class16lecturenotes-1

class16lecturenotes-1 - Class 16, Wednesday, February 17,...

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Class 16, Wednesday, February 17, 2010 Reading: 1,*,[ Sound waves Several classes ago we discussed how a sound wave propagates through a gas. Last class we analyzed the displacement and the pressure variation associated with a sound wave. Here we will focus on the speed of sound. In general, the speed of any mechanical wave depends on an inertial property of the medium (the capacity of the medium to store kinetic energy), and on an elastic property (the capacity of the medium to store potential energy). For the wave in a string, the inertial property is represented by the linear density, while the elastic property is represented by the tension in the string. tr- J~LI1"'r£ w - lit/diAl IY~"? lill For sound waves, which quantities might play the role of the tension and linear density? The inertial property is obviously the mass density. Thinking about our thought- experiment involving a gas in a long container closed by a piston, the restoring property must have something to do with the periodic expansion and contraction of a small volume element of the gas. So, pressure and volume must play some role. The physical quantity that describes the extent to which a medium changes its volume, !l.V, as the pressure applied to it varies by an amount !l.P, is called the bulk modulus, B. B ::; - bY _ It is measured in Pa. We can s:':iaYliqUidS and solids, which are less compressible than gasses, must have a larger bulk modulus. The sound speed then is: V;.w,J:: J-f In solids the sound speed must be larger than in liquids and gasses (for a given density), because they are less compressible. The material is very resistant to being squeezed, and furthermore it can communicate this resistance very quickly to whatever is trying to squeeze it This quick feedback results in a large speed of sound. In air the sound speed is 343 mls at 20°C. In gases the sound speed varies with the temperature. In water it is 1402m1s and in granite it is 6OOOmls. The variation of the sound speed with the temperature in gases can be shown on the example of an ;~~ g";( .t T =) ? = N; I ,,) 1t = - tff[ ~ > 8 IT =- > V;"UMJ.:: IW ;; > If; )/,t/ N k r:. . t{,UA-ftl::: v L---~--- M lb. !
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One more example that illustrates the importance of the boundaries with regards to wave propagation is the case of a rope with suddenly changing linear density, i.e. a rope that becomes thick or thin all of a sudden. 1) Pulse propagating from a thick into a thin rope. Where does the Rulse move fasteL? A The thick rope has a higher linear density, hence a lower wave speed. The pulse speeds up when it travels into the slim part of the rope. At this boundary. the pulse behaves similarly to the case where the end of the rope was allowed to slide freely across a pole. A reflected pulse gets sent back into the thick part
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class16lecturenotes-1 - Class 16, Wednesday, February 17,...

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