Chapter 48: Beats – Adding and Modulating Waves

When two cosine waves are added, the resultant wave features a rapid oscillation at the average frequency, but its overall amplitude varies slowly at the much smaller difference frequency. This periodic fluctuation in amplitude is called a "pulsation". The intensity of the resulting wave is analyzed, showing how it swells and falls between maximum and minimum values, which depend on the relative sizes of the individual amplitudes. The principles governing beats are then applied to amplitude modulation (AM), a technique used in radio broadcasting. In AM, a high-frequency carrier wave is modified by an audio signal, and the resulting complex wave can be mathematically understood as a combination of three distinct frequencies: the carrier frequency, plus two surrounding frequencies known as the side bands. The chapter then shifts focus to localized wave trains, introducing the crucial distinction between phase velocity and group velocity. The phase velocity is the speed at which individual wave crests move, while the group velocity represents the speed of the overall modulating envelope, which is the physical location where the energy and information are transmitted. For waves where the ratio of frequency to wave number is not constant, the phase velocity and group velocity differ. The concept of group velocity is demonstrated to be the rate of change of frequency with respect to the wave number. This concept is highly significant as the chapter applies it to quantum mechanics by considering the probability amplitude of a particle as a wave packet. By relating the particle's energy and momentum to the wave's frequency and wave number, it is established that the group velocity of the quantum wave packet corresponds exactly to the velocity of the particle itself, providing a key link between classical mechanics and wave theory. Finally, the discussion is extended to generalize the standard wave equation to three dimensions, a necessary step for accurately describing wave phenomena in space, such as sound waves and quantum probability amplitudes.