Physics 604
Homework #10
Fall ‘11
Dr. Drake
1. The inhomogeneous Airy equation is given by
d
2
y
dx
2

xy
= 1
(1)
This equation arises in mode conversion problems in plasma where an electro
magnetic wave drives an electrostatic wave at its cutoff. The ”1” on the right
hand side represents the electromagnetic driver and the left side the electrostatic
response. The time dependence
e

iωt
has been factored out of the equation.
(a) For

x

>>
1 write down approximate expressions for the particular and
homogeneous solutions to the equation. The complete solution must cor
respond to a linear combination of these solutions for

x

large. Which of
these large

x

solutions are physical (think about causality with a wave
source around
x
= 0)?
(b) To obtain the full solution, write
y
(
x
) =
Z
C
dkY
(
k
)
e
ikx
,
(2)
where the contour C can be chosen so that one of the endpoints of the
integral yields the ”1” on the right hand side of Eq. (2). The other endpoint
of the contour must be chosen to yield the physical boundary conditions
as
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 Fall '11
 drake
 Physics, Work, Steady State, Homogeneity, Dr. Drake, inhomogeneous Airy equation

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