When a guitar string is plucked, there are many frequencies of sound which are emitted. The lowest frequency is the note we associate with the string, while the mix of other frequencies gives the sound its timbre, or sound quality. The lowest frequency is the fundamental, while the higher frequencies make up the harmonic series. The next lowest frequency is the second harmonic; the next to lowest, the third harmonic, and so on. The timbre depends on the material of the string (steel or plastic or catgut), on the way it is plucked (middle or at the end), and on the sounding board.
Sometimes some of the frequencies may be suppressed, for example, by lightly holding a finger at a point along the string to force a node there. This is not the same as fretting the string, which involves holding the string all the way down to the neck in order to effectively change the length of the string.
The wave velocity is given by
where T is the tension in the string, and µ is the linear mass density, which is the product of material density and cross-sectional area.
For the following questions, consider an E string (frequency 660 Hz) which is made of steel. It has a mass of 0.66 grams for each meter of wire and has a circular cross section of diameter 0.33 mm. The string length when strung on a guitar is 0.65 m.
Also note that the D string has a wave velocity of 382 m/s.
If we want to increase the frequency of the fundamental of a string by 3%, by how much do we want to change the tension in the string?