Sound waves are mechanical waves, and in a piano the majority of the mechanical waves produced are transverse, other than the occasional longitudinal wave produced by tension in the string.
Longitudinal waves in a piano are caused from the local increase in tension caused by the wire’s elongation. The longitudinal waves caused by the wire’s tension has frequencies determined by the length of the wire and the speed of its sound.
Higher pitches on a piano will have a smaller wavelength and smaller instruments will also have a smaller wavelength. The reason that pianos like a grand piano will have such a larger sound than something like an upright piano is because of the size of their strings inside of the piano.
One unique fact about sound is that frequency is nearly independent of speed which becomes important in many things such as recitals or concerts, where you wouldn’t want people in the back of the audience to hear a different sound from the piano as those in the front of the audience.
Another unique fact about pianos is that when a higher pitch is played, as opposed to a lower one, the higher pitch will dissipate much faster than the lower pitches, even if they are played the exact same way. This is because a higher frequency requires more energy and therefore it will dissipate more rapidly when the same amount of energy is expended on both pitches.
A piano naturally has dampers which reduce the amount of time it takes for a note to dissipate. This is generally useful as it allows for the notes to not get muddled with each other when a pianist is trying to play a piece. This does not, however, allow for a pianist to play legato with ease, which is what the damper pedal is for. The damper pedal raises all of the dampers off the strings, allowing the strings to continue to vibrate after the key has been released and until the pianist either releases the pedal or the notes naturally dissipate.
The amplitude of a string doesn’t affect the tone of the piano, but hitting a piano key harder will increase the amplitude of the oscillation, creating a louder tone, and also shorten the contact time which increases the proportion of upper harmonics which creates a brighter timbre. Timbre can be defined as “the character or quality of a musical sound or voice as distinct from its pitch and intensity.”
The medium of a piano greatly changes the speed of sound. The more rigid the medium, the faster the speed of sound. A slower speed of sound would be found in a greater density medium. Since air is a compressible or less rigid medium, the speed of sound in air is fairly low.