Standing waves on a guitar string

Standing wave resonances in some shape or form are the basis for most acoustic instruments which produce tones. We have a prototype guitar which is not great for its sound quality but illustrates the point. The tension in the strings is determined by masses hanging from them. When I pluck the string I set up a standing wave resonance with a fundamental frequency

Note that all waves generated by the strings have the same wavelength on the string namely set by the boundary conditions there. However the frequencies are different because the speed of the wave along the string is different. In the expression T=mg is the tension in the string and is the mass per length unit of the string. Checking out the guitar we note that

• We get different frequencies when strings with different tension are subject to the same tension (this is why light strings are easier on the fingers when playing the guitar: a smaller tension will produce the same frequency as a heavier string at higher tension)
• Quadruple the mass or halve the length of the string to produce a tone with double frequency (an octave).