10: Signals and Fourier Analysis

Preview

In this chapter, we will see how this works in the context of a forced oscillation problem. In the process, we will find a subtlety in the notion of the speed with which a traveling wave moves. The phase velocity may not be the same as the velocity of signal propagation.

1. We begin by studying the propagation of a transverse pulse on a stretched string. We solve the problem in two ways: with a trick that works in this special case; and with the more powerful technique of Fourier transformation. We introduce the concept of “group velocity,” the speed at which signals can actually be sent in a real system.
2. We discuss, by example and then in general, the counterpoint between a function and its Fourier transform. We make the connection to the physical concepts of bandwidth and fidelity in signal transmission and to Heisenberg’s uncertainty relation in quantum mechanics.
3. We work out in some detail an example of the scattering of a wave packet.
4. We discuss the dispersion relation for electromagnetic waves in more detail and explore the question of whether light actually travels at the speed of light!