Doubt about the behavior of a string of a piano

[Bruno Tolentino]

Two doubts:

First, when the hammer hits any string, the string begins to vibrate, until here, no problems, but, the vibration decays over time and this decay is linear, exponential or assume another form?

Second, when the hammer hits a string and the string begins to vibrate, it vibrates in fundamental frequency, in resonance frequency, and we heard this vibration. But we heard other vibrations too, that is mutiples of the fundamental frequency.

So, when the hammer hits the string of 440 Hz, for example, we heard vibrations in 440 Hz, 880 Hz, 1320 Hz, 1760 Hz, 2200 Hz and so on. But in terms of sound intensity, what is the relationship between the fundamental frequency and the multiple of the fundamental frequency? I ask this because the amplitude of the multiple of the fundamental frequency is not equal the amplitude of the fundamental frequency. Probably, shoud exist a decaying relationship between the amplitude of the fundamental and its multiples.

EDIT: See this ilustration (https://upload.wikimedia.org/wikipedia/commons/a/a4/Espectro_harmónico.jpg), but I don't know if this graphic is correct or not.

 

[sophiecentaur]

The decay in loudness will be basically exponential but it will not be a simple exponential decay because the various overtones may well decay at different rates.
As for the actual spectrum of the sound from a piano string, there have been many threads on PF about this. The one in this link is very long and rambling but could be worth reading through before continuing here on this thread. Basically, a plucked (or struck) string will start off in a certain shape (triangular, for instance) and the resulting vibrations will be a combination of all the natural modes of vibration of the string. (no other frequencies will be sustained) The proportions of each of the frequencies will depend upon where the string was actually struck. In a piano, you have no control over that but a guitar can be plucked anywhere and a whole range of tonal colours can be obtained. The amplitudes of the overtones will depend on the initial shape but the fundamental is normally the highest. The decay rates will depend very much on the design of the piano and strings; it won't be calculable.
I use the word Overtone, rather than Harmonic because the frequencies of the modes are not necessarily exact harmonics of the fundamental. This is what makes a real instrument sound real, as opposed to some of the more crude synthesised sounds that are based on harmonics. As it happens, the overtones and harmonics on strings are not too different. In brass instruments, they are wildly different - so much so that a good brass player has to 'pull' the higher notes significantly for them to sound right when playing in high registers.

 

[Integrand]

The departure of overtones from exact harmonics mentioned by sophiecentaur is discussed in the linked thread, but as far as I could see, not named there. For the record, it is called inharmonicity: https://en.wikipedia.org/wiki/Inharmonicity