is a sine whose argument is squared. The further travels, the faster that argument changes, so the oscillations bunch ever closer together. A wave whose frequency rises as it goes is called a chirp, after the rising call of a bird1.
The domain is all real numbers and the range is the interval . Since the function is even, and the graph is symmetric about the -axis, bunching up in the same way to the left. It is not periodic.
The phase is , so its rate of change gives the angular frequency at each point.
Since it grows in direct proportion to , this wave is called a linear chirp.
We have when , that is at .
| Zero at | Gap from the previous zero | |
|---|---|---|
For large the gap between consecutive zeros shrinks roughly like : a sharp initial contraction followed by a gradual one.
The derivative is , which vanishes at . Close to the origin the approximation applies, so the curve is all but indistinguishable from a parabola.
Oscillation sets in only once has grown somewhat, the first crest appearing where , at .
This function has no elementary antiderivative; its integral defines the special function known as a Fresnel integral2. Taken out to infinity the value is as follows.
The integrand does not tend to , yet the integral is finite, because the oscillations quicken and the positive and negative parts cancel: the convergence is conditional, not absolute. Plotting the two Fresnel integrals as coordinates traces the Cornu spiral, used in computing the diffraction of light.
Radar systems transmit a long chirp instead of a short powerful pulse and compress it on reception, obtaining fine range resolution without high transmit power3.