Applications of the Roots of Bessel Functions of the First Kind

This project explores how Bessel functions of the first kind (Jn) and their roots appear in real-world physics, from vibration modes of circular drumheads to frequency modulation (FM) synthesis and planetary motion in Kepler’s equation.
Using Python’s SciPy library, we numerically compute and visualize the first five roots for orders n = 0, 1, 2.

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Authors

Madeline Renee Boss
Samhitha Devi Kunadharaju
University of Texas at Austin
Course: CS 323E – Elements of Scientific Computing

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Overview

The Bessel differential equation defines oscillatory solutions that naturally arise in problems with cylindrical or radial symmetry.
We compute the first five roots of J₀(x), J₁(x), and J₂(x) using SciPy’s quad() and fsolve() methods, then visualize how these roots relate to physical systems such as:

1. Vibrations of a circular drumhead
2. Frequency modulation (FM) signal synthesis
3. Harmonic suppression in Kepler’s planetary equation

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Included Files

File	Description
bessel_roots_analysis.ipynb	Jupyter/Colab notebook that computes and visualizes Bessel functions and their roots
P2.pdf	Full paper detailing mathematical background and physical applications
images/bessel_functions_plot.png	Visualization of J₀(x), J₁(x), and J₂(x)
images/bessel_roots_plot.png	Visualization showing computed roots for n = 0, 1, 2
requirements.txt	List of dependencies (NumPy, Matplotlib, SciPy)

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Visualizations

Bessel Functions	Computed Roots

Left: Plots of J₀(x), J₁(x), and J₂(x) showing their oscillatory behavior.
Right: The same functions with their first five roots marked using SciPy’s fsolve().

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Setup

Install dependencies:

pip install -r requirements.txt