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18PDE Waves: String, Quantum Packet, and E&MIn this chapter we explore the numerical solution of a number of PDEs known as waveequations. We have two purposes in mind. First, especially if you have skipped thediscussion of the heat equation in Chapter 17, “PDES for Electrostatics & HeatFlow,” we wish to give another example of how initial conditions in time are treatedwith a time-stepping or leapfrog algorithm. Second, we wish to demonstrate that oncewe have a working algorithm for solving a wave equation, we can include considerablymore physics than is possible with analytic treatments. Unit I deals with a number ofaspects of waves on a string. Unit II deals with quantum wave packets, which havetheir real and imaginary parts solved for at different (split) times. Unit III extendsthe treatment to electromagnetic waves that have the extra complication of beingvector waves with interconnected E and H fields. Shallow-water waves, dispersion,and shock waves are studied in Chapter 19, “Solitons and Computational FluidDynamics.”18.1 Unit I. Vibrating StringProblem: Recall the demonstration from elementary physics in which a string tieddown at both ends is plucked “gently” at one location and a pulse is observed totravel along the string. Likewise, if the string has one end free and you shake itjust right, a standing-wave pattern is set up in which the nodes remain in placeand the antinodes move just up and down. Your problem is to develop an accuratemodel for wave propagation on a string and to see if you can set up traveling- andstanding-wave patterns. 118.2 The Hyperbolic Wave Equation (Theory)Consider a string of length L tied down at both ends (Figure 18.1 left). The string hasa constant density ρ per unit length, a constant tension T , is subject to no frictionalforces, and the tension is high enough that we may ignore sagging due to gravity.We assume that the displacement of the string y(x, t) from its rest position is in thevertical direction only and that it is a function of the horizontal location along thestring x and the time t.1 Some similar but independent studies can also be found in [Raw 96].−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 478