Wave Propagation in Linear Media | re-examined

1.1 Phase and group velocity
!
!c
vp = !
k
kc
vg = @!
@k
Figure 1.1: Brillouin diagram.
!(k)
Using the Fourier representation of the signal, we can write the envelope as
0(x; t) =
Z 1
,1
A(k) e j((k,kc)x,(!(k),!c)t) dk : (1.14)
The dispersion relation !(k) may be expanded in a Taylor series about kc, and if terms of
higher than linear order can be neglected, we nd (!(k) , !c)t = vg(k , kc)t. This nally
yields
0(x; t) =
Z 1
,1
A(k) e j(k,kc)(x,vgt) dk ; (1.15)
which is exactly the initial waveform 0(x; 0), but displaced by an amount vgt. In other
words: the envelope preserves its shape and moves with the group velocity. This result is
exact if and only if the dispersion relation has the form of a straight line ! = vgk + !p [2]. In
most cases, however, the dispersion relation will have a more general form, and (1.15) is then
only a rst-order approximation that applies to a su ciently small region of the spectrum
centred at kc and !c, respectively. If the spectrum of the envelope becomes broader, the
group velocity di erences of the harmonic components will cause the packet to spread.
Remark (Modulated versus baseband signals) Note that for modulated signals a
constant group velocity is a prerequisite so that the pulse shape is not deformed. For
baseband signals, on the other hand, the phase velocity must be invariant within the
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k