If strings are solid, why don't they have longitudinal waves? A string is a solid but it does not show longitudinal waves. Well it is known that a string cannot be compressed but only be given tension but a answer with a sound scientific reasoning will be accepted. 
 A: Longitudinal waves do propagate in string.  That is how "tin can phones" work.
A: As Ben51 said, longitudinal waves do travel in strings. These don't really care how thick the material is, whether it's p-wave sound propagation through a solid block of steel or along a steel string.
There are two main reason that longitudinal modes aren't very important in string instruments:


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*They are much faster, and thus higher frequency, than the transversal modes. The p-wave velocity in iron is $5120\:\mathrm{\tfrac{m}s}$ (unlike for the transversal modes, this doesn't change much with tension/thickness), so a $650\:\mathrm{mm}$ guitar string will have a longitudinal fundamental mode at $7880\:\mathrm{Hz}$. Ok, that's still audible to most humans, but it's far away from the range of fundamentals where you'd actually play notes† fundamental mode at $3940\:\mathrm{Hz}$, much higher than the 80-300 Hz range where the strings have their musical pitches. So at most this will contribute some extra percussion to the timbre, not actual tone that's heard as such.

*There's no effective way to excite them. The string may be physically able to vibrate longitudinally, but how do you get it to do so? Plucking, hitting, bowing all excite mostly the transversal modes. To excite a longitudinal mode, you'd need to somehow pull along the string in either direction and release the extra tension, but to pull you'd need to really grab it and then it's very hard to release quickly enough to not immediately damp the vibration again. So even a longitudinal mode that's well in the audible range will usually barely sound during playing.


I do think longitudinal modes happen and are audible in some cases; I suspect they contribute to the somewhat bell-like sound of very low tones on a grand piano. In actual tubular bells, they are probably quite loud, even. But they are hardly musically useful.

†The $7880\:\mathrm{Hz}$ figure was a thought mistake. The wave actually travels twice the length of the string in each cycle (from the bridge to the nut and back).
A: Why would you expect it not to have longitudinal waves?
If you have a steel bar and you hammer on one end, you get compression waves. They travel as fast as the inter-atomic forces transmit them from one atom to another.
If a string is under tension, and you hammer backward against whatever it's tied to, I'd expect the string to transmit tension waves. The amount of tension would increase and decrease, and the waves would travel as fast as the inter-atomic forces transmit them from one atom to another.
It makes perfect sense that when you pluck a string under tension, you make both transverse and longitudinal waves. Beginning physics students pay attention to the longitudinal waves because they are a metaphor for transverse light waves, and they can be visible.
So when you make a guitar or a violin, does the sound come from the transverse waves from the string causing the air inside the sound box to vibrate, which causes the front panel to vibrate, which vibrates the air outside the soundbox?
Or is is from the longitudinal waves from the string causing the neck and bridge of the instrument to vibrate, which causes the front panel to vibrate, which vibrates the air inside and outside the soundbox.
If it's the latter, you could make a stringed instrument that had the soundbox at one end with the strings attached to it normal to the surface, and it would make sound even though the strings' transverse motion doesn't have much opportunity to affect the soundbox or the air inside it.
And you can.
So is it more likely that this instrument has its sounding board vibrated by the longitudinal tension of the strings, or the air moved by their transverse motion?
