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Unformatted text preview: Physics 604 Midterm #2 Fall ‘11 Dr. Drake
1. (35 points) The Coulomb Wave equation is given by
(£210
ppm—2mm =0 (1) where n is real and p > 0. This equation describes a wave propagating towards a turning point and an adjacent singularity. (a) Classify the singular points of this equation (don’t worry about inﬁnity). (1)) Find the behavior of the solutions for ,0 very close to zero. Show that the solutions are linearly independent. (c) Plot the square of the WKB wavevector k2(p) versus p. Calculate the behavior of the solutions for )9 very iarge. 2. (35 points) A wave in a onedimensional medium is driven by a 5 — function source
at :r = 0 that is turned on at t = O as follows: __.
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where y x 0 for t < 0 and H (t) is zero for t < O and one for t > 0. Solve for the
space/time dependence of y($, t) by completing Laplace and Fourier transforms of
the equation and then completing the subsequent inverse Laplace transform. Finally
complete the inverse Fourier transform to obtain y(:r, t). Plot the spatial dependence
of y at a ﬁxed time. Interpret the result on the basis of causality.
Hint: To reduce the required algebra evaluate y(m, t) explicitly only :1; > O and use the symmetry of the problem to sketch the solution for a: < 0. 3. (30 points) An integral representation for the Couiomb wave equation in (1) can be written as we) : f0 simmers (3) (a) Derive a differential equation for Y(k) by inserting (3) into (1). Explicitiy give the conditions on the contour C in order that the integral solution satisfy (1). Show that the solution of your equation for YUc) is given by
YUC) = (1 + klilimﬂ — WW” (4) (b) Choose two contours in the complex k plane which satisfy the constraint given
in part (a) and therefore give two linearly independent solutions to Draw them. Hint: You must deﬁne cuts for the singularities of YUC) at k = ii. The contours
wrap around the cuts. Where do they end? Recall that p > 0. Had116mm #2 Sv/wams é) % @ €W€€+C€’l”?)uﬂ:0
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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.
 Fall '11
 drake
 Physics

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