Fundamentals of Plasma Physics

15.2 Manley-Rowe relations 433

A corresponding set of relations for the action can be obtained by multiplying pairs
of equations in Eq.(15.15) byωjAjand then either adding or subtracting. For example,
multiplying the first pair byωjAjand subtracting gives

1

2

d

dt

(

ω 1 A^21 −ω 2 A^22

)

=0. (15.18)

Appropriate adding and subtracting in this manner gives

ω 1 A^21 −ω 2 A^22 = const.

ω 1 A^21 +ω 3 A^23 = const.

ω 2 A^22 +ω 3 A^23 = const. (15.19)

These relationships can be expressed in a manner analogous to quantum principles by defin-
ing the effective “quantum number” of a mode as its ratio of energy to frequency,

Nj=

Wj

ωj

=

m

2

ωjA^2 j. (15.20)

It is clear thatNjis the same as the action except for an unimportant constant factor. Thus,
the action conservation rules can be recast as

N 1 −N 2 = const.

N 1 +N 3 = const.

N 2 +N 3 = const. (15.21)

or if changes in action are considered

∆N 1 = +∆N 2

∆N 1 = −∆N 3

∆N 2 = −∆N 3. (15.22)

This provides an action accounting scheme such that a change∆N 3 =− 1 can be consid-
ered as a mode 3 "photon" decaying (or equivalently disintegrating) into a mode 1 photon
(∆N 1 = +1) and a mode 2 photon (∆N 2 = +1), all the while satisfying conservation of
energy;this is sketched in Fig. 15.2.