Normal modes of a musical drumhead under non-uniform tension

The effect of non-uniform tension on the normal modes of musical drumheads is examined. Using the standard $(m,n)$ designation for the number of nodal diameters and circles, ideal circular membrane modes with $m>0$ are doubly degenerate. These degeneracies can be lifted by perturbations to the circular symmetry. Of practical interest to drummers are perturbations caused by non-uniform tension applied at the rim of the drum, leading in some cases to audible frequency splitting. The role of the (1,1) mode in practical drum tuning is analyzed using data obtained using time-averaged electronic speckle-pattern interferometry along with finite element analysis. The resulting model is then generalized to include all modes, using symmetry arguments along with a selection rule taken from group theory. The model compares favorably with both perturbation theory and finite element analysis, and is consistent with experimental observations.