222 Chapter 7. Waves in inhomogeneous plasmas and wave energy relations
Integration of Eq.(7.56) taking into account the prescription given by Eq.(7.60)has the
form
w(t) = w(−∞)+∫t−∞dt〈
E(x,t)·(
J(x,t)+ε 0∂E(x,t)
∂t)
+
1
μ 0
B(x,t)·∂B(x,t)
∂t〉
= w(−∞)+1
4
∫t−∞dt〈 ̃
Ee−iωt·ε 0∂
∂t(←→
K(ω)· ̃Ee−iωt)∗
+
1
μ 0B ̃e−iωt·∂
∂t(
B ̃e−iωt)∗
+c.c.〉
(7.64)
where c.c. means complex conjugate. The term containing the rates of changeof the
electric field and the particle kinetic energy can be written as
〈
E·
(
J+ε 0∂E
∂t)〉
=
ε 0
4{
̃Ee−iωt·(
iω∗←→
K∗·E ̃∗eiω
∗t)
+c.c.}
=
ε 0
4{
̃E·iω∗←→K∗·E ̃∗−E ̃∗·iω←→K·E ̃}
e^2 ωit=
ε 0
4
iωr[
E ̃·←→K∗·E ̃∗−E ̃∗·←→K·E ̃
]
+ωi[
̃E·←→K∗·E ̃∗+E ̃∗·←→K·E ̃
]
e^2 ωit.(7.65)
To proceed further it is noted that
E ̃·←→K∗E ̃∗ =∑
pqE ̃pK∗pqE ̃∗q=∑
pqE ̃pKqp∗tE ̃q∗=E ̃∗·←→K†·E ̃ (7.66)where the superscripttmeans transpose and the dagger superscript†means Hermitian
conjugate, i.e., the complex conjugate of the transpose. Thus, Eq.(7.65) can be re-written
as
〈
E·
(
J+ε 0∂E
∂t)〉
=
ε 0
4[
iωrE ̃∗·(←→
K†−
←→
K
)
·E ̃+ωiE ̃∗·(←→
K†+
←→
K
)
·E ̃
]
e^2 ωit.
(7.67)
Both the Hermitian part of the dielectric tensor,
←→
Kh=1
2
(←→
K+
←→
K†
)
, (7.68)
and the anti-Hermitian part,
←→
Kah=1
2
(←→
K−
←→
K†
)
, (7.69)
occur in Eq. (7.67). The cold plasma dielectric tensor is a function ofωvia the functions
S,P,andD,
←→
K(ω)=
S(ω) −iD(ω) 0
iD(ω) S(ω) 0
0 0 P(ω)