Worksheet #10 – PHY102 (Spring 2011)The wave equation and the diffusion equationIn this worksheet we study two partial differential equations that are very important in physics.Many wave motions can be described by the linear wave equation. We shall do problems concerningwaves on a string, but the equation we study has many other applications. For example atomicvibrations in solids, light waves, sound waves and water waves are all described by similar equations.The linear wave equation for waves on a string is the partial differential equation1v2∂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 and time t, and v isthe speed of the waves. For transverse waves on an actual string, the wave speed is related to thetension T and mass-per-unit-length µ of the string by v =pT/µ (see e.g., Halliday and Resnickfor the derivation).A second partial differential equation that is very important in physics is the diffusion equation.Atoms in a gas diffuse in a manner described by this equation. Similarly, pollutants in the groundoften diffuse through the soil. This motion is very different from 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 and over long distances (e.g., sound waves), butdiffusive on long times and short distances (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. You can imagine puttinga drop of ink in water and watching the color spread. In that case, c(x, t) is the density of ink. Dis the “diffusion constant” which sets the rate at which the spreading occurs.Problem 1 – Wave Phenomena(i) Standing waves. Consider waves on a string of length L. The transverse displacement at eachend of the string is fixed at zero. Check that the two solutions: y1(x, t) = ymsin(kx − ωt), andy2(x, t) = ymsin(kx + ωt), both satisfy the wave equation, provided that k and ω are related inthe appropriate way. Show that y1(x, t) + y2(x, t) satisfies the boundary condition y(0, t) = 0 forall t. Find the condition required to also satisfy the boundary condition y(L, t) = 0 for all t (youcan use units where L = 1.)If we seek the “fundamental mode” (i.e., the lowest frequency mode of vibration), how are k andω related to v and the length of the string? Set k = k0and ω = ω0(i.e., the values for the1fundamental mode). Show that y(x, t) = y1(x, t) + y2(x, t ) gives rise to standing waves. Animatethe solution y1 and the solution y = y1 + y2.(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 − δω Check that the linear superposition of thesetwo propagating waves produces a beat pattern. How does the beat frequency depend on δω?(iii) Superposition. Almost all functions can be written as a superposition of sine and cosine waves.As an example, consider the linear superposition of sine waves such that;y(x) =nmaxXn=11nsin(2 π n x/L) (3)Check the evolving pattern as nmaxis increased. Make plots of y(x) for nmax= 3, 10, 100 terms.Can you ident ify limiting curve as nmaxbecomes large?Problem 2 – Diffusion.Show that c(x, t) =1√2 D texp(−x24 D t) satisfies the diffusion equation. Animate the plots of c(x, t)for different values of t. Notice that the amplitude of c(x, t) decays with time: this is the essenceof “diffusion.” In contrast, in the linear wave equation, the wave amplitude remained constant: itpropagates instead of spreading.(In reality there is usually some “damping” of waves. This is often modeled by adding a “diffusionterm” to the wave equation—analogous to t he damping term we sometimes add to the equation ofmotion of a harmonic
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