stain for lower notes.
You can test this phenomenon on any string instrument. Just play a bass note and then stop it using your hands/bow, and repeat the same for a high note. You will hear that the bass note resonating longer on the board.
Also note that pianos have bigger hammers and dampers for the lower notes for the same reason. You need to generate more energy and then consume it back.
Another example is, on a piano you can hear low notes sustaining longer than high notes when you lift your finger from the key.
Let’s consider a wire strung between two posts. After you hit it with a hammer two waves pulses propagate up and down the string, one in each direction. They hit the end posts, bounce in the other direction, and so on; two pulses racing back and forth along the length of the string.
The fundamental frequency, ie the pitch of the string, is the inverse of of the round trip time.
First, consider the case where the internal friction of the string is very low. Then the losses occur when the pluses hit the end posts. For higher frequency vibrations, this happens more rapidly. Assuming that the same fraction of the wave energy is lost from the string (and transferred to the sound board) each time, you’d expect the sound to dissipate more rapidly for higher frequencies.
Similar considerations apply when thinking about internal losses — a given short segment of the string flexes and then flattens as the pulse moves through it. And again, if each flex dissipates some energy, then more energy is lost per unit time for higher frequency strings. But this gets confounded by the fact that there are more such short segments in a longer string. A naive application of this logic leads one to conclude that this might result in a frequency-independent term of the losses. (From there you can think about whether the internal losses depend on the rate of change of the shape...)
So, to first order, keeping everything other than the length of the string fixed, you’d expect that higher pitched strings would lose energy more rapidly than lower pitched ones.
Of course, in a real piano not everything else is held fixed, and then when you start to consider the fact that the perception of loudness depends on frequency, then things getting more complicated fast.
I'm guessing, but because it vibrates at a higher frequency (being a higher pitch) the energy put into the string by striking it with the key, simply is used quicker. All keys, or strings get the same amount of energy when hit with a key (artificially, I get into it next paragraph), thus a quicker vibration vibrates and moves more in a given period of time, releasing energy quicker, thus running out.
The string size differs, the smaller strings being higher pitch, allowing and creating it to be moved or vibrate more (quickly). Allowing the large stings to store more energy and release it slower, in a pitch with a higher frequency (or less vibration, or movement back and forth in a given time). Small things take less energy to move, so the strings move more, and because of the movement being quicker, or a faster pace all energy dissipates before that of the large/wide/thick low notes. You will notice either; thicker strings should be harder to move or take more energy, or that Pianos have 'hammers' that vary in size, the lower the note the more large the 'hammer'. Those two things actually balance each other out. Large hammers are used to strike the thick strings because to get the volume/amplification of the high strings you need more energy. This in a way makes up for the fact that the strings are larger, leaving you with the simple fact that less mass takes less energy, thus the energy is used much faster, in the form of vibrations (at a quicker rate).
The size doesn't actually matter, what matters is how fast the sting is vibrating, that is what creates the sound waves. If you concentrated more energy into a single point and hit the thicker stings, they would vibrate more violently than less energy. The more vibration the higher the frequency, witch equals higher pitch.
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