As weve seen, Eq (7 33) describes the beat pattern Lets now derive a different version of that expression assuming that the two overlapping equal amplitude cosine waves have angular spatial frequencies of k c Icirc k and k c Icirc k, and angular temporal frequencies of Iuml c Icirc Iuml and Iuml c Icirc Iuml , respectively Here k c and Iuml c correspond to the central frequencies Show that the resultant wave is then Explain how each term relates back to 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, Icirc Iuml Icirc k E 2E01 cos (Akx Awt) cos (kx w t) Figure P 7 19 (a) E Eo Eo (b)

Question: As weve seen, Eq. (7.33) describes the beat pattern. Lets now derive a different version of that expression assuming that the two overlapping equal-amplitude cosine

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+ Î”k and k_c- Î”k, and angular temporal frequencies of Ï‰_c+ Î”Ï‰ and Ï‰_c- Î”Ï‰, respectively. Here k_cand Ï‰_ccorrespond to the central frequencies. Show that the resultant wave is then

Explain how each term relates back to
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, Î”Ï‰/Î”k.

E = 2E01 cos (Akx Awt) cos (kx w.t) Figure P.7.19 (a) E Eo --Eo (b)

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