Wave Propagation in Linear Media | re-examined

1.3 Signal velocity
dipoles with a single resonance frequency !0 and some damping coe
relation is thus given by
cient . The dispersion
k =
s ! 2
c 2
!p 2
1 ,
! 2 , !0 2 +2j!
; (1.29)
!p being the plasma frequency. Sommerfeld then showed that the wave front of any signal
always travels at the velocity of light c.
Proof The proof is based upon function theoretical arguments. Since the Fourier integral for the
propagated wave, R 1
d!
ej(kx,!t)
,1 , does not converge if the integration is carried out along the real
!,!c
axis due to the pole at !c, one can move the contour of integration to the upper half of the complex
plane, such that ! 7! ! + j . The integral converges if the argument of the exponential function has
a negative real part at in nity. Since lim!!1 k = !=c, the argument at in nity is,j!(t , x=c), and
its real part is negative if and only if t , x=c < 0. The contour of integration may be closed with
a semicircle to the right at in nity, and as this area contains no singularities, the integral is zero.
Hence the medium is at rest wherever t , x=c < 0, which means that the wave front propagates at
the velocity of light. This statement applies to all media whose dispersion relation is characterised by
kc = !j!!1.
The work of Sommerfeld was complemented by a paper by Brillouin [19], who elaborated the
behaviour of the signal. He used the relatively new method of saddle point integration and
steepest descents to obtain an approximate evaluation of the integral. This method relies on
the fact that a Fourier-type integral of a complex variable, R e '(!) d!, is dominated by those
areas of the complex plane where the path of integration passes over a saddle point of'(!).
Thus an approximate evaluation of the integral may be con ned to the environment of such
points, which are crossed following the path of steepest descent. In fact, this method is a
generalisation of the method of stationary phase.
Remark (Method of stationary phase) This approximation method was introduced
by Lord Kelvin. Consider a Fourier integral R 1
,1 f(!)ej(k(!)x,!t) d!, where the spectral
function f(!) is only slowly varying. If x or t are large, then the main contribution to
the integral comes from areas where the argument of the exponential function does not
change, i. e. k0 (!)x , t =0. Outside the vicinity of such points, the integrand oscillates
strongly and the contribution is practically zero. It is therefore su cient toevaluate the
integral around the points of stationary phase, which maybe accomplished by expansion
techniques (see, for example, Lighthill [7] or Wait [20]).
From a purely mathematical point of view, the evolution of the wave (x; t) is largely determined
by the location of the saddle points, and the contour of integration must follow
these points as they move through the complex plane. This is what Brillouin did, and he
nally found that the saddle points | of which there are two pairs due to the structure of the
dispersion relation (1.29) | cause two distinct so-called precursors to appear after the arrival
of the wave front. The rst precursor had already been identi ed by Sommerfeld by means of
an asymptotic expansion of the dispersion relation, thus valid only for small values of t , x=c
or high frequencies. Therefore, subsequent authors often referred to the rst precursor as
Sommerfeld and to the second as Brillouin precursor.
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