Fundamentals of Plasma Physics

8.5 Analysis of linear mode conversion 241

8.5 Analysis of linear mode conversion

8.5.1 Airy equation

The procedure for analyzing linear mode conversion was developed by Stix (1965)and is
an extension of the standard method for connecting solutions of the Airy equation

y′′+xy= 0 (8.54)

from thex< 0 region to thex> 0 region. The Airy problem involves a second order
ordinary differential equation with a coefficient which goes through zero. Themode con-
version problem involves a fourth order equation which also has a coefficientgoing through
zero. The Airy connection method will first be examined to introduce the relevant concepts
and then these will be applied to the actual mode conversion problem.
The sequence of steps for developing the solution to the Airy problem are asfollows:

1. Laplace transform the equation: Equation (8.54) cannot be Fourier transformed be-
   cause of the non-constant coefficientxin the second term;however it can be Laplace
   transformed by make use of the relation that the Laplace transform ofxy(x)is−d ̃y(p)/dp.
   Thus, the Laplace transform of Eq.(8.54) is

p^2 y ̃(p)−

d ̃y(p)

dp

= 0. (8.55)

Strictly speaking, Laplace transform ofy′′will also bring in terms involving initial

conditions, but these will be ignored for now and this issue will be addressed later.

Equation (8.55) is easily solved to give

y ̃(p) =Aexp(p^3 /3) (8.56)

whereAis a constant. The formal solution to Eq.(8.54) is the inverse transform

y(x) = (2πi)−^1

∫

C

̃y(p)epxdp=a

∫

C

ef(p)dp (8.57)

wherea=A/ 2 πi,f(p) =p^3 /3 +px,andCis some contour in the complex plane.

By appropriate choice of this contour Eq.(8.57) is shown to be a valid solution to the

original differential equation. In particular, if the endpointsp 1 andp 2 of the contour

Care chosen to satisfy

[

ef(p)

]p 2

p 1

= 0 (8.58)

then Eq.(8.57) is a valid solution to Eq.(8.54). The reason for this may beseen by

explicitly calculatingy′′andxyusing Eq.(8.57) to obtain

y′′=

d^2

dx^2

a

∫

C

ep

(^3) /3+px
dp=a

∫

C

p^2 ep

(^3) /3+px
dp (8.59)