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pde waves: string, quantum packet, and e&m 487In Figure 18.3 we show the resulting motion of a string plucked in the middle whenfriction is included. Observe how the initial pluck breaks up into waves travelingto the right and to the left that are reflected and inverted by the fixed ends. Becausethose parts of the wave with the higher velocity experience greater friction, thepeak tends to be smoothed out the most as time progresses.Exercise: Generalize the algorithm used to solve the wave equation to nowinclude friction and check if the wave’s behavior seems physical (damps intime). Start with T =40N and ρ =10kg/m and pick a value of κ large enoughto cause a noticeable effect but not so large as to stop the oscillations. As a check,reverse the sign of κ and see if the wave grows in time (which would eventuallyviolate our assumption of small oscillations).18.4 Waves for Variable Tension and Density (Extension)We have derived the propagation velocity for waves on a string as c = √ T/ρ. Thissays that waves move slower in regions of high density and faster in regions of hightension. If the density of the string varies, for instance, by having the ends thickerin order to support the weight of the middle, then c will no longer be a constant andour wave equation will need to be extended. In addition, if the density increases,then so will the tension because it takes greater tension to accelerate a greater mass.If gravity acts, then we will also expect the tension at the ends of the string to behigher than in the middle because the ends must support the entire weight of thestring.To derive the equation for wave motion with variable density and tension, consideragain the element of a string (Figure 18.1 right) used in our derivation ofthe wave equation. If we do not assume the tension T is constant, then Newton’ssecond law gives⇒∂ [T (x)∂xF = ma (18.25)]∂y(x, t)∆x = ρ(x)∆x ∂2 u(x, t)∂x∂t 2 (18.26)⇒∂T(x)∂x∂y(x, t)∂x+ T (x) ∂2 y(x, t)∂x 2 = ρ(x) ∂2 y(x, t)∂t 2 . (18.27)If ρ(x) and T (x) are known functions, then these equations can be solved with justa small modification of our algorithm.In §18.4.1 we will solve for the tension in a string due to gravity. Readers interestedin an alternate easier problem that still shows the new physics may assume−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 487