Whenever a vibrating structure has two (or more) modes at exactly the same frequency, these are called *degenerate* modes. For the ideal circular drum most of the modes appear in degenerate pairs. The exception is the modes with only nodal circles, but no nodal diameters. These modes, which include the lowest-frequency mode of the drum, are single.

For the drum mode discussed in the main text, with a single nodal diameter, the displacement around a circle of constant radius on the drum varies like $\cos \theta$, where $\theta$ is the usual polar coordinate angle. The second incarnation of the mode varies like $\sin \theta$. (Recall that both are vibrating at the same frequency, so we don’t need to write down the time dependence because it is the same for everything.) Now suppose we have the mixture $a\cos\theta+b\sin\theta$. Choose $r$ and $\phi$ so that $a=r\cos\phi$ and $b=r\sin\phi$. Then the combination is $r \left(\cos\phi \cos\theta+\sin\phi \sin\theta\right)=r\cos\left(\theta-\phi \right)$, the same pattern rotated by angle $\phi$.

As an aside, if you measure the natural frequencies of a real drum (or any other structure for which simple theory predicts degenerate modes) you will almost invariably find pairs of frequencies that are very close, but not exactly equal. The degeneracies are *split*, by the action of some physical mechanism that does not precisely satisfy the assumption of perfect circular symmetry. Some examples: the distribution of tension in a real drum head will never be precisely uniform, the membrane itself will have properties that vary slightly from point to point, and the shape of the rim of the drum will not be a perfect circle. Nothing has absolutely perfect symmetry, every aspect has some finite precision associated with it.