Worksheet #10 - PHY102 (Spr. 2007)The wave and diffusion equationsDue Thursday March 29thIn this worksheet we will study two partial differential equations that arevery important in physics.Many wave motions can be described by the linear wave equation. Weshall do problems concerning waves on a string, but the equation we studyhas many other applications. For example atomic vibrations in solids, lightwaves, sound waves and water waves are all described by similar equations.The linear wave equation for the waves on a string is the partial differentialequation,1v2∂2y(x, t)∂t2=∂2y(x, t)∂x2, (1)where y(x, t) is the distance by which the string is displaced at location x, attime t. v = (T /µ)1/2is the wave speed and is related to the tension T andmass density µ of the string(see Halliday and Resnick for the derivation).A second partial differential equation that is very important in physics isthe diffusion equation. Atoms in a gas diffuse around in a manner describedby this equation. Similarly pollutants in the ground often diffuse throughthe soil. This motion is very different than wave motion. In general eachphysical system has ranges of parameters where the motion is “diffusive”or “wavelike”. In solids for example motion is wavelike at short times andover long distances (e.g. sound waves), but diffusive on long times and shortdistances (atomic hops). The diffusion equation is given by,∂c(x, t)∂t= D∂2c(x, t)∂x2. (2)Here c(x, t) is the concentration of diffusing atoms at position x at time t.Here you can imagine putting a drop of ink in water and watching the colorspread. In that case, c(x, t) is the density of ink. D is the “diffusion constant”which sets the rate at which the spreading occurs.1Problem 1 - Wave Phenomena(i) Standing waves. Consider waves on a string of length L = 1. The trans-verse displacement at each end of the string is fixed at zero. Check that thetwo solutions: y1(x, t) = ymSin(kx − ωt) and y2(x, t) = ymSin(kx + ωt),satisfy the wave equation. If we seek the “fundamental mode”, how are kand ω related to v and the length of the string? Set k = k0and ω = ω0(ie.the values for the fundamental) and show that y(x, t) = y1(x, t) + y2(x, t)gives rise to standing waves. Animate the solution y1 and the solution y.Show your animation to a TA (don’t try to print it out)(ii) Beats. Now consider two solutions of the form y1(x, t) = ymCos(kx −ω1t), and y2(x, t) = ymCos(kx − ω2t), where ω1= ω + δω and ω2= ω − δωCheck that the linear superposition of these two propagating waves producesa beat pattern. How does the beat frequency depend on δω?(iii) Superposition. Almost all functions can be written as a superposition ofsine and cosine waves. As an example, consider the linear superposition ofsine waves such that;y(x, t) =maxXn=1−1nsin(nx) (3)Check the evolving pattern as max is increased. Make plots of y(x, t) formax = 3, 10, 100 terms. Can you identify the curve as max becomes large.Problem 2 - Diffusion.Check that c(x, t) =1√2Dtexp(−x24Dt) satisfies the diffusion equation. An-imate the plots of c(x,t) for different values of t. Notice that the amplitudeof c(x, t) decays with time, this is the essence of “diffusion”. In contrast,in the linear wave equation, the wave amplitude remains constant, it propa-gates instead of spreading. In reality there is some “damping” of waves, andthis is modeled by adding a “diffusion term” to the wave equation (like the“damping term” we sometimes add to Newton’s equation). Show a TA youranimation, but don’t try to print it
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