S18_Analyzing a Vibrating Guitar String using High-Speed Photography

Analyzing a Vibrating Guitar String Using High-Speed Photography

Kaitlin Boedigheimer & Anthony Hark

University of Minnesota

Methods of Experimental Physics II, Spring 2018

Abstract

High-speed photography and audio were used to study the motion and sound produced by two different vibrating guitar strings, and their decay rates. By plucking the string at three different locations along the string's length, we find that each Fourier component decays exponentially within 0.15 seconds, but the expected decay rate proportional to each Fourier component was not observed due to several limitations.

Introduction

This specific project of studying standing wave patterns in guitar strings is of interest for its insight into the physics of musical acoustics and stringed instruments. For experimental breadth, we also tested two materials of strings, the D’addario XL nickelwound string with 2.70 mm thickness, and the Elixir Nanoweb coated string with 2.54 mm thickness to see the effects of different materials on the vibrations. With the availability of high-speed photography and digital image analysis software, we are able to study the vibrational motion of a plucked guitar string. When a guitar string is plucked it vibrates and sets the surrounding air molecules into vibrational motion. The frequency at which the air molecules vibrate is equal to the frequency of vibration of the string [1]. When we pluck the string and observe its vibrations, we are seeing it in the spatial domain, but we would like to analyze the vibrations in their frequency domain by performing a Fourier transform. Additionally, this experiment will also seek to measure the rate of decay of these frequency components through the sound of the vibrating guitar string by using a spectrogram, where it is then compared to the results that were collected using the camera.

Theory

Let us start with the simple model of the vibrational pattern of a string fixed at both ends and plucked at an arbitrary position along its length. We are able to test the vibrations related to the linear case by plucking the string with a small initial amplitude where the oscillations are small enough that the length L and tension on the string remain constant over time. Under these conditions, the linear one-dimensional wave equation can hold since the tension in the string remains constant during the oscillations [2]. The initial shape of the string is then given by the Fourier series equation:

In this case, L is the length of the string and Ak is the coefficient of the kth sine function. The coefficient determines the amplitude of the specific wave, and since we are trying to determine the rate at which these waves decay over time it is necessary to know how to calculate them. In order to determine these coefficients we use the equation given below:

Finally, from the first equation we can construct a traveling wave, but we have to make a few substitutions. First, we substitute where v is the speed of the wave and is defined by where f0 is the fundamental frequency. This gives us two traveling waves, one moving right and one moving left and their sum is given by:

This equation defines the displacement of the string at any point at any time. There are several different mechanisms that lead to the decay which involve; the motion of the string through the air, internal damping (which depends on the material and construction of the string), and energy losses. In order to correctly model the motion of the string over time, we need to multiply the traveling wave equation by an exponential factor.

This is the simplest model for amplitude decay where all the factors mentioned above that contribute to the decay are accounted for with a single dimensionless constant C.

Experimental Set-Up/Methods

Instead of a real guitar, a wooden box with holes cut out of the sides served as our 'acoustic guitar'. White cardboard was used to separate the string from any background noise, and the two lights mounted to the poles on the sides served to illuminate the string clearly against the background. A Casio EX-F1 camera was used in its 60 fps burst-mode setting. A Snark SN-1 tuner was also used to check the frequency of the string (E1).

A bass guitar string was used since it is thicker and therefore less prone to breaking, and easier to see when it came to the analyzing process. The following process was repeated for the string at three different lengths (1/2, 1/4, and 1/5): The string was plucked and its motion was captured for 60 frames over the period of one second. Approximately 20 'good' snapshots were edited in ImageJ, in order to isolate the string from its background and obtain its X and Y coordinates. The images were then transferred individually to Excel to graph (and for error analysis), and the Excel data was transferred into a Fourier transform program in Matlab.

Data Collection

The data shown below is for the standard nickel-wound string, plucked at 1/4 of the total length. The top image is a snapshot at t=0. Below that is its ImageJ edit, and below that is the Excel graph.

Results

The figure below is the result of the Fourier transform performed by Matlab for each individual image of the string at 20 different times. This is data for the standard nickelwound string plucked at 1/4 of its length.

The different colors represent different snapshots in time. Going from left to right, we see the first harmonic (located at approximately 1 on the (1/x) axis) has the highest amplitude. The decay of this first harmonic can be seen as the triangular peaks begin to decrease with time, which is represented by the downward arrow. If we look at the second component we can also see that it is behaving in a similar fashion. The third frequency component is smaller than the first two but its decay can still be seen.

Now we can see the data above represented on a 3D graph where the third axis is the time axis.

The representation of our data in the spectrogram form is portrayed below. The spectrogram gives us a visual representation of the frequencies as they vary with time for different sounds. In this case, we created spectrograms based on the data collected from the images to compare them to the spectrograms produced by the sounds of the strings. The y-axis represents the frequency components, the x-axis is the time and the color variation determines the amplitude or intensity of that specific frequency component at a specific time. This graph is based on the collected data.

The spectrogram below was recorded using audio data of the string. The frequency amplitudes are strongest in the red zone, and decay in the order of a standard rainbow to blue/violet being the weakest. The first red 'row' represents the first harmonic, and the second more orange row as the second harmonic and so forth. If we compare the audio spectrogram images to the photography spectrogram images we can see some similarities in the decay patterns of the individual graphs but the graphs will look different due to the number of data points taken for each.

The first harmonic of the coated string appears to take longer to decay than that of the standard string, although it starts out weaker. The rest of the harmonics also appear to display this trend. This could represent that while a coated string requires more plucking effort to be at the same frequency strength as a standard string, it holds the note with more sustain over time.

Conclusion

This experiment gave us the opportunity to understand the harmonic composition of standing waves in the strings of musical instruments and how it is dependent on the way the string is played. We visually observed how other harmonics are also present and how they affect the sound when the string is plucked at different locations. We observed the values of the coefficients for each frequency component and we did see each term decay with time as we had expected, but we also expected to see a certain proportional decay trend happening where each kth coefficient decayed k times the coefficient of the fundamental frequency, or at least with some clear rate which we did not.

Future Improvements

Previous projects took a video of one string and modeled it to a simple string- vibration model. This employs the principle of superposition but neglects the actual composition of the string. By doing this, they then altered the value for C to determine its value by the fit. What we tried to do is create a Fourier transform code in Matlab to see the different harmonics that result in the final wave that we are seeing. By doing this, we can analyze the different decay rates of the individual harmonics with time and see how it causes the string to decay, and we studied it using two different strings plucked at three different positions. If future students want to analyze the string using our method we recommend that they use a higher fps on the camera. They can use the video of 1200 fps but they will face the same problem we did with the pixelated images. There are two ways to make this work; the first solution is to simply use a higher quality camera. If that is not possible, they can still use the 1200 fps but once the images are generated, expand them and perform a Gaussian fit on the string's position in order to determine the exact pixel value of the amplitude. This may also cause a large error in the end but it would be interesting to try since we already proved our method also has its limitations.

Acknowledgments

We would like to thank our advisor Professor Ke Wang and our instructor Professor Kurt Wick for assisting us in the development and data analysis of this experiment.

References

• 1. “Guitar Strings and Longitudinal Waves.” The Physics Classroom, www.physicsclassroom.com/mmedia/waves/gsl.cfm.

  2. Scott B. Whitfield and Kurt B. Flesch. An experimental analysis of a vibrating guitar string using high-speed photography. American Journal of Physics 82, 102 (2014); doi: 10.1119/1.4832195 .

  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. Gulla,Jan. “Modeling the Wave Motion of a Guitar String.” (2011).

  5. “Discrete Fourier Transform”.Wikipedia, Wikimedia Foundation, 10 Apr.2018, en.wikipedia.org/wiki/Discrete_Fourier_transform.