10: Signals and Fourier Analysis
- Page ID
- 34401
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Traveling waves with a definite frequency carry energy but no information. They are just there, always have been and always will be. To send information, we must send a nonharmonic signal.
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.
- 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.
- 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.
- We work out in some detail an example of the scattering of a wave packet.
- 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!