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**Unformatted text preview: **of the Bessel boundary value problem (12.5455) are the squares of the roots of the Bessel function of order m . The corresponding eigenfunctions are w m,n ( r ) = J m ( m,n r ) , n = 1 , 2 , 3 , . . ., m = 0 , 1 , 2 , . . ., (12 . 112) defined for 0 r 1. Combining (12.112) with the formula (12.53) for the angular com- ponents, we conclude that the separable solutions (12.51) to the polar Helmholtz boundary value problem (12.49) are v ,n ( r ) = J ( ,n r ) , v m,n ( r, ) = J m ( m,n r ) cos m, hatwide v m,n ( r, ) = J m ( m,n r ) sin m, where m, n = 1 , 2 , 3 , . . . . (12 . 113) These solutions define the normal modes for the unit disk; Figure 12.5 plots the first few of them. The eigenvalues ,n are simple, and contribute radially symmetric eigenfunctions, whereas the eigenvalues m,n for m > 0 are double, and produce two linearly independent separable eigenfunctions, with trigonometric dependence on the angular variable. Recalling the original ansatz (12.48), we have at last produced the basic separable eigensolutions u ,n ( t, r ) = e 2 ,n t v ,n ( r ) = e 2 ,n t J ( ,n r ) , u m,n ( t, r, ) = e 2 m,n t v m,n ( r, ) = e 2 m,n t J m ( m,n r ) cos m, m, n = 1 , 2 , 3 , . . . . hatwide u m,n ( t, r, ) = e 2 m,n t hatwide v m,n ( r, ) = e 2 m,n t J m ( m,n r ) sin m, (12.114) to the homogeneous Dirichlet boundary value problem for the heat equation on the unit...

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