Wave Propagation in Linear Media | re-examined

8.1 Preparation of the wave integrals for quadrature
Now we are ready to formulate the integrals in an appropriate way for the implementation
with Mathematica.
The equations (8.1) to (8.6) still permit no straightforward application of our standard quadrature
routine. Remembering that we can extrapolate only to +1, we must make a variable
transformation ! , rst for the integrals over the negative real axis. Having done so,
we nd that the coe cient functions for the incident, re ected, and transmitted part of the
wave now become unique for both integrals, whereas the exponential arguments as well as the
coe cient functions of the wave spectra now are di erent in the two integrals. The coe cient
functions (8.8), (8.10), and (8.12) thus reduce to their positive branches
cinc( )=1
cref( )= ,p 2 ,1
+ p 2 ,1
ctun( )=
2
+ p :
2 ,1
For the triangular wave packet, we can combine (8.5) and (8.1 { 8.3) and nd
p
3
Ctria =
2k 2p
s +
1
tria ( )=
1, 1 p
s , tria ( )=
1
1+ 1
p
2
2
(8.16)
(8.17)
(8.18)
With these de nitions we can put the parts together to eventually obtain the nal form of
the integrals that is suitable for quadrature:
inc,tria
p l = Ctria
+ Ctria
+ Ctria
Z 1
1
Z 1
1
Z 1
,1
s +
tria ( ) 2ej(,T 2 + k 1 p +X , k)
,
, e j(,T 2 +2 k 1
p +X ,2 k) , e j(,T 2 +X ) d +
s , tria ( ) 2ej(,T 2 , k 1
p ,X , k) ,
, e j(,T 2 ,2 k 1
p ,X ,2 k) , e j(,T 2 ,X ) d +
s +
tria ( ) 2ej(,T 2 + k 1
p +X , k)
,
, e j(,T 2 +2 k 1
p +X ,2 k) , e j(,T 2 +X ) d
193
(8.19)