Mathematical Methods for Physics and Engineering - Matematica.NET

PDES: SEPARATION OF VARIABLES AND OTHER METHODSaω =2.40c/aω =3.83c/aω =5.14c/aω =5.52c/aFigure 21.8 The modes of oscillation with the four lowest frequencies for acircular drumskin of radius a. The dashed lines indicate the nodes, where thedisplacement of the drumskin is always zero.Helmholtz’s equation in cylindrical polarsGeneralising the above method to three-dimensional cylindrical polars is straightforward,and following a similar procedure to that used for Laplace’s equationwe find the separated solution of Helmholtz’s equation takes the form[ (√ ) (√ )]F(ρ, φ, z) = AJ m k2 − α 2 ρ + BY m k2 − α 2 ρ× (C cos mφ + D sin mφ)[E exp(iαz)+F exp(−iαz)],where α and m are separation constants. We note that the angular part of thesolution is the same as for Laplace’s equation in cylindrical polars.Helmholtz’s equation in spherical polarsIn spherical polars, we find again that the angular parts of the solution Θ(θ)Φ(φ)are identical to those of Laplace’s equation in this coordinate system, i.e. they arethe spherical harmonics Yl m (θ, φ), and so we shall not discuss them further.The radial equation in this case is given byr 2 R ′′ +2rR ′ +[k 2 r 2 − l(l +1)]R =0, (21.56)which has an additional term k 2 r 2 R compared with the radial equation for theLaplace solution. The equation (21.56) looks very much like Bessel’s equation.In fact, by writing R(r) =r −1/2 S(r) and making the change of variable µ = kr,it can be reduced to Bessel’s equation of order l + 1 2, which has as its solutionsS(µ) = J l+1/2 (µ) and Y l+1/2 (µ) (see section 18.6). The separated solution to740