Lecture (30 minutes)

This lecture demonstrates how to solve the non-dispersive wave equation \(\frac{\partial ^{2}\psi \left( x,t \right)}{\partial t^{2}}=v^{2}\frac{\partial ^{2}\psi \left( x,t \right)}{\partial x^{2}}\) by the technique of separation of variables. Tell the students that this equation results from application of Newton's law to a rope under tension or the Maxwell equations to fields in free space, and that they will shortly study this, but that at present, we want to show that the harmonic, wave-like quantities are the solutions to this equation.

(The approach may seem backwards to the expert, but the students seem to be happy with it. It seems that when they finally show that the wave equation results from the application of Newton's law to a system, they think, “Oh, OK. And I know what to do with this equation!” If I try the “forwards” approach, they see that the wave equation results, but don't know what to do with it, and by the time we solve the equation, they've forgotten how we got to that point.)

Notes on the NDWE and the separation of variables ndwe_sepvar_wiki.ppt

  • Start with the NDWE and discuss the dimensions of both sides.
  • Explain the technique of separation of variables.
  • Once $x$ appears on one side only and $t$ on the other side only, there is no possible arbitrary variation of two independent parameters that can produce equality at all positions and times unless that variation is a constant.
  • With that, the linear 2nd order partial differential equations separates into two linear second order equations, with whose solutions the students are already familiar (a sum of a sin and a cosine of the relevant variable). The full solution follows easily.
  • A discussion of dimensions is in order. We need \(kx\) and \(\omega t\), and we must have \(v=\frac{\omega }{k}\)

Now the initial conditions activity follows, after a short introduction.

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