There is no such thing as the instantaneous period of a wave. The Heisenberg uncertainty principle guarantees this. Oh you can find a set of estimated frequencies for a given window over the signal, but that is frequencies for the whole window, not for that instant. You need to figure out what size of time-frequency window satisfies your needs and then calculate something the Fourier transform for that window over the entire signal (and maybe search the resulting frequencies for the peak).
Two things to remember, the larger the window, the higher the frequency resolution (how many frequencies you can see), but the smaller the time resolution (when something happened--and this is a bit hard to explain as it's been a few years since I've had to learn and deal with it extensively so I may not be saying it right).
Another factor is that amplitude (multiplication in the time domain) is convolution in the frequency domain. To remove some artifacts of that you may have to do some filtering (that is, convolution in the time domain or multiplication in the frequency domain.) It might even make sense to filter with one widow size then do analysis with another window size. There's a lot of finicky business when it comes down to signal sampling.
Ok, this "Heisenberg uncertainty principle" with regard to signal sampling is not some random thing I pulled out of my ear one day. It is, AFAIK, a well established concept within Wavelet literature. (Wavelets involve a whole family of mathematical non-Fourier techniques to analyze a signal) "The World According to Wavelets: The Story of a Mathematical Technique in the Making" by Barbra Burke Hubbard (ISBN 1-56881-072-5) devotes at least one (if not two) chapters to the subject and has a proof in the Appendices.
According to her, (I'm going to have to paraphrase as my attempts to translate to Mathjax/TeX/LaTeX are failing, please also forgive as it's been years since I've looked at this anyway) For every $f(t)$ ($t$ a real number) such that the integral (from negative infinity to positive infinity) of $|f(t)|^2$ with respect to $t$ equals $1$, the product of the variance of $t$ and the variance of $\tau$ (the variable of $\hat f$) is at least $1/(16*\pi^2)$, and then gives a large, explicit inequality to state what exactly she means with further non mathematical explanation throughout the chapter.
I will make no attempt to explain further, partially for fear I'd likely mess up. But if you want, you can read that and/or its references and see that, yes, the Heisenberg uncertainty principle does indeed apply here.