Fundamentals of Plasma Physics

15.3 Application to waves 435

it is seen that
dA 2
dt

=

dτ

dt

dA 2

dτ

=sinθ

dA 2

dτ

(15.27)

and so
d^2 A 2
dτ^2

=

λ^2 A^23

16 m^2 ω 1 ω 2

A 2. (15.28)

Sinceω 1 ω 2 > 0 andω 1 +ω 2 =ω 3 ,ifω 3 is chosen to be positive, then bothω 1 andω 2
must both be positive and thereforeω 3 >ω 1 ,ω 2 .IfA 2 grows, then because of the action
rules,A 1 must grow in the same proportion. Furthermore, asA 1 andA 2 grow,A 3 must
decrease until the approximationA 3 >>A 1 ,A 3 fails. This process can be considered as
a high energy “photon” with frequencyω 3 decaying into anω 1 photon and anω 2 photon
where the latter two photons have lower energy thanω 1 .Equation (15.28) shows that the
decay of the mode 3 photon into mode 1,2 photons can be characterized by an exponential
growth of the mode 2 amplitude,
A 2 ∼eγτ (15.29)
where

γ=

λA 3

4 m

√

ω 1 ω 2

(15.30)

is the instability growth rate. Mode 3 is called the pump mode since it supplies the energy
for the instability. Modes 1 and 2 are called the daughter modes.

15.3 Application to waves

Nonlinearities in wave equations give rise to coupled systems of equations similar to Eqs.(15.3).
SupposeQnow refers to a plasma parameter, say density, and that the plasma can support
three distinct waves (modes) having respective linear densityfluctuations

Q 1 (x,t) = A 1 (t)cos(k 1 ·x−ω 1 t−δ 1 (t))

Q 2 (x,t) = A 2 (t)cos(k 2 ·x−ω 2 t−δ 2 (t))

Q 3 (x,t) = A 3 (t)cos(k 3 ·x−ω 3 t−δ 3 (t)); (15.31)

the densityQtherefore has the functional form

Q(x,t)=Q 1 (x,t)+Q 2 (x,t)+Q 3 (x,t). (15.32)

This situation is mathematically analogous to the coupled oscillator system discussed in
Sec.15.2 if the amplitudeQjof each of the three waves is considered to be an effective
canonical coordinate. Each wave satisfies a linear dispersion relationωj=ωj(k)and non-
linearities in the wave equations provide a mutual coupling between the modesanalogous
to the coupling between the different directions of motion for the oscillatingmass discussed
in Sec.15.2. Theexp(ik·x)dependence of the waves suggests the need for a wavenumber
selection rule
k 3 =k 1 +k 2 (15.33)
analogous to the frequency selection rule Eq.(15.10). Using the wavenumber selection rule
and generalizing the phase offset definition to beδ′j(t) =δj(t)−kj·x,it is seen that
the coupled wave equations become identical to the coupled oscillator equations withδ′j