Lecture (5 minutes)

This is really just an introduction to the activity. The students know that the position and velocity of an oscillator at, say, t = 0, determine the two arbitrary constants that appear in the equation of motion for the oscillator: \(B_{p}\cos \omega t+B_{q}\sin \omega t\) or \(A\cos \left( \omega t+\varphi \right)\) . They also know that this is the solution to a 2nd order ordinary differential equation.

Now that they have the equation of motion for a “collection of oscillators”, they need to understand how to determine the arbitrary constants that appear in the general form. They have also studied the PDE that leads to this general form, but they don't yet know what this has t do with physics. Keep reassuring them that Newton's law or the Maxwell equations (which are coming up soon) will lead to this form!

The mini-lecture, or introduction to the activity, is to use the example of a transverse wave on a rope, which comes closest to a collection of oscillators. The students now know that most general form of a wave is

\[\psi \left( x,t \right)=A\cos kx\cos \omega t+B\cos kx\sin \omega t+\sin kx\cos \omega t+D\sin kx\sin \omega t\]

then if they know the position and velocity of every point on the rope at, say, t = 0, they will be able to determine the 4 arbitrary constants. In other words, if they know \(\psi \left( x,0 \right)\) and \(\left. \frac{\partial \psi \left( x,t \right)}{\partial t} \right|_{t=0}\) , then A, B, C and D will be known.

It is important to recognize that some students have trouble distinguishing the arbitrary constants A, B, C and D from the system parameters $\omega$ and k. They all look arbitrary to the students. And in fact it's a subtle point. There's really only one system parameter, and that's \(v_{ph}=\frac{\omega }{k}\) . The frequency $\omega$, at this stage in the absence of spatial boundary conditions, can be chosen at will. The students need help classifying all these different “constants”.

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