I have some questions for you about the resonance phenomenon in a guitar:
Why resonance frequency are only full integer multiples of the natural frequency, such as 2 x, 3 x, or 4 x or 3 x, 5 x, or 7 the natural frequency?
Why a vibrating string tend to vibrate at the natural resonant frequency(x1) and not x2 or x3?
Is it true that in a perfect world, in a string of a guitar the resonance frequencies(x1,x2,x3,x4,ecc...) will never die out? Why non-integer multiples, otherwise, will die out?
I think the question applies not specifically to a guitar, but to any string. (To be precise, there are some differences in how sound is generated between guitars and instruments like violin, which I address in the end.)
Oscillations of a guitar string are a standing wave, which has its regions of high amplitude and nodes with zero amplitude. The nodes are separated by half wavelength distance. Since the ends of the string are fixed, the only waves that can be excited in such a string are those that have nodes at the ends of the string, which means that the length of the string is equal to a half integer number of wave lengths. Note that not only these tones coincide with the natural scale - this is precisely the reason why it is called natural scale. Very similar arguments can be applied to other types of musical instruments.
Obviously, when we touch a string, we excite many modes. The lowest usually has the highest energy and dominates the sound, while the rest determine the timber/quality of the sound. This is why the sound quality can be changed by touching a string in different places - an effect used in playing classical acoustic guitar: by placing the right hand closer or farther from the middle of the string the musician can extract sounds that are deeper or more dry. Are more sophisticated techniques is by extracting harmonics sounds with one's left hand, giving them a particular overtone.
From the answer to point 1 it should be already clear why non-integer waves die out. By "ideal world" one often means the world without dissipation. This is hardly applicable to a guitar, whose point is to generate sound, i.e., to transfer the energy from the string to the air oscillations. The string oscillations in vacuum however would not die out indeed. The damping of the string oscillations of a guitar is one of the reasons why (before appearance of electrical amplifiers) this instrument was impractical for anything but chamber music. Bowed string instruments (violin, cello, etc.) have the advantage that the sound can be excited continuously while the musician moves the bow along the string.
