The Wave Equation and the Timpani: An Exploration into Various Models for the Vibration of a Timpani Drumhead Pubblico
Li, Jack (Spring 2022)
Published
Abstract
The vibration of a timpani can be approximately modeled using the 2-dimensional wave equation, and various other factors (such as damping or viscoelastic effects) can also be quantitatively factored into the formulation of the equation.
Various initial conditions (specifying an initial displacement, an initial velocity, and various forcing terms) and versions of the wave equation were used to model the vibration of a kettledrum, and animations, periodograms of the vibration at the point of impact, and synthesized sounds using the point of impact were used to assess the accuracy of each model and the effects that each model had on the vibration of the drum. Moreover, points of impact were obtained from a survey response and their resulting animations, periodograms and sounds compared to the literature point of impact of 0.75r.
It was found that, with regards to initial conditions, while it was possible for the amplitudes of vibrational modes to change significantly between various choices of initial conditions (with a Dirac forcing term at 0.75r appearing to give the most accurate sound and vibration out of all chosen conditions), the frequencies of these modes did not seem to be significantly affected by these initial conditions. First-order damping also did not seem to significantly affect frequencies, and the impact of viscoelastic effects was similarly slight, only slightly raising the frequencies of higher frequency modes. These modeled frequencies all ended up being significantly higher than what was found in reality - it was deduced using results from Gallardo et. al 2020 and Fletcher et al. 1997 that the effect of air-loading, which was not accounted for in any of the equations, was responsible for this effect.
Table of Contents
1 Introduction 7
1.1 The Closed-Form Solution for the 2-Dimensional Wave Equation . . . . . . . . . . . . . . . . . . 7
1.2 Fourier-Bessel Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10
1.3 Vibrational Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12
1.4 Limitations of the Standard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .14
1.5 Numerical Methods: The Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . . . .15
2 Methods and Materials 15
2.1 The Mallet Radius Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17
2.2 The Initial Displacement Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .18
2.3 The Initial Velocity Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19
2.4 The Forced Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19
2.5 The Forced, Modified Wave Equation Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20
2.6 Adding a "Dirac"-like Element to The Forced, Modified Wave Equation Model . . . . . . . . . .21
2.7 The Dirac-Forced, Modified Wave Equation Model applied to Surveyed Impact Points . . .22
3 Results 23
3.1 Animation Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23
3.2 Periodograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23
3.3 Synthesized Sounds and Audio Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24
3.4 Frequencies and Relative Amplitudes of Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . .25
3.5 Examining the Forced Model on Drummer Survey Results . . . . . . . . . . . . . . . . . . . . . .26
3.6 Comparison of Frequencies to the Undamped Wave Equation . . . . . . . . . . . . . . . . . . . .27
4 Discussion 29
4.1 Effect of Initial Conditions, Damping and Viscoelastic Effects . . . . . . . . . . . . . . . . . . . .29
4.2 Effect of the Point of Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30
4.3 Effect of Air Loading on the Vibration of the Membrane . . . . . . . . . . . . . . . . . . . . . . .31
4.4 Future Steps - Accounting for Air Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32
4.5 Near-harmonic ratios in an unloaded membrane . . . . . . . . . . . . . . . . . . . . . . . . . . .33
4.6 Other Future Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34
5 Appendices 35
5.1 Appendix A: Meshes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35
5.2 Appendix B: Selected Animation Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .37
5.3 Appendix C: Periodograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41
5.4 Appendix D: Audio Waveforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .44
5.5 Appendix E: Frequency Raw Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .45
5.6 Appendix F: Amplitude Raw Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47
5.7 Appendix G: Bessel Function Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .49
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Primary PDF
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The Wave Equation and the Timpani: An Exploration into Various Models for the Vibration of a Timpani Drumhead () | 2022-04-07 18:03:47 -0400 |
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Supplemental Files
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Thesis Presentation (Presentation including animations and sounds for all relevant models) | 2022-04-07 18:03:54 -0400 |
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