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 $\phi$.