A **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.

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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.

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

In some cases the equation can be simplified to:

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.

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.

## 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.