hwk8 - the equation ( y ∼ e-iωt ), d 2 y dx 2 + k 2 ( x...

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Physics 604 Homework #8 Fall ‘11 Dr. Drake 1. Arfken 9.6.11 2. Bessel’s equation is given by z 2 d 2 y dz 2 + z dy dz + ( z 2 - ν 2 ) y = 0 . Convert this equation to the standard form d 2 y dz 2 + k 2 ( z ) y = 0 and calculate the WKB solutions for large | z | . Where are the solutions valid? Compare your results to the asymptotic form of the Hankel function obtained from the integral representation (in the book). 3. The parabolic cylinder equation is given by d 2 y dz 2 - ( 1 4 z 2 + a ) y = 0 . Repeat the above for this equation. 4. A wave is incident on the region x 0 from the left with an amplitude unity and a frequency ω . The medium is nonuniform around x = 0. The wave satisfies
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Unformatted text preview: the equation ( y ∼ e-iωt ), d 2 y dx 2 + k 2 ( x ) y = 0 with k 2 = ω 2 c 2 (1 + 1 π L 2 x 2 + L 2 ) (a) Write an expression for the space/time dependence of the wave when x is negative and | x | ± L (to lowest order). (b) Describe what happens to the wave when ωL/c ± 1 and when ωL/c ² 1 (qualitative statement). (c) Calculate the amplitude of the wave everywhere for ωL/c ± 1. (d) Calculate the amplitude of the wave everywhere for ωL/c ² 1. Hint: In this case you can approximate π-1 L/ ( x 2 + L 2 ) = δ ( x ). Why?...
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This note was uploaded on 12/30/2011 for the course PHYSICS 604 taught by Professor Drake during the Fall '11 term at Maryland.

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