Wave Propagation in Linear Media | re-examined

8.1 Preparation of the wave integrals for quadrature
8.1 Preparation of the wave integrals for quadrature
We recall from section 4.1 the integrals describing the wave function in the di erent regions
of a step potential barrier:
inc = !p
c
ref = !p
c
+ !p
c
+ !p
c
tun = !p
c
+ !p
c
+ !p
c
Z 1
A( ) e
,1
jX ,j 2T d ; (8.1)
Z ,1
,1
Z 1
,1
Z 1
1
Z ,1
,1
Z 1
,1
Z 1
1
A( ) + p 2 , 1
, p ,jX ,jT 2
e d +
2 , 1
A( ) , jp1 , 2
+ j p 2 ,jT
e,jX d +
1 , 2
A( ) , p 2 , 1
+ p ,jX ,jT 2
e d ;
2 , 1
A( )
A( )
A( )
2
, p e
2 , 1 ,jX
p
2,1,jT 2
d +
2
+ j p 1 ,
2 e,X
p 1, 2 ,jT 2
d +
2
+ p e
2 , 1 jX
p
2,1,jT 2
d :
In section 4.2, we found the spectra of the initial wave forms,
A rect ( )
p l = j
4 2 k
A tria ( )
p l
A gauss( )
p l
p
3
=
4 3k2 =
1
1 , 1
p
1
1 , 1
p
1
p 2 n p 2 e
,
0
,j2 k 1, p1
e
,j k 1, p
1
2e 2
@ 2 k p 1
,1
2n
191
1
A 2
(8.2)
(8.3)
, 1 ; (8.4)
,j k 1, 1
p
,j2 k 1, p
1
, e
, 1 ; (8.5)
: (8.6)