10.6: Chapter Checklist

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You should now be able to:

Solve a forced oscillation problem for a stretched string with arbitrary time dependent displacement at the end;
Decompose an arbitrary signal into harmonic components using the Fourier transformation;
Compute the group velocity of a dispersive system;
Understand the relations between a function and its Fourier transform that lead to the relation between bandwidth and fidelity;
Be able to describe the scattering of a wave packet;
Understand the effect of free charges on the dispersion relation of electromagnetic waves.

Problems

10.1. Is it possible for a medium that supports electromagnetic waves to have the dispersion relation \(\omega^{2}=c^{2} k^{2}-\omega_{0}^{2}\) for real \(\omega_{0}\)?

Why or why not?

10.2. A beaded string has neighboring beads separated by \(a\). If the maximum possible group velocity for waves on the string is \(v\), find \(T / m\).

10.3. In the next chapter, we will derive the dispersion relation for waves in water (or at least an idealized picture of water). If the water is deep, the dispersion relation is \[\omega^{2}=g k+\frac{T k^{3}}{\rho}\]

where \(g\) is the acceleration of gravity, 980 in cgs units, \(T\) is the surface tension, 72, and \(\rho\) is density, 1.0. Find the group velocity and phase velocity as a function of wavelength. When are they equal?

10.4. Consider the longitudinal oscillations of the system of blocks and massless springs shown below:

Each block has mass \(m\). Each spring has spring constant \(K\). The equilibrium separation between the blocks is \(a\). The ring on the left is moved back and forth with displacement \(B \cos \omega t\). This produces a traveling wave in the system moving to the right for \(\omega<2 \sqrt{K / m}\). There is no traveling wave moving to the left.

The dispersion relation for the system is \[\omega^{2}=\frac{4 K}{m} \sin ^{2} \frac{k a}{2} .\]

Suppose that \(\omega=\sqrt{K / m}\). Find the phase velocity of traveling waves at this frequency.
For \(\omega=\sqrt{K / m}\), find the displacement of the first block at time \(t=\pi / 2 \omega\). Express the answer as \(B\) times a pure number.
Find the group velocity in the limit \(\omega \rightarrow 2 \sqrt{K / m}\).
Find the time average of the power supplied by the force on the ring in the limit \(\omega \rightarrow 2 \sqrt{K / m}\).
Explain the relation between the answers to parts c. and d. You may be able to do this part even if you have gotten confused in the algebra. Think about the physics and try to understand what must be going on.