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Forgive me if this is more a physics question than a music question! My question is: When a note is struck on a piano and held down (or a tuning fork is struck, or any instrument where there is just a one-time impact that then rings until dampened), what is the rate of decay in volume (aka velocity)? I strongly suspect that it is not linear; that the volume after two seconds is not half the volume after one second.

By extension, is it possible to calculate, or even just estimate, how long a note continues to reverberate on a piano if struck at maximum impact, versus lightly pressed, when held down in both cases?

Background, not essential:

I have a beautiful AvantGrand N2, which includes a standard MIDI output via a UX16 capble that allows one to record on a computer when and how forcefully a note was struck. While MIDI will never capture the complete soul of a piano, I want to use this just to see how accurately I'm expressing certain notes in a chord versus others.

The way MIDI works, it just sends a signal when a note was struck and how forcefully (known as "velocity"), and a second signal when that note was released, even if one removes one's finger 30 seconds later, long after the note has decayed. (It can read the pedals too, but that's a future endeavor.) So my question is: When a note is struck, and I want to visualize the velocity over time, what sort of function should I apply to the initial velocity?

Thank you!

You needn't calculate or estimate or do fancy math. Others have measured piano string decay rates, and they're messy:

Measurements have been made on a high‐quality spinet piano to determine the initial amplitude and decay characteristics of the principal harmonic components of notes covering the entire scale. Decay rates of individual components varied from 1 to 100 db/sec as a function principally of frequency and string position in the scale rather than of harmonic number. An exception was noted in the third to fifth octaves, where the fundamental of a given string decayed at a considerably higher rate than the harmonics. This pattern appears to have a strong influence in determining typical piano tone in this region. Frequency spectra of the initial amplitudes of individual notes over the entire scale show, on the average, a sharp cutoff below 100 cps, a fairly flat region between 100 and 1000 cps, and a roll‐off of at least 12 db per octave above 1000 cps, with negligible energy above 8000 cps.

Here's another technical paper whose abstract confirms your hunch that the decay is nonlinear.