7.5 Cold-plasma wave energy equation 221taking of the real part is often not explicitly stated in linear relationships where its omission
does not affect the mathematical logic. However, taking the real part isa critical step in
nonlinear relationships, because for nonlinear relationships omitting thisstep causes seri-
ous errors. In particular, when dealing with a product of two oscillating physical quantities,
sayψ(t)=Re
[
ψ ̃e−iωt]
and χ(t)=Re[
̃χe−iωt]
,it is essential to write the product
as
ψ(t)χ(t)=Re[
ψ ̃e−iωt]
×Re[
χ ̃e−iωt]
; (7.60)
that is the real part of the factors must always be establishedbeforecalculating the product.
Ifω=ωr+iωiis a complex frequency, then the product in Eq.(7.60) assumes the form
ψ(t)χ(t) =(
ψ ̃e−iωt+ψ ̃∗eiω∗t
2)(
̃χe−iωt+ ̃χ∗eiω
∗t2)
=
1
4
[
ψ ̃χ ̃e−2iωt+ψ ̃∗ ̃χ∗e2iω∗t
+ψ ̃ ̃χ∗e−i(ω−ω
∗)t
+ψ ̃
∗
̃χe−i(ω−ω
∗)t]
. (7.61)
When considering the energy density of a wave, we are typically interested in time-
average quantities, not rapidlyfluctuating quantities. Thus the time average of the product
ψ(t)χ(t)over one wave period will be considered. Theψ ̃χ ̃andψ ̃
∗
χ ̃∗terms oscillate at
the fast second harmonic of the frequency and vanish upon time-averaging. In contrast, the
ψ ̃ ̃χ∗andψ ̃∗χ ̃terms survive time-averaging because these terms have no oscillatory factor
sinceω−ω∗=2iωi.Thus, the time average (denoted by〈〉) of the product is
〈ψ(t)χ(t)〉 =1
4
(ψ ̃χ ̃∗+ψ ̃
∗
χ ̃)e^2 ωit;=1
2
Re(
ψ ̃χ ̃∗)
e^2 ωit; (7.62)this is the desired rule for time-averaging products of oscillating quantities.
7.5 Cold-plasma wave energy equation
The current densityJin Ampere’s law consists of the explicit plasma currents which are
frequency-dependent. This frequency dependence means that care is required wheninte-
gratingE·J.In order to arrange for this integration we express Eq.(6.9) and Eq.(6.10)
as
1
μ 0∇×B ̃ = ε 0∂
∂t(←→
K(ω)· ̃Ee−iωt)
= ̃Je−iωt+ε 0∂
∂t(
E ̃e−iωt)
(7.63)
where ̃Jconsists of the plasma currents and the time dependence is shown explicitly.