15.3 Application to waves 443Although the two velocities are oscillating 90^0 out of phase relative to their respective elec-
tric fields, the non-linear product of the velocities has the same phase behavioras the elec-
tric fields. This is because the nonlinear product of the velocities scales as〈cosω 2 tcosω 3 t〉
and the nonlinear product of the electric fields scales as〈sinω 2 tsinω 3 t〉so that the dif-
ference between these scalings is〈cosω 2 tcosω 3 t−sinω 2 tsinω 3 t〉=〈cos(ω 2 +ω 3 )t〉
which vanishes becauseω 2 +ω 3 is non-resonant. The system of equations thus becomes
(
∂^2
∂t^2+ω^2 pe−c^2 ∇^2)
E ̃ 3 = −ω^2 pe ̃n^1
nE ̃ 2
(
∂^2
∂t^2+ω^2 pe−
3 κTe
me∇^2
)
E ̃ 2 = −ω^2 pe ̃n^1
nE ̃ 3
(
∂^2
∂t^2
−c^2 s∇^2)
n ̃ 1
n=
qe^2
mimeω 2 ω 3∇^2
(
E ̃ 2 ·E ̃ 3
)
. (15.76)
If different modes are used, e.g., a Langmuir mode decaying into another Langmuir mode
or if the low frequency mode is a Langmuir wave, the left hand side wave terms will be
changed accordingly, but the right hand coupling terms will stay the same exceptif the low
frequency wave is a Langmuir wave in which case the productmimemust be replaced by
m^2 ein the denominator of the right hand side of the third equation.
The right hand coupling terms can be made identical by defining a renormalized density
perturbation
ψ ̃=αn ̃^1
n(15.77)
so that the equations become
(
∂^2
∂t^2+ω^2 pe−c^2 ∇^2)
E ̃ 3 = −ω^2 pe
ψ ̃
αE ̃ 2
(
∂^2
∂t^2+ω^2 pe−3 κTe
me∇^2
)
E ̃ 2 = −ω^2 peψ ̃
αE ̃ 3
(
∂^2
∂t^2−c^2 s∇^2)
ψ ̃ = − αq2
ek
2
1
mimeω 2 ω 3E ̃ 2 ·E ̃ 3. (15.78)
Equating the coefficients of the second and third terms gives
α=ωpe√
mimeω 2 ω 3
qek 1(15.79)
so the equations become
(
∂^2
∂t^2+ω^2 pe−c^2 ∇^2)
E ̃ 3 = −λψ ̃ ̃E 2
(
∂^2
∂t^2
+ω^2 pe−3 κTe
me∇^2
)
E ̃ 2 = −λψ ̃ ̃E 3
(
∂^2
∂t^2−c^2 s∇^2)
ψ ̃ = −λE ̃ 2 ·E ̃ 3. (15.80)