 it helps.
Using the simple ideal model for a vibrating string, vibrating plates, etc, the linear damping force is proportional to the velocity of a mass element of the string. When the equations are expressed in the frequency domain this is proportional to the frequency of the wave propagating on the string. From this it is reasonable to conclude that the higher harmonics in a single wave packet will die faster than the fundamental. This is commonly observed in isolated systems. After some time the fundamental is the only noticeable frequency left. The conclusion is also valid comparing fundamentals of different strings.
One has to understand where this relation comes from. There are at least two sources of damping that I can think of for the near ideal string mounted on ideal rigid supports. The first is air resistance of the string moving through the air. This second is internal damping due to the vibrations of the material within the string. In other words energy of the transverse mode (the ideal model) is lost in longitudinal modes in the material and heats it up, increases entropy etc. These are both pretty small but not completely zero.
The first critique of this is that true strings also have stiffness in them and obey higher order differential equations than the ideal string. This does not change the above arguments but contributes to dissonant overtones that are not in the harmonic sequence, fn = n*f1.
Energy is eventually lost from the string to the body of the instrument and eventually to the air as acoustic sound. If this were not possible we would not be able to hear the instrument. This introduces a whole new set of equations, couplings, and physics to consider. The top of a guitar for example would obey a set of equations for stiff plates. They have their own natural harmonics that may or may not be aligned with those of the strings. Part of Luthier's art is optimizing this. So depending on the quality of the instrument and its condition some notes may get amplified more than others. This is a very common occurrence with acoustic stringed instruments and something we test for when purchasing an expensive instrument. You check for buzzing, dead spots, and RESONANCE. You want resonance to some degree as that adds to the sound but you don't want anomalous resonance which might show up as Bb4 is always 3dB louder than any other note (just a silly example but not impossible).
This brings me to an important point. That the rest of the instrument will vibrate in sympathetic resonance to the note being played and its harmonics.
The harmonic content in the string depends on the attack. Not all strings are the same. In fact one could argue that this is the most important part of sound and the hardest part of learning an instrument, learning proper attack and for guitar learning variety of attacks. Each attack produces a completely different "tone". This makes the guitar a great mimic and has a reputation for versatility. In contrast your piano hammers are fixed. You can control amplitude (strength of attack) and with pedals you can control sustain but you cannot control the initial attack profile of the string(s). Keep in mind that each "key" strikes several strings not just one.
Now typically (but not always) the fundamental is the strongest note, has the highest amplitude or volume among the spectrum of the string. And linear systems do NOT excite sub-harmonics. They don't even excite harmonics for that matter. The other strings will vibrate in sympathetic resonance to the string you play but only if the harmonics of the string are present in the one you played. And they will only vibrate at the frequency of that harmonic. A caveat to this is that coupling with other parts of the instrument could cause coupling between different modes due to a non-linearity, perhaps a joint in the wood etc. thus causing coupling between harmonics. But for the most part the linear model works well. For example if I play the high E string on my guitar, and assuming I attack it so that only the fundamental is present (close to possi