Wave Propagation in Linear Media | re-examined

4.2 Initial wave forms
-60 -50 -40 -30 -20 -10
1
0.5
-0.5
-1
Ψ0
X
0.15
0.125
0.1
0.075
0.05
0.025
A
-2 -1 1 2 ξ
-2 -1 1 2 ξ
Figure 4.1: Wave function and spectrum of a rectangular pulse for k = 6 and = 0:8. The left graph
shows the real part of the wave function (thin line) and the corresponding probability density (thick
line). The right graph gives the absolute value jA( ) l ,1=2 j of the wave number spectrum (4.22). The
evanescent part is the dark grey area, and the peak of the spectrum lies at = 1=2 .
which immediately gives
Arect( )= p l 1
2
sin( 0 , ) l
2
( 0 , ) l
e
2
,j( 0, ) l 2 : (4.19)
We prefer, however, to write the spectrum with the normalised variable (4.13). To this end,
we add the additional de nitions
= !0
!p
; 2 k = 0 l: (4.20)
The length l of the original wave packet is thus given in a multiple k of the wavelength of
the modulation frequency. From the dispersion relation (4.6) and (4.12) we nd the identity
!p=c = 0= p , which together with (4.13) yields
l =2 kp : (4.21)
If we furthermore write the sine function in its Euler form, we obtain the normalised spectrum
of the rectangular wave packet,
A rect( )
p l = j
4 2 k
1
1 , 1
p
,j2 k 1, p
1
e
, 1 : (4.22)
Fig. 4.1 shows both the rectangular wave packet and its spectrum (4.22). Note that in the
scaled notation, the length l changes to 2 k= p and the phase function 0 x becomes p X.
The second initial wave packet we consider has a triangular or tent shape,
0 (x) =
8
><
>:
0 if x