16.7:
Problem-Solving: Tuning of a Guitar String
A six-string guitar with a standard effective string length plays the fundamental frequency when the low E string is plucked. What is the tension in the low E string if its linear density is known?
The fundamental wavelength is twice the effective string length.
The wave speed is the fundamental wavelength multiplied by the fundamental frequency.
Now, the string's tension is the string's linear mass density multiplied by the square of the wave speed.
The expression for wave speed is substituted in the tension equation.
The values of the effective string length, fundamental frequency and linear mass density are plugged into the equation to obtain the required tension at which the low E string needs to be tuned.
Note that the frequency is inversely proportional to the effective string length.
So, on shortening the effective string length by pressing it on any fret, one can play frequencies higher than the set frequency.
16.7:
Problem-Solving: Tuning of a Guitar String
In the case of stringed instruments like the guitar, the elastic property that determines the speed of the sound produced is its linear mass density or the mass per unit length. This is simply called the linear density. If the string's linear density is constant along the string, then the linear density is simply the total mass divided by the total length.
The string's wave speed can be regulated by varying the linear density. Tension is the other property that determines the speed of transverse waves along the string.
Note that the fundamental frequency of the string also depends on the string's length. If the string is held on a fret, for example, then its effective length decreases. That is, by changing the length of the string, the frequency can be regulated. The positions of the frets are made so that the different frequencies produced occur in a progression that is pleasing to the human ear. These different frequencies are the different musical notes. Since decreasing the string's effective length increases the frequency, the notes progress from lower to higher as different frets are held along the string.
Different strings are arranged and regulated so that their combination follows a particular pattern of fundamental frequencies. By playing different notes on the strings, different harmonics are obtained, which is the craft of playing the guitar.
Suggested Reading
- OpenStax. (2019). University Physics Vol. 1. [Web version]. Retrieved from https://openstax.org/books/university-physics-volume-1/pages/1-introduction: section 16.3; pages 805–808.