We shall show the 3 first order modes vibrating in a free-undamped way.
The behavior of this system can be explained in terms of standing waves going forth and back because of reflections at the ends of the string. These waves propagate at a speed given by v=sqrt(t/m), t being the tension of the elastic string and m the mass per unit length of the string. The natural frequency of vibration for this system is f 0 =v/(2L). So, the higher the tension of the string the higher the frequency of the vibration for a given length. For a given tension, the shorter the string is the higher the frequency. These are, for example, the main rules for changing the frequency of a guitar or violin string.
Shown below are the two following higher order modes (n=2,3). In general, for the nth. mode the natural vibrating frequency is f n =nf 0.
The tone of a sound depends mainly on the frequency of the fundamental mode, but when a string of a musical instrument is played, usually a combination of all the above and more modes take place at the same time. This gives the timbre of the sound. To characterize a sound it is important not only the fundamental frequency it has but also the harmonic content (how many higher order modes are involved, the intensities they have, etc.).