Any vibration decomposes into weighted sums of a system’s modal oscillations

Evidence

Modal frequency vibrations, such as those characterized in Figures 2.5 and 2.6, are only a small subset of possible vibrations in these systems. The special role of modes is that all other vibrations can be expressed as linear combinations of the modes, or, in other words, as movements where the modal oscillations at each point are each summed together with proper amplitude scaling. — Ville Pulkki, Matti Karjalainen, Communication Acoustics- An…, 170

Categorization

• DEFINE: Explains why modes are mathematically fundamental: they form a basis for all possible vibrations.
• XREF: The linear-combination basis idea parallels Fourier decomposition and eigen-basis expansions familiar from linear algebra.