Unformatted text preview: (b) Construct a set of basis functions Φ n ( r ) such that Φ n ( a ) = 0 which are appropriate for solving for T ( r,t ). Write the basis functions as a linear combination of the two solutions of Bessel’s equation Y ν ( kr ) and J ν ( kr ). Write down expressions for the eigenvalues of your basis functions and normalize Φ n ( r ) so that Z a dr w Φ 2 n ( r ) = 1 . You do not have to prove that the eigenfunctions are orthogonal, but state why you are conﬁdent that they are. Sketch the lowest three eigenfunctions. (c) Write the space/time dependence of T as T ( r,t ) = ∞ X n =1 c n ( t )Φ n ( r ) and solve for c n ( t ). At late time ﬁnd an approximate expression for T ( r,t ). How does T decay at late time?...
View
Full Document
 Fall '11
 drake
 Physics, Linear Algebra, Space, Work, Manifold, Spacetime

Click to edit the document details