Damping Retention Multiplies Each Cycle - Oscillations and Coupled Resonance Step by Step

A finite damping window uses exact repeated multiplication. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Retention multiplies each cycle

Initial amplitude 8 metres is multiplied by 1/2 for each of 3 cycles.

A_n=8\cdot\left({1\over 2}\right)^{3}

After three cycles, one metre remains

The retained amplitude is 1 metre, so the lost amplitude is 7 metres.

A_n=1\ \mathrm{m},\quad \mathrm{lost}=7\ \mathrm{m}