As we've seen, Eq. (7.33) describes the beat pattern. Let's...

As we've seen, Eq. (7.33) describes the beat pattern. Let's now derive a different version of that expression assuming that the two overlapping equal-amplitude cosine waves have angular spatial frequencies of $k_{c}+\Delta k$ and $k_{c}-\Delta k,$ and angular temporal frequencies of $\omega_{c}+\Delta \omega$ and $\omega_{c}-\Delta \omega,$ respectively. Here $k_{c}$ and $\omega_{c}$ correspond to the central frequencies. Show that the resultant wave is then $$E=2 E_{01} \cos (\Delta k x-\Delta \omega t) \cos \left(k_{c} x-\omega_{c} t\right)$$ Explain how each term relates back to $$E=2 E_{01} \cos \left(k_{m} x-\omega_{m} t\right) \cos (\bar{k} x-\bar{\omega} t)$$ Prove that the speed of the envelope, which is the wavelength of the envelope divided by the period of the envelope, equals the group velocity, namely, $\Delta \omega / \Delta k$.

As we've seen, Eq. (7.33) describes the beat pattern. Let's now derive a different version of that expression assuming that the two overlapping equal-amplitude cosine waves have angular spatial frequencies of $k_{c}+\Delta k$ and $k_{c}-\Delta k,$ and angular temporal frequencies of $\omega_{c}+\Delta \omega$ and $\omega_{c}-\Delta \omega,$ respectively. Here $k_{c}$ and $\omega_{c}$ correspond to the central frequencies. Show that the resultant wave is then
$$E=2 E_{01} \cos (\Delta k x-\Delta \omega t) \cos \left(k_{c} x-\omega_{c} t\right)$$
Explain how each term relates back to
$$E=2 E_{01} \cos \left(k_{m} x-\omega_{m} t\right) \cos (\bar{k} x-\bar{\omega} t)$$
Prove that the speed of the envelope, which is the wavelength of the envelope divided by the period of the envelope, equals the group velocity, namely, $\Delta \omega / \Delta k$.

Key Concepts

Beat Phenomenon

The beat phenomenon occurs when two waves of similar, but not identical, frequencies interfere, leading to a modulation in amplitude at a frequency equal to the difference of the two original frequencies. In this problem, the envelope term embodies this phenomenon, as its spatial and temporal variations—characterized by ?k and ??—define the beat frequency and wavelength, giving rise to the observable modulation pattern.

Envelope and Carrier Waves

In the expression derived, the 'cos(?k x - ?? t)' term represents the envelope, which modulates the amplitude of the fast oscillating 'cos(kc x - ?c t)' carrier wave. The carrier wave oscillates at the central or average frequency, while the envelope oscillates at a much lower frequency corresponding to the difference between the two original waves. Recognizing these two components is crucial for understanding phenomena such as beating and amplitude modulation in wave behavior.

Group Velocity

Group velocity is the speed at which the overall shape (or envelope) of the wave propagates through space. It is typically defined as the derivative of the frequency with respect to the wave number (d?/dk) and, in this example, is approximated by the ratio ??/?k. This concept is particularly important in understanding how wave packets and modulated signals travel, as it carries energy and information, in contrast to the phase velocity of the carrier wave.

Superposition of Waves

This concept refers to the principle that when two or more waves travel through the same medium, the resultant wave is the algebraic sum of the individual waves. In the context of this problem, the superposition of two cosine waves with slightly different angular spatial and temporal frequencies leads to an interference pattern that displays both a fast oscillatory component (the carrier) and a slower modulating envelope (the beat pattern).

Trigonometric Sum-to-Product Identity

This mathematical concept involves converting the sum or difference of trigonometric functions into a product. Specifically, the sum of two cosine functions with different arguments can be expressed as a product of a cosine function representing the envelope (or beat modulation) and another cosine function representing the carrier wave. This identity is key to transforming the sum of two waves into the form that clearly separates the modulation from the carrier, as seen in the derived expression.

Please give Ace some feedback