F16GuitarString

Analysis of Vibrating Guitar String

Hayden McCormick

University of Minnesota - TC

Introduction:

The field of physics pertaining to musical acoustics relies on the mathematical understanding of waves with proper boundary conditions modelling the physical behavior of vibrating strings. Confirmation of these models for high-frequency phenomena, which were difficult to inspect visually, leveraged other observations such as the corresponding tone of stringed instruments. However, the recent popularity and affordability of high-speed photography allows physicists to observe visually the motion of plucked musical strings. Additionally, the availability of digital image analysis software provides an effective means of reading and analyzing large volumes of data. By taking advantage of both developments, this research conducted an experiment into analyzing the transience of plucked guitar strings to confer the ability of Fourier Series to model their behavior and identify the rate of decay for each term as a function of the initial fractional displacement. The high-speed capabilities of a Casio EX-F1 camera recordings provide documentation of a guitar string undergoing oscillation. The tuned string is then plucked and recorded for both the transient and steady-state responses from an initial triangular waveform as it undergoes many oscillations for range of displacements. The subsequent images are digitally processed into data sets containing the vertical position of the string along the length of the apparatus on a particular frame using the scientific image software Image-J. This positional data is then compared to the standard mathematic model for vibrating strings, a Fourier Series constructed using the experiment’s boundary and initial conditions. The experiment shows the validity of these models in describing the oscillation of strings in the time-domain within the accuracy of the camera’s resolution and continues to evaluate the decay behavior of each term within the Fourier Series.

Theory:

To mathematically model any waveform, the Fourier series states that any function can be represented as the sum of a number of sine and cosines. Considering the perimeters of an oscillating guitar string where nodes exist at the end points, we can simplify this case into only sines as a function of only of position.

𝑓(𝑥) = ∑ 𝐴𝑛sin( 𝑛𝜋𝑥 / 𝐿 )

Creating a model for the motion of a vibrating string involved creating a model that reflected the time-evolution of the Fourier series as the wave propagated and an exponential decay term to account for the decay of the waveform over time.

𝑓(𝑥,𝑡) = 𝑒 (−𝐶𝑓0𝑡) ∑ 𝐴𝑛 sin ( 𝑛𝜋𝑥 / 𝐿 ) cos(2𝑛𝑓0𝜋𝑡)

This model is insufficient to account for the decay of each term within the amplitude, which will allow the experiment to study the effect of the decay on each harmonic so this final obstruction is solved by moving the decay term in to the summation.

𝑓(𝑥,𝑡) = ∑ 𝐴𝑛 𝑒 (−𝐶𝑛𝑛𝑓0𝑡) sin ( 𝑛𝜋𝑥 / 𝐿 ) cos(2𝑛𝑓0𝜋𝑡)

This final model is used to model our vibrating guitar string and observe the harmonic decays.

Experimental Set-Up

For this experiment, a resonance box is used to house and control the tension of the guitar string during observations. Using an actual guitar would be inadvisable because the musical instrument is not set up in a way to easily determine the string’s position using the stated methods. In addition, the construction would not allow the experiment to measure a full range of positions and amplitudes to test. The resonance box is a simple open wooden box that can securely hold the winded string taut to the desired tone. The resonance box is illuminated with powerful 500 W Halogen lamps to obtain high contrast between the string and the white background and the camera is positioned parallel to the open length of the string. The high contrast is necessary for the software to discern the string from the backdrop. zRecording at maximum speed produces individual images at a resolution of 96 x 336 pixels. Unfortunately, this resolution is the most influential factor in fitting the model to the data using the decay constants.

Data Collection

The string is plucked from half and the a quarter of the length, and recorded until the string has nearly settled. The images are processed into positional data sets reflecting the pixel along the horizontal axis where the string was registered and the corresponding vertical pixel. The data sets are converted into units of meters by correlating the pixel size to a real world size of 2 mm by 2 m.

Results

The model equation is compared to the resulting data sets with the error attributed to a uniform distribution. A number of these graphs for both displacements are shown below as they oscillate in the first half period.

I order to observe the behavior of the decay of each term, we fit the model to the data using the decay constant, C, as variables as well as time. The cumulative error in time and the inaccuracy of measuring time by number of frames is a problem throughout measing the model to the data. By fitting these parameters, we can observe what harmonic terms dominate over time. A collection of these graphs are shown below.

The exponential decay constants change dramatically from frame to frame but we can observe the fit of the data and general behavior of the harmonics. For the half-plucked string, the wave decays to the first term sine, and for the quarter-plucked string it decays to the 1st harmonic sine.

Conclusion:

By utilizing the capabilities of high-speed photography and digital image software, we’re able to confer the validly of Fourier series method in describing the motion of a vibrating string considering the initial and boundary conditions. The ability of the model equation derived after the fitting to determine improved values of time and the decay constants, considering the values of the calculated reduced chi square which average to 4.2, supports this claim. Additional, this method of modeling vibrating strings allows for analyzing in the frequency-domain to observe the individual decay of each harmonic and thus the decay from transience to steady-state

Acknowledgement

The author would like to thank Dr. Kurt Wick, Dr. Dan Dahlberg, and graduate TA Kevin Booth for their assistance in the development the experiment, methodology, and data analysis throughout the research.

References

[1]Churchill, Ruel V. Fourier Series and Boundary Value Problems. New York: McGraw-Hill,

1963. Print.

[2] Benade, Arthur H. Fundamentals of Musical Acoustics. New York: Oxford UP, 1976. Print.

Walker, James S. Fourier Analysis. New York: Oxford UP, 1988. Print.

[3] The physics of guitar string vibrations Perov, Polievkt and Johnson, Walter and Perova-Mello,

Nataliia, American Journal of Physics, 84, 38-43 (2016),

[4] Uniform distribution (continuous). (2016, November 11). In Wikipedia,

https://en.wikipedia.org/w/index.php?title=Uniform_distribution_(continuous)

Understanding nonlinear effects on wave shapes: Comment on “An experimental analysis of a

vibrating guitar string using high-speed photography” [Am. J. Phys. 82(2), 102–109 (2014)