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A damped sine wave is a sinusoidal function whose amplitude approaches zero as time increases.[1]

Damped sine waves are commonly seen in science and engineering, wherever a harmonic oscillator is losing energy faster than it is being supplied.

## Definition

Sine waves describe many oscillating phenomena. When the wave is damped, each successive peak decreases as time goes on.

A true sine wave starting at time = 0 begins at the origin (amplitude = 0). A cosine wave begins at its maximum value due to its phase difference from the sinewave. In practice a given waveform may be of intermediate phase, having both sine and cosine components. The term "damped sine wave" describes all such damped waveforms, whatever their initial phase value.

The most common form of damping, and that usually assumed, is exponential damping, in which the outer envelope of the successive peaks is an exponential decay curve.

## Equations

The general equation for an exponentially damped sinusoid may be represented as:

${\displaystyle y(t)=A\cdot e^{-\lambda t}\cdot (\cos(\omega t+\phi )+\sin(\omega t+\phi ))}$

where:

${\displaystyle y(t)}$ is the instantaneous amplitude at time t
${\displaystyle A}$ is the initial amplitude of the envelope.
${\displaystyle \lambda }$ is the decay constant, in the reciprocal of the time units of the X axis.
${\displaystyle \phi }$ is the phase angle at some arbitrary point.
${\displaystyle \omega }$ is the Angular frequency.

which can be simplified to

${\displaystyle y(t)=A\cdot e^{-\lambda t}\cdot (\cos(\omega t+\phi ))}$

Where:

${\displaystyle \phi }$ is the phase angle at t = 0.

Other important parameters include:

Period ${\displaystyle \tau }$, the time it takes for a single cycle, in units of time t. It is the reciprocal of frequency (see below), i.e. ${\displaystyle f^{-1}}$.
Frequency ${\displaystyle f}$. is the number of cycles per time unit, and is equal to ${\displaystyle \omega /(2\pi )}$. It is the reciprocal of period, i.e. ${\displaystyle \tau ^{-1}}$. and is expressed in inverse time units ${\displaystyle t^{-1}}$.
Half-life is the time it takes for the exponential amplitude envelope to decrease by a factor of 2. It is equal to ${\displaystyle \ln(2)/\lambda }$ which is approximately ${\displaystyle 0.693/\lambda }$.
Wavelength of a travelling wave is the distance between adjacent peaks and varies according to the speed of the wave's travel.