How is wavelength defined when it's changing continuously?

Question

Take an observer, who is receiving an electromagnetic wave signal, which is constantly changing. It can be for example from a source of light falling into a black hole, so the observed wavelength is increasing all the time.

The only definition I know is that the wavelength is the distance over which the wave's shape repeats. But to be precise, continuously changing wave does not repeat itself.

So how do we define the wavelength for "an almost sinusoidal" wave? My guess is just to take a small (even infinitesimal?) fraction of the changing wave and match it with a sine function and take the wavelength of that function. Another idea is to average over the wave for example by taking the distance between maxima. But the problem with this method is that then the wavelength won't be a continuous function. Which is the correct way of doing this, especially for an electromagnetic wave?

Thank you!

Answer 1

Those are good ways. Though you can't make the time interval infinitesimal because you then can match any sine wave.

If the function is slowly varying, the frequency doesn't change fast. You get a good estimate of the frequency at a given time from the wave in the near future and near past. So average over an interval.

To get an estimate at short time later, slide the interval forward. There will be a lot of overlap, so you won't see much change. But that is what you expect for a slowly varying frequency.

Another approach is to take the Fourier transform. You often get interesting information from how the Fourier components vary with time.