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Calculating the Natural Frequency of a String Under Tension



  • A string, or wire, stretched under tension will have a natural frequency that is a function of the length of the string, L, the tension applied to the string, T_0, and the mass per unit length of the string, μ. This is the principle that allows stringed instruments to have a specific pitch for each string. The formula for the natural frequency of a string is given by [1]:





  • It can be seen from this equation that the natural frequency increases with increased tension and decreases with increased string length or increased mass per unit length. We will use a guitar's High E string as an example. Many guitars have a scale length (the free vibration length of the string [2]) of 25.5 inches and a typical diameter for a High E string is 0.011 inches:







  • Using this diameter, we can express the mass per unit length as the product of the mass density of the string material, ⍴, and the cross sectional area of the string, A:







  • We need to choose the string material to set the density. The table below allows us to switch between a steel string a nylon string:



  • Steel Music Wire [3]
    Nylon Music Wire [4]


  • Finally, we need to set the value for the string tension:





  • We are now able to query the value of the frequency variable, f, to determine the resulting natural frequency for the string. For the steel string, the above parameters give us the correct frequency for a guitar's High E string of approximately 330 Hz [5].





  • References:

    [1] “String Vibration.” In Wikipedia, April 27, 2022. https://en.wikipedia.org/w/index.php?title=String_vibration&oldid=1084999613.

    [2] “Scale Length (String Instruments).” In Wikipedia, September 26, 2022. https://en.wikipedia.org/w/index.php?title=Scale_length_(string_instruments)&oldid=1112524340.

    [3] Norton, Robert. Machine Design. 5th edition. Boston: Pearson, 2013.

    [4] Lynch-Aird, Nicolas, and Jim Woodhouse. “Mechanical Properties of Nylon Harp Strings.” Materials 10, no. 5 (May 4, 2017): 497. https://doi.org/10.3390/ma10050497.

    [5] “Guitar Tunings.” In Wikipedia, February 27, 2023. https://en.wikipedia.org/w/index.php?title=Guitar_tunings&oldid=1141849805.