7.5 Cold-plasma wave energy equation 223Ifωi= 0thenS,P, andDare all pure real. In this case
←→
K(ω)is Hermitian so that
←→
Kh=
←→
Kand←→
Kah=0.However, ifωiis finite but small, then←→
K(ω)will have a small
non-Hermitian part. This non-Hermitian part is extracted using a Taylor expansion in terms
ofωi,i.e.,
←→
K(ωr+iωi)=
←→
K(ωr)+iωi[
∂
∂ω←→
K(ω)]
ω=ωr. (7.71)
The transpose of the complex conjugate of this expansion is
[←→
K(ωr+iωi)]†
=
←→
K(ωr)−iωi[
∂
∂ω←→
K(ω)]
ω=ωr(7.72)
since
←→
Kis non-Hermitian only to the extent thatωiis finite. Substituting Eqs.(7.71) and
(7.72) into Eqs.(7.68) and (7.69) and assuming smallωigives
←→
Kh=←→
K(ωr) (7.73)and
←→
Kah=iωi
[
∂
∂ω←→
K(ω)]
ω=ωr. (7.74)
Inserting Eqs. (7.73) and (7.74) in Eq.(7.67) yields
〈
E·(
J+ε 0∂E
∂t)〉
=
2 ε 0 ωi
4[
ωr ̃E∗·[
∂
∂ω←→
K(ω)]
ω=ωr· ̃E+E ̃∗·
←→
K(ωr)·E ̃]
e^2 ωit=
2 ε 0 ωi
4E ̃∗·
[
∂
∂ω
ω←→
K(ω)]
ω=ωr· ̃Ee^2 ωit.
(7.75)
Similarly, the rate of change of the magnetic energy density is
〈
1
μ 0
B·
∂B
∂t〉
=
1
4 μ 0[
2 ωiB ̃∗·B ̃]
e^2 ωit. (7.76)Using Eqs.(7.75) and (7.76) in Eq.(7.64) gives
w=w(−∞)+
{
ε 0
4E ̃∗·
[
∂
∂ω(
ω←→
K(ω))]
ω=ωr· ̃E+
1
4 μ 0[
B ̃∗·B ̃
]
}∫
t−∞dt 2 ωie^2 ωit(7.77)
which now may be integrated in time to give the total energy densityassociated with bring-
ing the wave into existence
w ̄ = w−w(−∞)={
ε 0
4E ̃∗·
[
∂
∂ω(
ω←→
K(ω))]
ω=ωr·E ̃+
1
4 μ 0[
B ̃∗·B ̃
]
}
e^2 ωit. (7.78)In the limitωi→ 0 this reduces to
w ̄=ε 0
4̃E∗·∂
∂ω[
ω←→
K(ω)]
·E ̃+
|B ̃|^2
4 μ 0