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Sound travels in all directions. High pressure waves expand in all directions. But in string instruments you hear mostly the vibrating instrument that holds the string, which is why it's not the string that costs a million dollars, but the wood that holds it.

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Bingo. If the string itself was the only thing propagating the sound then a guitar would just be a 2x4 with strings attached.

If you want a very simple visualisation: imagine the soundwave as people, leaving an elevator into a big room. They will tend to spread out in a half circle. Like a soundwave does if it has to pass a narrow slit. The string vibrates the soundbox, and from there, sound goes in every direction.

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That's an interesting question.

On the surface of a liquid or in the volume of a gas, a disturbance impacts every adjacent point, in every direction. A little pressure wave goes out in every direction from the point of disturbance. On the 2-D surface of a liquid like a pond, you get an ever expanding circle. In a 3-D volume like a gas you get an ever expanding sphere.

There are ways to guide or focus this energy initially, but as soon as the little pressure wave is in free space, away from the thing that disturbed it, it starts behaving this way (in every direction.) Even things as tightly focused as lasers diffuse as they pass through gas and liquids, for the same reasons sound does.

Now think of your string not as one thing but as a collection of an infinite number of points (which it is). As it moves, it's disturbing the air around it at every single one of those points, initiating an ever expanding sphere from each. The disturbance does not just travel perpendicular to the string. It goes out in all directions from everywhere on the string. Again, this can be directed initially like we do with speakers and guitar bodies, but as soon as the wave is away from the surface of the speaker or out of the guitar, it starts going in every direction.

Hope this helps. Again, thoughtful question.

Thanks, this is the first answer I got that explains what is actually happening on a particle level.

If your string was stretched between two anchors on a solid piece of metal bar, such that the string was vibrating, but nothing else was acting as a sounding board... and then you took the construction into an anechoic room and plucked the string you might find that the sound is directional - though the direction would be quite broad, you might notice "nulls" quiet directions when the vibration was in a tangential (sideways) as opposed to back-and-forth towards you.

Although there might still be enough turbulence around the string for the sound to be radiated in most directions. A vibrating ribbon, rather than a string may demonstrate the effect better.

In a normal (non-anechoic) room you get enough sound bouncing chaotically off the walls/floor/ceiling, you won't really detect "quiet" directions from a sound source.

Changing subject slightly, if you have a tuning fork, the direction of the sound from that is typically somewhat directional I recall.