Modelando una solución de la ecuación de la onda en coordenadas polares utilizando Python

Eric Antonio Acevedo; María T. Ortega O.

doi:10.48204/j.colonciencias.v13n2.a8744

Submitted November 24, 2025

Published 2026-07-07

Artículos

Vol. 13 No. 2 (2026): Revista Colón Ciencias, Tecnología y Negocios

Modeling solutions to the wave equation in polar coordinates with Python

Eric Antonio Acevedo⁺⁻
María T. Ortega O.⁺⁻

DOI https://doi.org/10.48204/j.colonciencias.v13n2.a8744

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DOI: 10.48204/j.colonciencias.v13n2.a8744

Published: 2026-07-07

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Acevedo, E. A., & Ortega O., M. T. (2026). Modeling solutions to the wave equation in polar coordinates with Python. Revista Colón Ciencias, Tecnología Y Negocios, 13(2), 38–66. https://doi.org/10.48204/j.colonciencias.v13n2.a8744

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.

Abstract

The wave equation is a classic problem in physics and mathematics, widely applied to the study of vibrations, acoustics, and wave propagation in circular or radial media. This equation describes how a disturbance travels through space and time. Regionally, most work has focused on Cartesian geometry; however, expressing it in polar coordinates incorporates additional terms that reflect radial and angular symmetry, allowing the modeling of phenomena such as the vibration of circular membranes or propagation in cylindrical media. Therefore, the objective of this research was to model the solution of the wave equation in polar coordinates using the explicit finite difference method implemented in Python, thus developing a reproducible scheme that allows the simulation and visualization of the time evolution of waves in media with radial symmetry. To achieve this, the methodology employed the finite difference method, discretizing both the radial and angular variables and constructing a mesh that facilitates the time simulation. In addition, Python was used as the main tool because of its flexibility and availability of scientific libraries (NumPy, Matplotlib), which allow the implementation of algorithms and the generation of visualizations.

The numerical scheme includes initial and boundary conditions, essential for ensuring stability and realism in the simulation. The results show propagation patterns consistent with the theory, revealing nodal structures and oscillatory dynamics characteristic of circular media. The visualizations obtained allow for the analysis of the influence of parameters such as wave number and angular mode on the solution's behavior.

This research helps strengthen the understanding of wave phenomena in non-Cartesian geometries, provides valuable pedagogical resources for teaching and research, and promotes access to advanced methodologies through open-source software, thus contributing to the democratization of knowledge and fostering applications in engineering, acoustics, optics, and computational physics.

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References

Allain, R. (2024). Modeling waves with numerical calculations using Python. Springer. https://doi.org/10.1007/978-3-031-78291-6
Amadeusferro. (2023). Finite-Difference Wave Equation Simulator (Python) [Repositorio en GitHub]. https://github.com/amadeusferro/Finite-Difference-Wave-Equation-Simulator
Behera, S., & Behera, D. K. (2024). Use of various finite difference methods for solving PDE. International Journal of Physics and Mathematics, 6(2), 48–56. https://doi.org/10.33545/26648636.2024.v6.i2a.96
Belu, R., & Belu, A. C. (2007). Using finite difference methods instead of standard calculus in teaching physics. En Proceedings of the American Society for Engineering Education Annual Conference & Exposition. https://peer.asee.org/using-finite-difference-methods-instead-of-standard-calculus-in-teaching-physics.pdf
Berg, I. (2025). Solving the 2D wave equation with the finite difference method. Recreational Mathematics. Beltoforium https://beltoforion.de/en/recreational_mathematics/2d-wave-equation.php
Duarte González, E. Y., Pichardo Silva, M. J., & Ramírez Mairena, L. V. (2019). Solución de las ecuaciones diferenciales parciales mediante diferencias finitas [Informe técnico]. Universidad Nacional Autónoma de Nicaragua, León.
Holman, B., & Kunyansky, L. (2010). A second-order finite difference scheme for the wave equation on a reduced polar grid (Technical Report). University of Arizona. https://math.arizona.edu/~leonk/papers/polarFD7.pdf
Krishna, G., Kalyanrao, T., & Shrikisan, G. (2021). New technique for solving time fractional wave equation: Python [Manuscrito no publicado]. Academia.edu. https://www.academia.edu/76599169
Langtangen, H. P., & Linge, S. (2017). Finite difference methods for 2D and 3D wave equations. En Finite difference computing with PDEs (pp. 241–276). Springer.
LeVeque, R. J. (2007). Finite difference methods for ordinary and partial differential equations: Steady-state and time-dependent problems. SIAM.
Maas, M. (s.f.). Apuntes de diferencias finitas [Apuntes de cátedra]. Universidad de Buenos Aires.
Mohanty, R. K., Kumar, R., & Dahiya, V. (2012). Cubic spline method for 1D wave equation in polar coordinates. ISRN Computational Mathematics, 2012, 1–6. https://doi.org/10.5402/2012/302923
Morton, K. W., & Mayers, D. F. (2005). Numerical solution of partial differential equations: An introduction (2nd ed.). Cambridge University Press.
Morse, P. M., & Feshbach, H. (1953). Methods of theoretical physics. McGraw-Hill.
Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (2007). Numerical recipes: The art of scientific computing (3rd ed.). Cambridge University Press.
Qingkai, K., Timmy, S., & Alexandre, B. (2020). Finite difference method. En Python Numerical Methods. University of California, Berkeley. https://pythonnumericalmethods.studentorg.berkeley.edu
Smith, G. D. (1985). Numerical solution of partial differential equations: Finite difference methods (3rd ed.). Oxford University Press.
Strikwerda, J. C., & Nagel, Y. (1986). Finite difference methods for polar coordinate systems (Technical Summary Report No. 2934). Mathematics Research Center, University of Wisconsin-Madison.
Zhu, N., & Zhao, M. L. (2019). High-order finite difference method for Helmholtz equation in polar coordinates. American Journal of Computational Mathematics, 9(3), 174–186. https://doi.org/10.4236/ajcm.2019.93013
Zill, D. G., & Cullen, M. R. (2009). Ecuaciones diferenciales con problemas con valores en la frontera (A. E. García Hernández, Trad.; 7ª ed.). Cengage Learning Editores.

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[1] Allain, R. (2024). Modeling waves with numerical calculations using Python. Springer. https://doi.org/10.1007/978-3-031-78291-6

[2] Amadeusferro. (2023). Finite-Difference Wave Equation Simulator (Python) [Repositorio en GitHub]. https://github.com/amadeusferro/Finite-Difference-Wave-Equation-Simulator

[3] Behera, S., & Behera, D. K. (2024). Use of various finite difference methods for solving PDE. International Journal of Physics and Mathematics, 6(2), 48–56. https://doi.org/10.33545/26648636.2024.v6.i2a.96

[4] Belu, R., & Belu, A. C. (2007). Using finite difference methods instead of standard calculus in teaching physics. En Proceedings of the American Society for Engineering Education Annual Conference & Exposition. https://peer.asee.org/using-finite-difference-methods-instead-of-standard-calculus-in-teaching-physics.pdf

[5] Berg, I. (2025). Solving the 2D wave equation with the finite difference method. Recreational Mathematics. Beltoforium https://beltoforion.de/en/recreational_mathematics/2d-wave-equation.php

[6] Duarte González, E. Y., Pichardo Silva, M. J., & Ramírez Mairena, L. V. (2019). Solución de las ecuaciones diferenciales parciales mediante diferencias finitas [Informe técnico]. Universidad Nacional Autónoma de Nicaragua, León.

[7] Holman, B., & Kunyansky, L. (2010). A second-order finite difference scheme for the wave equation on a reduced polar grid (Technical Report). University of Arizona. https://math.arizona.edu/~leonk/papers/polarFD7.pdf

[8] Krishna, G., Kalyanrao, T., & Shrikisan, G. (2021). New technique for solving time fractional wave equation: Python [Manuscrito no publicado]. Academia.edu. https://www.academia.edu/76599169

[9] Langtangen, H. P., & Linge, S. (2017). Finite difference methods for 2D and 3D wave equations. En Finite difference computing with PDEs (pp. 241–276). Springer.

[10] LeVeque, R. J. (2007). Finite difference methods for ordinary and partial differential equations: Steady-state and time-dependent problems. SIAM.

[11] Maas, M. (s.f.). Apuntes de diferencias finitas [Apuntes de cátedra]. Universidad de Buenos Aires.

[12] Mohanty, R. K., Kumar, R., & Dahiya, V. (2012). Cubic spline method for 1D wave equation in polar coordinates. ISRN Computational Mathematics, 2012, 1–6. https://doi.org/10.5402/2012/302923

[13] Morton, K. W., & Mayers, D. F. (2005). Numerical solution of partial differential equations: An introduction (2nd ed.). Cambridge University Press.

[14] Morse, P. M., & Feshbach, H. (1953). Methods of theoretical physics. McGraw-Hill.

[15] Press, W. H., Teukolsky, S. A., Vetterling, W. T., & Flannery, B. P. (2007). Numerical recipes: The art of scientific computing (3rd ed.). Cambridge University Press.

[16] Qingkai, K., Timmy, S., & Alexandre, B. (2020). Finite difference method. En Python Numerical Methods. University of California, Berkeley. https://pythonnumericalmethods.studentorg.berkeley.edu

[17] Smith, G. D. (1985). Numerical solution of partial differential equations: Finite difference methods (3rd ed.). Oxford University Press.

[18] Strikwerda, J. C., & Nagel, Y. (1986). Finite difference methods for polar coordinate systems (Technical Summary Report No. 2934). Mathematics Research Center, University of Wisconsin-Madison.

[19] Zhu, N., & Zhao, M. L. (2019). High-order finite difference method for Helmholtz equation in polar coordinates. American Journal of Computational Mathematics, 9(3), 174–186. https://doi.org/10.4236/ajcm.2019.93013

[20] Zill, D. G., & Cullen, M. R. (2009). Ecuaciones diferenciales con problemas con valores en la frontera (A. E. García Hernández, Trad.; 7ª ed.). Cengage Learning Editores.

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