Highwayman above is right. Even the most solid of solid body guitars will flex under the varying loads of the vibrating strings.
Pythagoras discovered that a string fixed solidly at both ends and plucked carefully in the middle of its length would produce tones related to a wavelength equal to the full length, half the length, a third of the length and a quarter of it.
A single string, playing one note thus produces its fundamental pitch, the octave above it, the fifth above that and a further octave above that. Clearly all these frequencies are in tune with each other and sound smooth and harmonious.
If the string is struck with a plectrum close to one of the ends, other pitches, much higher and less simply related to the fundamental are generated, though they die away very rapidly and the string settles into the simpler mode of vibration above. These high overtones are called transients and they give that metallic ping of clang for the first few milliseconds of a guitar note.
If the ends of the string are fixed that's all you can get.
But in practice they are not. The neck of the guitar flexes. The bridge and top of an acoustic guitar breathes in and out- much more so with a banjo! This means two things.
Firstly, that as the neck and body flex, the tension of the strings alters, and makes tiny alterations to the tuning which vary over time.
(That's the bo-inggg sound of a banjo as every note goes flat then sharp before it settles to pitch because of the flex of the skin)
That's the 'pitch envelope'.
Secondly, if the ends of the string are moving, then there are a whole series of extra, less harmonic overtones that the string can produce.
It's the way in which these harmonic and non harmonic overtones arrive and decay that gives an instrument its character.
This link illustrates the way in which the shape of solid body guitars effects the way in which they vibrate.
PDF file on guitars
On amplifiers, a basic characteristic of valve amps is their tendency to create overtones when driven to distortion which are even number multiples of the fundamental signal frequency- so they sound smooth, and actually richer than the original signal. A basic solid state amp tends to generate overtones which are odd number multiples- which after the third, become more and more discordant, and thus sound harsher.