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# hwk10 - Physics 604 Homework#10 Fall 11 Dr Drake 1 The...

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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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