I just read that coffee table book full of mathematicians' biographies: "Mathematicians: An Outer View of the Inner World". The most striking part of it was the variety of backgrounds. While the majority of featured mathematicians were interested in math early, or had parents with analytic/technical backgrounds, a significant fraction of them only became interested in mathematics while in college, and some later than that.

BTW, you said you came to mathematics from music and acoustical applications, and that you're starting to feel overwhelmed by the math of those waveforms. There are actually many other connections between music and mathematics, and some of these other areas may be fruitful to you. For example, look at Combinatorics for the study of melody or harmony. But along the path you've described, of Fourier and DST and Bessel functions, take a look at this free online edition of "Music: a Mathematical Offering", by Dave Benson and Cambridge University Press: http://www.maths.abdn.ac.uk/~bensondj/html/maths-music.html . It looks terrific, and I'm going to download it myself now.

Also, it's very common to not understand some new mathematics the first time. Professional mathematicians will often plow through a difficult paper on the first read, skipping over equations to get the general idea of where it's going. Even Feynman recommended this technique. Don't let yourself get stuck trying to understand an equation, but keep going. Then you can come back and read it again, either immediately or later, with more understanding.