21.3 SEPARATION OF VARIABLES IN POLAR COORDINATES
In the above example, on the equator of the sphere (i.e. atr=aandθ=π/2)the potential is given by
v(a, π/ 2 ,φ)=v 0 / 2 ,i.e. mid-way between the potentials of the top and bottom hemispheres. This is
so because a Legendre polynomial expansion of a function behaves in the same
way as a Fourier series expansion, in that it converges to the average of the two
values at any discontinuities present in the original function.
If the potential on the surface of the sphere had been given as a function ofθandφ, then we would have had to consider a double series summed overand
m(for−≤m≤), since, in general, the solution would not have been axially
symmetric.
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 the
polar axis, the angular part of the solution is given by
Θ(θ)Φ(φ)=Pm(cosθ)(Ccosmφ+Dsinmφ),whereandmare integers with−≤m≤. This general form is sufficiently
common that particular functions ofθandφcalledspherical harmonicsare
defined 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 subsection
can be employed to solve other equations in which the∇^2 operator appears. Since
we have discussed the general method in some depth already, only an outline of
the solutions will be given here.
Let us first consider the wave equation∇^2 u=1
c^2∂^2 u
∂t^2, (21.52)and look for a separated solution of the formu=F(r)T(t), so that initially we
are separating only the spatial and time dependences. Substituting this form into
(21.52) and taking the separation constant ask^2 we obtain
∇^2 F+k^2 F=0,d^2 T
dt^2+k^2 c^2 T=0. (21.53)The second equation has the simple solution
T(t)=Aexp(iωt)+Bexp(−iωt), (21.54)whereω=kc; this may also be expressed in terms of sines and cosines, of course.
The first equation in (21.53) is referred to asHelmholtz’s equation; we discuss it
below.