I am attempting to measure temperature using a signal through an optical fiber. If I send a waveform through an optical fiber, how would temperature affect it? I've looked here (Calculating Temperature of an Optical Fiber) on this site, but haven't managed to find any actual equations (if any) representing this phenomenon. The closest I've gotten is temperature causing a change in the index of refraction of water, which changes the speed of the wave in accordance with v = c/n as found here: (https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6640572/)
The temperature will change the refractive index and in consequence the speed of the wave. But experimentally, how do you measure it? You will need short pulse and a long fiber to have some resolution. And it will not be distributed sensing.
The normal way to measure temperature with fiber is to use the change of frequency shift of the brillouin scattering. The idea is that the Brillouin scattering frequency is dependant of temperature and strain. You can use this dependancy to have a temperature distributed sensor. The name of this technic is BOTDR (Brillouin Optical Time Domain Reflectometry). I advise you to first understand how an OTDR (https://www.rp-photonics.com/optical_time_domain_reflectometers.html) is working before trying to understand BOTDR.