Analysis of Patterns in the Harmonics of a String with Artificially Enforced Nodes

This study examines the higher harmonics in an oscillating string by analyzing the sound produced by a guitar with a spectrum analyzer. Higher harmonics of a string are simultaneously oscillating modes which have frequencies that are integer multiples of the fundamental frequency of the string. These harmonics can be viewed on an audio spectrum analyzer. They are always present in an oscillating string and contribute to its timbre and tonal quality. Specific higher harmonics can be produced directly by placing nodes (points where the string cannot oscillate) at different lengths along a string. The tone thus produced lacks the fundamental frequency but also has a very different harmonic structure. In a guitar string, for example, it is this harmonic structure which gives rise to the very different tonal quality of a plucked harmonic as compared to the directly excited pitch of the same frequency. We mathematically hypothesized that the higher harmonics in the series of the directly excited 2^nd harmonic contain the alternate frequencies of the fundamental series, the higher harmonics of the directly excited 3^rd harmonic series contain every third frequency of fundamental series, and so on. We also verify a simple mathematical relationship between two different harmonic series arising from two different boundary conditions that feature the same fundamental mode. To test our hypotheses, we enforced artificial nodes to excite the 2^nd, 3^rd, and 4^th harmonics directly, and analyzed the resulting spectrum to verify the mathematical hypothesis. The data analysis corroborates both hypotheses.