Damplitude

Damplitude is a term used in the study of damped dynamical systems and signal processing to refer to the effective or envelope amplitude of a waveform as it evolves under damping. The concept emphasizes how energy dissipation reduces amplitude over time, distinguishing the instantaneous oscillation from its decaying envelope.

For a linear damped harmonic oscillator governed by x'' + 2ζω0 x' + ω0^2 x = 0, the solution is x(t) = A0 e^{-ζω0 t} cos(ωd t + φ), with ωd = ω0 sqrt(1-ζ^2). The envelope A(t) = A0 e^{-ζω0 t} is often described as the damplitude, representing the time-varying amplitude of the oscillation under damping.

Applications include mechanical engineering, electrical circuits with resistance and inductance and capacitance (RLC circuits), acoustics, and seismology, where understanding the damplitude helps in predicting response, designing damping strategies, and assessing structural integrity. Damplitude can be estimated by fitting decay envelopes to measured signals or by extracting decay rates from the logarithmic decrement.

Terminology and usage vary; some authors treat damplitude as synonymous with envelope or decay amplitude, while others reserve it for a specific time-varying envelope. The term remains uncommon outside niche contexts, and standard references typically discuss damping, envelope, and decay without using the word damplitude.