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This is an important question - however, it has been asked and answered (with different content compared to above) here. Be that as it may, I think you should certainly compare what's been provided here to the link (and other resources too!). I'll further supplement everything you can find with some insight that's helped me understand what happens with the ...

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NMO stretch and associated mathematical relation i.e. $$\frac{\Delta f}{f} = \frac{\Delta T_{NMO}}{T_0}$$ are precisely explained in the book- Seismic Data Analysis by Yilmaz. Mathematical part can be found in Appendix C2 The book is freely available.!

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