Wave Propagation in Linear Media | re-examined

1.5 Other velocity de nitions
For negligible losses, the Fourier integral of (x; t) is determined by frequency components
where the derivative ofthe phase function k(!)x , !t vanishes, which yields x=t = vg.
Thus ve takes the role of vg in the presence of losses.
In a subsequent paper, Oughstun and Shen [45] extended the analysis to multiple-resonance
media. In their most recent publication, Sherman and Oughstun [46] provided a model that
allows more physical insight into the pulse propagation than did the purely mathematical
treatment with the saddle point integration. It applies if the medium is not too highly
absorbing and if only the region of mature dispersion is regarded. At anygiven time and in
a small region of space, the pulse is then predominantly made up of two components. One
of these is always quasi-monochromatic, the other may either be also quasi-monochromatic
or non-oscillatory. The two components signify the Sommerfeld and Brillouin precursors,
respectively, and move together with the same energy velocity. As the pulse propagates,
these monochromatic components decay according to their attenuation coe cients, and in
addition, the entire pulse spreads because the partial waves travel at di erent velocities.
Note that since the authors set up the model for a Dirac delta pulse (0;t)= (t), there is no
third contribution from a steady-state solution like the one that appeared in the investigation
of the step-modulated input eld.
There are still other examples where ve 6= vg. Mainardi [32] mentions the case of a uniform
transmission line characterised by frequency-independent distributed inductance, resistance,
capacitance, and shunt conductance. These parameters form the classical equivalent circuit
of a transmission line we shall treat in more detail in chapter 3.1. Using this model, it can be
shown that the energy velocity equals the phase velocity. If the transmission line is assumed
to be loss-free, the validity of this statement is trivial since then the transmission line is also
dispersion-free. If, however, losses are taken into account, the wave number becomes complex,
and we obtain ve = != Re k = vp in accordance with the de nition of (1.26).
Like all other velocity de nitions, the energy velocity was criticised, too. Smith [8] complained
that although the de nition is meaningful, it gives no hint astohow one can actually measure
the velocity. In particular, unlike the signal velocity, it does not de ne a signal arrival.
Concerning laser phenomena, Schulz-DuBois [13] noted that for emissive dispersive media
the energy velocity in the classical de nition can exceed the velocity of light. He therefore
proposed an alternative approach that distinguishes the propagation of purely electromagnetic
energy from the energy stored in the resonant medium.
1.5 Other velocity de nitions
The aforementioned three di erent velocities of propagating waves are the oldest and thus
classical ones. As if this had not been enough, a number of additional de nitions emerged in
the course of time. Many of these approaches considered in one way or another the movement
of a moment of a distribution or, in allusion to classical dynamics, the motion of some centre
of gravity. We have already encountered one such de nition. It was discussed by Smith [8] as
a proposed substitute for the group velocity and concerned the temporal centre of gravity of
the wave packet's amplitude j (x; t)j. Smith himself suggested still another alternative: the
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