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__ v_{ph} = ω/k of the packet while
the speed of the modulating envelope is the __group velocity__ v_{gr} = dω/dk.
These different speeds are succinctly illustrated in the following videos by
meyavuz, where different relations between v_{ph} and v_{gr} are demonstrated by
playing with different __dispersion relations__ ω(k).
- Case 1 : v_{gr} > v_{ph}
- Case 2 : v_{gr} = zero
- Case 3 : v_{gr} negative
- Case 4 : v_{ph} = zero
- Case 5 : v_{ph} > 0 and v_{gr} < 0