sampleexam2 - Physics 604 Midterm #2 Fall ‘10 Dr. Drake...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 equation: ∂T ∂2 − D 2T = 0 ∂t ∂x 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. ...
View Full Document

This note was uploaded on 12/30/2011 for the course PHYSICS 604 taught by Professor Drake during the Fall '11 term at Maryland.

Ask a homework question - tutors are online