Mathematical Methods for Physics and Engineering - Matematica.NET

21.3 SEPARATION OF VARIABLES IN POLAR COORDINATESIn the above example, on the equator of the sphere (i.e. at r = a and θ = π/2)the potential is given byv(a, π/2,φ)=v 0 /2,i.e. mid-way between the potentials of the top and bottom hemispheres. This isso because a Legendre polynomial expansion of a function behaves in the sameway as a Fourier series expansion, in that it converges to the average of the twovalues at any discontinuities present in the original function.If the potential on the surface of the sphere had been given as a function of θand φ, then we would have had to consider a double series summed over l andm (for −l ≤ m ≤ l), since, in general, the solution would not have been axiallysymmetric.Finally, we note in general that, when obtaining solutions of Laplace’s equationin spherical polar coordinates, one finds that, for solutions that are finite on thepolar axis, the angular part of the solution is given byΘ(θ)Φ(φ) =P m l (cos θ)(C cos mφ + D sin mφ),where l and m are integers with −l ≤ m ≤ l. This general form is sufficientlycommon that particular functions of θ and φ called spherical harmonics aredefined and tabulated (see section 18.3).21.3.2 Other equations in polar coordinatesThe development of the solutions of ∇ 2 u = 0 carried out in the previous subsectioncan be employed to solve other equations in which the ∇ 2 operator appears. Sincewe have discussed the general method in some depth already, only an outline ofthe solutions will be given here.Let us first consider the wave equation∇ 2 u = 1 ∂ 2 uc 2 ∂t 2 , (21.52)and look for a separated solution of the form u = F(r)T (t), so that initially weare separating only the spatial and time dependences. Substituting this form into(21.52) and taking the separation constant as k 2 we obtain∇ 2 F + k 2 d 2 TF =0,dt 2 + k2 c 2 T =0. (21.53)The second equation has the simple solutionT (t) =A exp(iωt)+B exp(−iωt), (21.54)where ω = kc; this may also be expressed in terms of sines and cosines, of course.The first equation in (21.53) is referred to as Helmholtz’s equation; we discuss itbelow.737