Results and Conclusions
The Linear Case
Trials adhering to the linear model were fit to the first 10 terms of the Fourier series expressed in the Theory section with the relationship between the coefficients fixed according the succeeding equation on the same page. A single decay parameter and the phase parameter were also allowed to vary.
The fit was performed in ROOT in order to minimize chi-squared over every point in x and t. One example particularly illustrative example was plotted against its fit at three points in one cycle, shown in the following figures. Table 1 summarizes the results of all type 1 trials.
Nonlinear Vibrations
Trials of nonlinear vibrations (high amplitude) were first fit to the same Fourier series and relationship to coefficients as the linear case. The phase parameter was allowed to vary. The fit was performed in ROOT in order to minimize chi-squared at t=0. The resulting coefficients were catalogued. A second fit allowed the first 10 terms to vary individually, using the first fit as a starting point. This fit was performed in ROOT in order to minimize chi-squared over every point in x and t. The fit also included a single parameter for exponential decay.
The coefficients resulting from the first fit are considered the linear model’s prediction. The basis for nonlinear trials is a separable partial differential equation. The time-dependent contribution incorporates a changing tension, whereas the space-dependent contribution remains formulated as a Fourier series for which the coefficients are exact functions of initial displacement of magnitude hand position
. As such, it is sensible to extract predicted coefficient values where it is known that the time-dependent contribution is unitary, namely at t = 0. As an illustration, a plot of the partially processed data is contrasted with the fit function (for t = 0, the predicted function is also shown).
The scatterplots above show data collected for the nonlinear case with total length of L=57.5 cm plucked at x=L/3. The predicted waveform is shown in blue, best fit in red. The top plot shows the initial displacement at t=0, and the bottom image shows the waveform approximately halfway through its fundamental cycle.
Conclusions
Linear Vibrations
For trials of the low-amplitude type, a linear model was able to adhere to collected data producing a mean- squared error of 8.6% or less for any experiment performed. The high degree of correspondence between theory and experiment is a strong indicator of the validity of the model. This is one of very few experiments that has been able to demonstrate this correspondence in both space and time using high speed photography. However, the model has been prolifically validated by other experimental methods. In fact, every common stringed instrument relies on the tendency of strings to exhibit a linear frequency response.
Nonlinear Vibrations
The analysis of high-amplitude trials is more interesting. The least-squared minimization produced fit coefficients |An| that got smaller with increased mode number relative to prediction. This pattern was not violated once between the first five non-zero modes for any trial. This suggests that deviation from linear behavior is proportional to a positive function of mode number.
Although the application of a linear model provides only limited phenomenological insight into nonlinear vibrations, we were able to perform a meaningful comparison between prediction and experiment for high amplitude vibrations, exploiting the separability of the perturbed problem to produce a normalized linear prediction without a priori knowledge of the magnitude of the initial displacement. This is direct advantage of using high-speed photographic analysis to obtain multidimensional waveforms.
References
[1] 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 .
[2] Oplinger, Donald W., Frequency Response of a Nonlinear Stretched String, The Journal of the Acoustical Society of America, 32, 1529-1538 (1960), DOI:http://dx.doi.org/10.1121/1.1907948
[3] Rowland, David R., 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), 102109 (2014)] American Journal of Physics, 83, 979-983 (2015), DOI:http://dx.doi.org/10.1119/1.4931714