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.

Watch the video for an overview, the formula, and a demonstration of what happens when you change the damping factor:

What is a Damped Sine Wave?
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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, which has the general formula N(t) = A e(kt). Draw a curve from peak to peak, and you’ll see the exponential function.

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
general equation damped sine wave

In some cases the equation can be simplified to:
Simplified Equation of a Damped Sine Wave


  • 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.

Notice that the curve in the above example crosses the y-axis at x = 1 (since A = 1 in this particular example) and that the amplitude (height) goes to zero as x goes to infinity.


Guido, Mueller. Damped Simple Harmonic Motion. Retrieved from on April 18, 2019.
Townsend, Lee. Analyzing Damped Oscillations. Retrieved from on April 18, 2019.

Stephanie Glen. "Damped Sine Wave: Definition, Example, Formula" From Calculus for the rest of us!

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