Wave Propagation in Linear Media | re-examined

8 Application of the quadrature routine
If we take a closer look at the integrands, we recognise that they all have a similar structure.
The integrands of the wave functions have the general form
f( )=CrA( )cr( )e jpr( ) ; (8.7)
with a constant Cr, a coe cient function cr( ) that is de ned by the transmission and reection
coe cients and a function pr( ) describing the phase of the integrand. All these
functions depend on the region where the wave function is evaluated and on the direction
of propagation, respectively. In addition, they might be slightly di erent for positive and
negative wavenumbers because of the dispersion relations.
For the step potential barrier, these functions are given below. Note that for the evanescent
part of the spectrum in the interval ,1 < 1ifwe take the principal value of the complex root. The
constant Cr = !c=c is independent of both the region and the direction of propagation, so we
disregard it in the sequel.
cinc( )=1 (8.8)
pinc( )=X , T 2
c ref( )=
8
><
>:
p
+ 2,1
, p 2 ,1
, p 2 ,1
p
+ 2,1
p ref( )=,X , T 2
ctun( )=
ptun( )=
8
><
>:
8
<
:
2
, p 2 ,1
2 p
+ 2,1
if ,1
if ,1
,X p 2 ,1,T 2 if ,1
(8.9)
(8.10)
(8.11)
(8.12)
(8.13)
The spectral functions, too, can be written in a uniform manner as the sum of exponential
functions,
A( )=Csss( )
mX
i=1
as;i e jgs;i( ) ; (8.14)
where Cs is some waveform-dependent constant and ss is a function that de nes the shape of
the spectrum. Taking equations (8.7) and (8.14) together, we obtain the complete structure
of the wave integrands,
f( )=Csss( )cr( )
mX
i=1
192
as;i e j(gs;i( )+pr( )) : (8.15)