude of the first three harmonics (lowest and higher two) in the tube showing the nodes and maxima. c) What is the air pressure as a function of x,t in the tube? Solution: Once the wave is reflected. The reflected wave travels in the –x direction and there is a phase change of π (180 degrees at the solid wall. Therefore: Forward : χ f = Ae i ω t − kx − φ ( ) reflected : χ r = Ae i ω t + kx + φ ( ) The total wave is the sum of the two: χ = Ae i ω t − kx − φ ( ) + Ae i ω t + kx + φ ( ) χ = Ae i ω t e ikx + ϕ + e...
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- Spring '10
- Friction, Wavelength, Trigraph, Partial differential equation, Fundamental physics concepts