Analyzing a Vibrating Guitar String Using High-Speed Photography

Sanjeev Mishra and Jackson Homstad

Abstract

The vibrational behavior of a guitar string was observed using high-speed photogra- phy and image processing. These observations were compared to a standing wave model rooted in Fourier analysis. Small amplitude oscillations were found to follow the model more closely than large amplitude oscillations, although both forms of oscillations were found to deviate signi.cantly from the model after six to seven periods. Vibrations at smaller initial fractional plucking lengths were also found to follow the model more closely than fractional plucking lengths near the middle of the string. The decay of amplitude over time was also found to be proportional to the fundamental frequency of oscillation.

Introduction

Standing waves and their behavior form the basis of many musical instruments, including stringed instruments, as well as brass and woodwinds. Inside these instruments, standing waves of different amplitudes and frequencies are created to produce notes of different volumes and pitches, respectively. In stringed instruments, these waves take the form of transverse waves with fixed nodes at each end of the strings. Many factors affect the behavior of string vibrations, such as string tension and oscillation amplitude.

In this study, Fourier analysis was employed to model the behavior of a vibrating guitar string. In order to observe the vibrational motion, a vibrating string was filmed using a high-speed camera with a maximum frame rate of 1200 frames per second, and its shape in space over time was determined using image-processing software. Once the string's vibration was observed, it was compared to a theoretical model to determine how effectively the model describes the string's behavior. Because non-linear vibrational behavior is well known to affect the timbre and harmonic capabilities of an instrument, higher order effects were also induced and studied.

Theory

Assume that a one-dimensional string of length L is plucked with some initial amplitude, and that its motion is constrained to a two-dimensional plane (the x-y plane.) Furthermore, assume that the initial amplitude of the oscillation is small, such that the tension and length of the string will remain approximately constant over time. The initial shape of the string is then given by

where the coefficients are given by

where F is the horizontal position of the initial pluck, expressed as a fraction of the total string length measured from the left, and H is an amplitude scaling factor. One can construct a traveling wave from this formulation by making the substitution

where v is the velocity of the wave. By superpositioning two traveling waves of equal and opposite velocity, a standing wave is created. The velocity can be expressed as

where f_0 is the fundamental frequency of the string. Two traveling waves, one moving right and the other moving left, are respectively given by

The sum of these waves is the desired standing wave, and is thus more succinctly given by

In order to account for the decay of the amplitude over time through energy losses, the above equation is multiplied by an exponential factor

where C is a dimensionless constant. The model developed above neglects effects due to changing tension and string length over time. These effects are noticeable during large amplitude oscillations, leading to higher order effects and complicated motion. Causes of this complicated motion include time-dependent tension, motion in the z direction. etc. To study these effects, the string was plucked at large amplitudes, and its vibrational behavior was compared to our simple model.

Methods

A bass guitar string of fixed length was mounted on an apparatus as shown below. An A-string of a bass guitar was chosen to ensure that the string's motion would be easily detectable and that the string would not break during large amplitude oscillations. The string was illuminated from above using two bright lamps to minimize shadows and mounted in front of a featureless, white background. The string was then plucked at various fractional lengths with both small and large initial amplitudes using the sharp edge of a flathead screwdriver. Care was taken to pluck the string as vertically as possible to constrain its motion to the x-y plane. This motion was captured using a Casio EX-F1 camera at a frame rate of 1200 frames per second. This motion was recorded for approximately ten seconds, at which point the oscillations of the string had decayed significantly.

Videos of the guitar string's motion were decomposed into their constituent frames and analyzed using ImageJ. First, each frame was cropped such that the left hand side of the image began at x = 0, and the right hand side was located at x = L. A cropped image is shown below. ImageJ's background subtraction and contrast enhancement routine were used to further isolate the string from its background. The image was then color thresholded to convert it to a black and white binary image. A frame that has undergone ImageJ processing is also given below. ImageJ then exported the X-Y pixel coordinates of each frame as an array. Elements of the data arrays corresponding to apparatus or the screwdriver were removed by a MATLAB function. Model parameters were determined through .chi-squared minimization. After these parameters were determined, theoretical amplitude values at each point along the string were computed using the model equation.

Results and Analysis

The reduced .chi-squared values as a function of time for small (left) and large (right) amplitude sizes at four diff.erent fractional plucking lengths are given below.

The measured decay constant C versus the fundamental frequency of motion f_0 is given above.

For both small and large amplitude motion, reduced chi-squared values increased signifi.cantly after about fi.ve periods, with the increase much more dramatic for large amplitude data. Reduced .chi-squared was also typically larger for larger initial fractional plucking lengths F for both small and large amplitude motion. However, for large amplitude motion, reduced .chi-squared values were much larger (3 - 30) than for small amplitude motion (0.5 to 2.5). This implies that small amplitude motion followed the behavior predicted by the model more closely than for large amplitude. It was also observed that the decay constant C increased as a function of the fundamental frequency f_0. Additionally, the fundamental frequency for large amplitude motion was found to be larger than the fundamental frequency of small amplitude motion, despite the fact that the tuning peg on the apparatus was not adjusted between trials.

For large amplitude motion, most of the deviation from the model occurred at one-quarter and three-quarters of a period, where the model predicted small amplitudes but the string instead exhibited substantial motion. One explanation for this phenomenon could be the existence of higher-order e.ffects that were not factored into our model. It has been shown that the presence of higher order terms in the equations of motion for vibrating strings leads to precessional behavior in its motion, as well as some trading of amplitude prominence between di.fferent modes over time. The fact that the decay constant was larger for large amplitude motion than small amplitude motion implies that frictional energy losses were more signifi.cant for large amplitude motion. Large amplitude oscillations occur at higher string velocities, so the energy loss could be partly due to air resistance which becomes more pronounced at larger velocities.

To improve upon the work in this study, one could use a camera stand that is better equipped to measure the rotational orientation of the camera. This would make it easier to position the camera properly during calibration. Additionally, one could use a camera with better resolution at higher frame rates to allow for a more precise determination of the observed amplitude of the string over time. The motion could also be analyzed for a time period larger than 200 frames, to see if other eff.ects become more apparent after motion has decayed signi.ficantly. To expand upon this study, one could develop a model that accounted for higher order eff.ects that become important at large amplitudes, and compare this more sophisticated model to large amplitude observations. The string's motion could also be observed at di.fferent frequencies (perhaps larger than 100 Hz) than the frequency range probed in this study, and strings of various gauges and materials could be studied to see how the composition of the string a.ffects its motion.

Acknowledgements

This study was conducted with the assistance of Jackson Homstad and with the help of our advisor, Peter Martin. Additional help was provided by Kurt Wick and Professors Clem Pryke and Elias Puchner.

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