It follows from the periodicity condition that Θ Θ = α α cos ⁡ ⁡ n θ θ + β β sin ⁡ ⁡ n θ θ , and that n must be an integer. The radial component R has the form R ( r ) = γ γ J n ( ρ ρ ) , where the Bessel function J n (ρ) satisfies Bessel's equation ρ ρ 2 J n ″ + ρ ρ J n ′ + ( ρ ρ 2 − − n 2 ) J n = 0 , and ρ = kr. The radial function J n has infinitely many roots for each value of n, denoted by ρ m, n. The boundary condition that A vanishes where r = a will be satisfied if the corresponding wavenumbers are given by k m , n = 1 a ρ ρ m , n .