User:Igor.Favorskiy/Sandbox

It follows from the periodicity condition that

\Theta = \alpha \cos n\theta + \beta \sin n\theta, \,

and that n must be an integer. The radial component R has the form

R(r) = \gamma J_n(\rho), \,

where the Bessel function Jn(ρ) satisfies Bessel's equation

ρ2Jn'' + 2ρJn' + (ρ2 − n2)Jn = 0,

and ρ=kr. The radial function Jn has infinitely many roots for each value of n, denoted by ρm,n. The boundary condition that A vanishes where r=a will be satisfied if the corresponding frequencies are given by

k_{m,n} = \frac{1}{a} \rho_{m,n}. \,

The general solution A then takes the form of a doubly infinite sum of terms involving products of

\sin(n\theta) \, \hbox{or} \, \cos(n\theta), \, \hbox{and} \, J_n(k_{m,n}r).

These solutions are the modes of vibration of a circular drumhead.