Supremum of two incommensurate harmonic modes (${\sup }_t|x(t)|=|A|+|B|$)

For $x(t)=A\cos {\omega }_{1}t+B\cos {\omega }_{2}t$ with irrational frequency ratio and nonzero amplitudes, $|x|\le |A|+|B|$. The phase trajectory is dense on the two-torus, so this bound is a supremum. The required exact phase alignment may never occur at finite time. Thus an exam's maximum-amplitude estimate can be a limiting bound rather than an attained displacement; finite observation intervals have smaller maxima.