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Ripples in a tranquil water body

The Enchanting Dance of Ripples: A Mathematical Exploration


The captivating sight of ripples spreading across a pond after a stone is dropped has intrigued scientists and mathematicians for centuries. Delve into the elegant dance of nature’s forces as we explore the multiscale mathematical model behind the formation and propagation of these mesmerizing ripples.

Governing Equations and Assumptions

We begin our exploration by introducing the Navier-Stokes equations for incompressible fluids, which serve as the choreographer of the dance:

(1)   \begin{equation*} \rho\left(\frac{\partial \boldsymbol{u}}{\partial t} + \boldsymbol{u} \cdot \nabla \boldsymbol{u}\right) = -\nabla p + \mu \nabla^2 \boldsymbol{u} + \boldsymbol{F}  \end{equation*}

(2)   \begin{equation*} \nabla \cdot \boldsymbol{u} = 0  \end{equation*}

To simplify our problem, we make a few key assumptions, such as considering the water as an ideal, incompressible fluid with negligible viscosity.

Linear Wave Theory and Dispersion Relations

To describe the water surface perturbations, we introduce the velocity potential \Phi and the free surface elevation \eta:

(3)   \begin{equation*} \Phi(x, y, z, t) = \Phi_0(x, y) \exp(ikz - i\omega t)  \end{equation*}

(4)   \begin{equation*} \eta(x, y, t) = \eta_0(x, y) \exp(-i\omega t)  \end{equation*}

Applying boundary conditions and substituting these expressions into the Navier-Stokes equations yields the dispersion relation:

(5)   \begin{equation*} \omega^2 = gk + \frac{\gamma k^3}{\rho}  \end{equation*}

Ripple Formation and Propagation

The initial disturbance generated by the stone can be described by a Fourier series, which represents a symphony of waves with different wavelengths and directions:

(6)   \begin{equation*} \eta_0(x, y) = \sum_n A_n \exp[i(k_n x + l_n y)]  \end{equation*}

The interference of these waves leads to the formation of multiple ripples. Dispersion plays a crucial role in the propagation, causing ripples to broaden as they move away from the impact point.

Numerical Simulations and Results

Through numerical simulations, we unveil the following insights:

  1. The formation of multiple ripples is primarily due to wave interference, as depicted in the Fourier series representation (6).
  2. Dispersion significantly impacts ripple propagation.
  3. The stone’s size, shape, and impact velocity influence ripple amplitude and pattern.
  4. Surface tension and fluid density affect the dispersion relation (5) and the ripple propagation speed and pattern.

What Does It All Mean

The phenomenon of ripples forming on a pond’s surface after a stone is dropped can be likened to the delicate dance of nature’s forces. Imagine the water as a vast, serene field of wheat, with each stalk representing a fluid particle. When a gust of wind blows across the field, the wheat stalks sway gracefully, interacting and influencing each other, much like the water particles responding to the stone’s impact.

The Navier-Stokes equations, in this context, act as the choreographer of the dance, dictating how the fluid particles move and interact. These equations describe the behavior of the fluid with respect to its density, velocity, and pressure, much like the tempo, rhythm, and intensity of the wind that sweeps across the wheat field.

Viscosity, a measure of a fluid’s resistance to flow, can be thought of as the thickness of the syrup that coats the stalks of wheat, making it harder for them to sway in response to the wind. In our mathematical model, however, we assume that the water is an ideal fluid with negligible viscosity, akin to the stalks of wheat moving freely without the sticky syrup coating.

Dispersion, another crucial factor in ripple formation, can be compared to the spreading of seeds carried by the wind. Shorter wavelengths, similar to smaller seeds, are more easily scattered and spread by the wind, resulting in a broadening of the ripples as they move away from the impact point.

The Fourier series representation of the initial disturbance is like the various musical notes of an orchestra, combining harmoniously to create a rich and intricate symphony. Each term in the series corresponds to a wave with a different wavelength and direction, and their interference with one another gives rise to the beautiful pattern of multiple ripples on the water’s surface.


Our mathematical model unravels the complex interplay of forces and factors responsible for the enchanting dance of ripples on a pond. The Navier-Stokes equations, acting as the choreographer, guide the water particles in their captivating spectacle as they respond to the stone’s impact. This deeper understanding of the underlying mechanisms has potential applications in fluid dynamics, remote sensing, and environmental monitoring.

For More Information

If you are interested in learning more about the mathematical modeling of ripple formation and propagation or fluid dynamics in general, consider exploring the following resources:

  1. Applied Hydrodynamics: An Introduction to Ideal and Real Fluid Flows by Hubert Chanson: This comprehensive textbook offers a detailed introduction to fluid dynamics, covering topics such as ideal and real fluid flows, surface waves, and wave propagation.
  2. Water Wave Mechanics for Engineers and Scientists by Robert G. Dean and Robert A. Dalrymple: This book provides an extensive discussion of water wave mechanics, including the theory and applications of water wave propagation, wave-structure interactions, and coastal processes.
  3. Online courses on fluid dynamics and applied mathematics: Many universities and online platforms offer courses on fluid dynamics, applied mathematics, and related topics. Coursera, edX, and MIT OpenCourseWare are great starting points to find suitable courses.
  4. Research articles and journals: To stay updated on the latest advancements in the field, consider following research articles and journals such as the Journal of Fluid Mechanics, the Journal of Applied Mathematics, and the Annual Review of Fluid Mechanics.
  5. Scientific conferences and workshops: Attending conferences and workshops in fluid dynamics and applied mathematics provides opportunities to learn from experts, present your research, and network with other professionals in the field.