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Unformatted text preview: Physics 604 Midterm #2 Fall ‘10
Dr. Drake 1. (40 pts) The modiﬁed spherical Bessel equation is given by
z 2 y + 2zy − [z 2 + n(n + 1)]y = 0
(a) Find the lowest order behavior of the solutions as z → 0. Show that the
solutions are linearly independent. Identify values of n for which they are
not linearly independent and also calculate the solutions for this special
value of n.
(b) Calculate the behavior of the solutions for z large.
Hint: To put the equation in standard form take y = f /z p and solve for p.
2. (60 pts) The temperature in a one-dimensional medium satisﬁes a diﬀusion
− D 2T = 0
At t = 0, the temperature in the medium is given by
T (0, x) = 0 x>0
1 x<0 (a) Solve for the space/time dependence of T (x, t) by FIRST completing a
Laplace transform and then a Fourier transform of the equation and completing the subsequent inverse of the Laplace transform. The inversion of
the Fourier transform can not be done exactly so you will have an integral
representation for the solution given by a k space integral.
Hint: The Fourier transform must be done carefully to insure that the
integral converges for x → −∞. What does this convergence requirement
imply about k ?
(b) You can evaluate the time dependence of T for | x | large by evaluating
the k space integral asymptotically. What is the large parameter? Do this
ﬁrst for x > 0 and then x < 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