Calculus How To

Damped Sine Wave: Definition, Example, Formula

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damped sin waveA damped sine wave is a smooth, periodic oscillation with an amplitude that approaches zero as time goes to infinity. In other words, the wave gets flatter as the x-values get larger.


Damped sine waves are often used to model engineering situations where a harmonic oscillator is losing energy with each oscillation. For example: a bouncing tennis ball or a swinging clock pendulum.


The term damped sine wave refers to both damped sine and damped cosine waves, or a function that includes a combination of sine and cosine waves. A cosine curve (blue in the image below) has exactly the same shape as a sine curve (red), only shifted half a period. Where a sine wave crosses the y-axis at y = 0, the cosine wave crosses it at y = 1.


Notice though, that the sine and cosine waves in the above image are not damped: they are a uniform height as they move from left to right.

Formula for a Damped Sine Wave

A sine wave may be damped in any of an infinite number of ways, but the most common form is exponential damping. If your sine curve is exponentially damped, drawing a line from peak to peak will result in an exponential decay curve. Draw a curve from peak to peak, and you’ll see the exponential function. Notice that the curve crosses the y-axis at x = 1 (since A = 1 in this function) and that the amplitude goes to zero as x goes to infinity.

exponential damping

Exponential damping y(t) = e-t · cos (2 π t), with the exponential decay curve shown in red.



We can write a general equation for an exponentially damped sinusoid as

Equation of a Damped Sine Wave

Simplified Equation of a Damped Sine Wave

Where:

  • A is the initial amplitude (the highest peak),
  • λ is the decay constant,
  • φ is the phase angle (at t = 0)
  • ω is the angular frequency.

References

Guido, Mueller. Damped Simple Harmonic Motion. Retrieved from http://www.phys.ufl.edu/~mueller/PHY2048/2048_Chapter16_F08_wHitt_Part1.pdf on April 18, 2019.


Townsend, Lee. Analyzing Damped Oscillations. Retrieved from http://uhaweb.hartford.edu/ltownsend/Analyzing_Graphs_of_Damped_Oscillation_Data.pdf on April 18, 2019.

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