A characterization of damped and undamped harmonic oscillations by a superposition property

Summary.

For $f:\mathbb{R} \to \mathbb{R}$ and $\beta ,B \in \mathbb{R},$ we call $f_{[\beta ]} :\mathbb{R} \to \mathbb{R}$ with $f_{[\beta ]} (x): = f(x + \beta )$ the β–translate of f and Bf the B–scaled of f. A function $f:\mathbb{R} \to \mathbb{R}$ has property $\mathcal{S}$ superposition property) iff $f + Bf_{[\beta ]}$ is again a scaled translate of f for all $\beta ,B \in \mathbb{R}.$ Our main result is the determination of all continuous functions f with property $\mathcal{S}$ and with at most countably many zeroes. These (and the null function which trivially has property $\mathcal{S})$ are given by $f(x) = e^{cx + d} {\text{cos}}(ax + b),\;a,b,c,d \in \mathbb{R}.$ Our characterization has some implication for the possibility of unisono playing by conventional music instruments. Two violins (undamped waves) or two drums (damped case) playing the same tone (C’ for example) will give the impression of that tone, but with eventually altered loudness. No other waves allow unisono playing!