Because the wave equation is linear, the sum of any number of solutions is also a solution. It is therefore possible to look for a basis for the solution set, and express any solution as a (possibly infinite) linear combination of elements of the basis. A convenient choice for the form of the basis elements is as a product of functions, each of which depends on just one of the coordinates. Choosing a __cylindrical coordinate__ system for symmetry, we find solutions, called modes, of the form:

${u}_{n\,k}\left(rcomma;\mathrm{\theta}comma;t\right)equals;{A}_{ncomma;k}\cdot {J}_{n}\left({\mathrm{\lambda}}_{ncomma;k}\frac{r}{a}\right)\cdot \mathrm{sin}\left(n\cdot \left(\mathrm{theta;}-{\mathrm{vartheta;}}_{ncomma;k}\right)\right)\cdot \mathrm{sin}\left({\mathrm{lambda;}}_{ncomma;k}\cdot c\cdot \left(t-{\mathrm{tau;}}_{ncomma;k}\right)\right)$,

where ${J}_{n}$ is the $n$ th order __Bessel function__ (of first kind), ${\mathrm{\λ}}_{n\,k}$ is its $k$ th root, $a$ is the radius of the drum head, and ${A}_{n\,k}comma;{\mathrm{vartheta;}}_{ncomma;k}comma;{\mathrm{tau;}}_{ncomma;k}$ are arbitrary constants depending on n and k. In fact the general solution can be expressed as an infinite sum of such functions:

$uequals;\sum _{nequals;0}^{\infty}\sum _{kequals;1}^{\infty}{u}_{ncomma;k}\left(rcomma;\mathrm{\theta}comma;t\right)$.