Wave Propagation in Linear Media | re-examined

1.3 Signal velocity
The arbitrariness that underlies this de nition of the signal velocity has caused considerable
irritation and dispute over the years. The main objection was the di culty to apply the
de nitions to any practical case. If one waits for the signal to reach a certain fraction of
its maximum amplitude, one must know the entire signal before being able to decide when
it has arrived [8], and still one might catch the precursor instead of the steady-state part.
The other problem is that the amplitude of the steady-state signal may be much less than
that of the transient part, so if one required the signal to exceed its precursor, one would
literally wait forever. Apart from these theoretical arguments, Trizna and Weber [23] carried
out a simulation based on an expansion technique and attempted an experimental veri cation
of the signal velocity. To this end, they had to investigate the square of the wave (x; t) 2 .
They observed a ringing e ect in accordance with the theory, but no pulse delay. Hence they
concluded that a separation of precursor and steady state signal is not as straightforward as
Brillouin suggested, and that the signal velocity de nition is not meaningful in practice.
Despite or perhaps just because of these discussions, the wave propagation in the Lorentz
medium was the subject of continued research. More than seventy years after the papers of
Sommerfeld and Brillouin, Oughstun and Sherman [24] published an article that reconsidered
the classical problem and provided an improved solution. They also used the saddle point
integration method, but did not follow the paths of steepest descents, which allowed for a
simpler contour of integration. In their interpretation of the signal velocity they followed
Baerwald in that they de ned the signal arrival to occur when the real part of the phase
function along the contour of integration equals that of the residue at ! = !c. Physically, this
means that before this point, the wave is dominated by the contribution of one of the saddle
points moving through the complex plane. After the signal arrival, the eld is dominated by
the steady state signal, and the precursors are negligible. The signal arrival is thus exactly
the moment when the attenuation of the stationary oscillation is the same as that of the
dominating saddle point at that time | hence the use of the real part of the phase function.
A di erent interpretation useful for measurements and numerical experiments is that at this
time, the wave begins to oscillate with the carrier frequency !c [25, 26].
The result of this analysis was that the rst or Sommerfeld precursor oscillates at a high
frequency, whereas the second or Brillouin precursor is low-frequent [27]. Whether they reach
a signi cant amplitude, and which of the two is more important, depends on the characteristic
of the input signal. In particular, the distortion due to the precursors becomes more severe as
the signal frequency is shifted towards the absorption band or the rise and fall times are made
smaller (i. e. the spectrum is broadened). Additionally, for rectangular pulses with su ciently
short pulse width the precursors of the leading and trailing edge may interfere [28, 29].
Since the signal velocity has a similar de nition to that of Baerwald, its dependence on the
signal frequency is similar, too. Consequently, Oughstun also found a minimum near resonance.
At frequencies well above resonance, however, the spacing between the two precursors
is large enough that the signal seems to break up into two parts separated by the Brillouin
precursor. Therefore, Oughstun de ned two distinct velocities for these two parts (namely
the prepulse and the actual main signal), which adds a certain amount of absurdity to the
entire concept. Balictsis and Oughstun extended the research to Gaussian-shape pulses [30]
and found that even such a pulse may evolve into a pair of pulses provided that the initial
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