Piano and PDEs

The story goes back to a conversation between me and analytique de maximus, Lothar. "Which instrument do you like the most Purushottam?", He asked. "Piano", I answered quickly. Though I played only guitar, piano was my favorite always. " And why is that?", he insisted. Why one likes anything is a different issue altogether and I am not going to dwell into that here. Important to this context was the answer I gave. "I like the pure sound, the notes, without mircrotones, not like violin, piano has a definite sound". I was able to satisfy his ultra-inquisitive mind with this. I was relieved for that.

Some years later, that is today, I was sitting in in my PDE class. And the prof. was teaching something about wave equation. It's not that I hate it or something. Being the geek I am, I was enjoying the mathematical aesthetics in the problem. Now, I am going to write something which you might not understand or like, but it's a graduate course and never mind :)

In the wave equation, the initial conditions on time are the initial velocity and the initial displacement on the string. The string is assumed to be bolted at the two ends so that given a displacement, it vibrates periodically. The solution generally is a fourier series, each term representing a definite mode of vibration of the string. If we solve a problem where the initial velocity is zero while the initial displacement is finite and linear, we find that the fourier coefficients die down as 1/n^2. While in the initial velocity (no displacement) case, the fourier coefficients die down as 1/n^4. Even this part was enough interesting for me. But then, the professor started talking about piano.

In a piano, sounds are produced by pressing the keys which in turn results in a small hammer striking the string. Hence an initial velocity is given to the string problem. While playing a guitar, you pluck the string and give it an initial displacement. But since in an initial velocity problem the coefficient which are nothing but the amplitude, decrease rapidly, the higher overtones of the frequency are not heard, while the higher overtones are powerful in guitar and what we hear is a mixture of frequencies. That is why a piano sounds pure and single frequency and the guitar doesn't.

Math is beautiful in her own right. But then you can see her in nature, and in things you love and then it feels like adding more stars in her crown.