Mathematical Methods for Physics and Engineering - Matematica.NET

21.3 SEPARATION OF VARIABLES IN POLAR COORDINATESAs in the two-dimensional case, single-valuedness of u requires that m is aninteger. However, in the particular case m = 0 the solution isΦ(φ) =Cφ + D.This form is appropriate to a solution with axial symmetry (C = 0) or one that ismultivalued, but manageably so, such as the magnetic scalar potential associatedwith a current I (in which case C = I/(2π) andD is arbitrary).Finally, the ρ-equation (21.34) may be transformed into Bessel’s equation oforder m by writing µ = kρ. This has the solutionP (ρ) =AJ m (kρ)+BY m (kρ).The properties of these functions were investigated in chapter 16 and will notbe pursued here. We merely note that Y m (kρ) is singular at ρ = 0, and so, whenseeking solutions to Laplace’s equation in cylindrical coordinates within someregion containing the ρ = 0 axis, we require B =0.The complete separated-variable solution in cylindrical polars of Laplace’sequation ∇ 2 u = 0 is thus given byu(ρ, φ, z) =[AJ m (kρ)+BY m (kρ)][C cos mφ + D sin mφ][E exp(−kz)+F exp kz].(21.35)Of course we may use the principle of superposition to build up more generalsolutions by adding together solutions of the form (21.35) for all allowed valuesof the separation constants k and m.◮A semi-infinite solid cylinder of radius a has its curved surface held at 0 ◦ C and its baseheld at a temperature T 0 . Find the steady-state temperature distribution in the cylinder.The physical situation is shown in figure 21.5. The steady-state temperature distributionu(ρ, φ, z) must satisfy Laplace’s equation subject to the imposed boundary conditions. Letus take the cylinder to have its base in the z = 0 plane and to extend along the positivez-axis. From (21.35), in order that u is finite everywhere in the cylinder we immediatelyrequire B =0andF = 0. Furthermore, since the boundary conditions, and hence thetemperature distribution, are axially symmetric, we require m =0,andsothegeneralsolution must be a superposition of solutions of the form J 0 (kρ)exp(−kz) for all allowedvalues of the separation constant k.The boundary condition u(a, φ, z) = 0 restricts the allowed values of k, sincewemusthave J 0 (ka) = 0. The zeros of Bessel functions are given in most books of mathematicaltables, and we find that, to two decimal places,J 0 (x) =0 forx =2.40, 5.52, 8.65, ....Writing the allowed values of k as k n for n =1, 2, 3,... (so, for example, k 1 =2.40/a), therequired solution takes the formu(ρ, φ, z) =∞∑A n J 0 (k n ρ)exp(−k n z).n=1729