18 CHAPTER 2. ADIABATIC THEORY
Since the derivation of the second-order oscillatory components is more extensive, we will
only go through the calculation of v2!
2 . The second and third term on the left hand side
of Eq. (2.39b) cancel due to rst-order continuity Eq. (2.24b) and the rule Eq. (2.17) for
the 2!-component of products of rst-order elds, as shown here
0
��
v1@yv1
2! = 1
20v1@yv1 = 1
20v1i!
1
0
��
1(��i!v1)
= ��
2!: (2.40)
Inserting Eq. (2.39a) in Eq. (2.39b) and applying the 2!-rule Eq. (2.17) yields
4k2
0 v2!
2 + (1 �� i4��)@2
yv2!
2 +
1
20
@2
y (1v1) = 0: (2.41)
Inserting the rst-order solution Eq. (2.28) the source term becomes
1
20
@2
y (1v1) =
1
20
@2
y
i0
k
!
v2b
c
cos2(k w
2 )
cos(ky) sin(ky)
= ��i
k3
!
v2b
c
cos2(k w
2 )
sin(2ky);
(2.42)
The governing equation (2.41) for v2!
2 thus becomes
4k2
0 v2!
2 + (1 �� i4��)@2
yv2!
2 �� i
k3
!
v2b
c
cos2(k w
2 )
sin(2ky) = 0: (2.43)
The solution to the homogeneous equation is
v2!;hom
2 = Acos(2qy) + B sin(2qy); (2.44)
q = k0(1 + i2��); (2.45)
whereas the guess for the inhomogeneous solution is
v2!;inhom
2 = C sin(2ky): (2.46)
Inserting the inhomogeneous solution Eq. (2.46) into the governing equation (2.43) yields
4k2
0C �� C(1 �� i4��)4k2 �� i
k3
!
v2b
c
cos2(k w
2 )
= 0; (2.47)
which to rst order in �� becomes
i8��Ck2
0 �� i(1 + i3��)
k3
0
!
v2b
c
cos2(k w
2 )
= 0 )
C =
1 + i3��
8��
k0
!
v2b
c
cos2(k w
2 )
=
1
8
1
��cs
v2b
c
cos2(k w
2 )
+ O
��
��
: (2.48)