Say C4 is played. There are the harmonics, and then other frequencies of lesser amplitude. Based on how a string vibrates, can we describe a pattern to the phases of these frequencies?
The first mistake here is that most notes on the piano do not consist of a single string. It would be impossible to tune the two or three strings to identical frequencies, and in fact they are intentionally tunes slightly differently, because if the tuning accuracy is too "tight" the instrument sounds "dead".
The second mistake is that because of the flexibility of the bridges on the sound board, even a single string has two different modes of vibration, one perpendicular to the soundboard and the other parallel to it, and each of these vibration modes has its own full set of harmonics, with slightly different frequencies from each other.
The third mistake is that because the piano wire is not perfectly flexible, the harmonics are not exact multiples of the fundamental frequency. The high harmonics are progressively sharper than the fundamental.
A further issue is that the rate of sound radiation from the vibration modes perpendicular to the soundboard is higher than from those parallel to it, and the sound from the "perpendicular modes" decays faster.
The combination of all those factors makes the development of the sound very complex. An over-simplified version of what happens is that the initial sound is dominated by the "perpendicular" vibration modes. They decay faster than the "parallel" modes and are vibrating at a slightly different frequency, so the two families of vibrations "beat" against each other, and at some point (typically about 0.5 to 1 sec after the start of the note) there is a minimum in the sound amplitude cause by the first beat. After that point, the "perpendicular" vibrations have decayed to a low amplitude and the sound level then rises (the so-called "blooming" of the piano tone, named by analogy with a flower bud opening) and finally decays slowly, over several seconds, or even over several minutes for low pitched notes.
To summarize, what you learned about vibrating strings in a first "sound and vibration" physics lab doesn't even begin to explain what is really going on in real musical instruments.
If you want to understand and experiment with these effects, an acoustic guitar is a better instrument to use than a piano, because (if you learn about classical guitar technique) you can control the relative amplitudes of the "perpendicular" and "parallel" vibration modes depending on how you pluck the string, and hear what effect that has on the sound. For example, this explains why "strumming" an acoustic guitar produces a very different sound from playing it with classical guitar technique, which plucks the strings away from the sound board, not from side to side.