Useful Links about Fourier Analysis and Sinusoidal Wave Packets • A Visual Introduction to the Fourier Transform : If you are not as comfortable with Fourier Analysis as you would like, this 20-minute video is a magnificently intuitive introduction. Thank you to Ian Call for this recommendation! • When you superpose sinusoidal waves with wavelengths that are close to, but not equal to, a central “carrier” wavelength, a wave packet is formed that looks like the central carrier wave “modulated” by an envelope. The speed of the carrier is the phase velocity vph = ω/k of the packet while the speed of the modulating envelope is the group velocity vgr = dω/dk. These different speeds are succinctly illustrated in the following videos by meyavuz, where different relations between vph and vgr are demonstrated by playing with different dispersion relations ω(k). - Case 1 : vgr > vph - Case 2 : vgr = zero - Case 3 : vgr negative - Case 4 : vph = zero - Case 5 : vph > 0 and vgr < 0