Worked example
Damped Oscillation: e^(-x/2) · cos(3x)
A plucked guitar string, a bouncing car suspension, and an RLC circuit all share the same mathematical shape: an oscillation whose amplitude decays over time. Multiplying cos(3x) by the decaying exponential exp(-x/2) produces exactly that — each swing is a fixed fraction of the previous one.
The two envelope curves, ±exp(-x/2), are the "rails" the oscillation rides between. The wave touches the upper envelope exactly at the crests of the cosine and the lower envelope at its troughs, and the envelopes themselves never oscillate. This separation of "how fast it wiggles" (the cosine) from "how fast it dies out" (the exponential) is why engineers analyze the two factors separately.
Plotted expressions
- y = exp(-x/2) * cos(3*x)
- y = exp(-x/2)
- y = -exp(-x/2)
The link preloads these exact expressions into the calculator — no typing needed.
What to notice
- The oscillation never crosses outside its exponential envelopes.
- Increasing the 3 in cos(3x) packs more wiggles into the same decay; increasing the 1/2 in the exponent kills the motion faster.
- Zoom out along the x-axis to watch the wave settle toward zero — the mathematical signature of damping.