Waves in Nature Through Particle and Fractal Simulations
by Andrei Barbu, Daniel Gavrila, Justin Teixeira and Charles-Antoine Vézina
Introduction to Ocean Waves in Nature
A wave is created by a disturbance and travels through a medium, away from its origin. Waves transfer an amount of energy given by the initial shock through the medium it is into. Waves can be observed everywhere, in music with sound waves, in light with light waves and more. However, for the purpose of the research we will observe waves in the ocean and how the are created. (UT Dallas Education , 2015)
Ocean waves, also known as wind waves, use fluid dynamics and are usually generated by wind blowing on a free area of water. When directly generated and affected by local winds, a wind wave system is called a wind sea. After the wind ceases to blow, wind waves are called swells. However, bigger waves can occur when two tectonic plates rub together and transfer energy to the water.
Waves can travel thousands of kilometers before hitting land. Multiple factors can affect a wave’s amplitude or period. The two main factors that can affect a wave are : the wind and the shore.
Wind
There are mainly two types of winds in surfing : onshore and offshore. Offshore wind is a wind traveling from the land towards the water while onshore is a wind traveling from the water towards the land. These winds heavily affect the period, the peak and the surfing style of the waves.
Onshore : Shorter period, breaking faster, more dangerous, shortboards are usually used to surf these conditions.
Offshore : Longer period, breaking continuously for a long period, easier to surf, for all types of surfboards.
Shore
The shore is the most important and most effective aspect to the formation of a surf wave. The angle of the slope of the shore in addition to the constitution of the soil affects wave formation like no other. In general, the higher the angle of inclination of the bottom, the more the wave will break and will have more power and push. (Fairclough, 2017)
Introduction to Fractals
Have you ever realized the self-similar patterns observed in snowflakes, tree and even coast-lines? In fact, these characteristics define a fractal, which is defined as a curve or geometric figure, each part of which has the same statistical character as the whole.
This word was coined by the mathematician Benoit Mandelbrot in 1975 based on its Latin origin meaning broken or fractured. However, the notion of fractals has been investigated by mathematicians from the late 17th century. Indeed, the German mathematician and philosopher Gottfried Leibniz stumbled upon fractals during his study on recursive self-similarity, however he and many other mathematicians of his period were unable to understand this concept with its current level of geometry.
Therefore, they simply used the term fractional exponents and referred to it as mathematical monsters (Jones 1991). It is not until Karl Weierstrass published his paper at the Royal Prussian Academy of Sciences in 1872 that a function with a graph of a fractal was displayed. Weierstrass function is the first example of a function that has the non-intuitive property of being everywhere continuous but nowhere differentiable and it is illustrated below (Zaleski 2012).
Not long after, many mathematicians such as Georg Cantor, Helge Von Koch, Wacław Sierpiński, and many others have started creating functions and geometric figures that show the fractal behaviour.
However, fractals were popularized by Benoit Mandelbrot, who wrote about self-similarity in papers such as “How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension” (Mandelbrot 1982). He was also the first to use computer graphics to illustrate fractals and its properties such as the famous Mandelbrot set which gave popularity to this notion.
Simulated Waves as Fractals
For our fractal simulation of ocean waves in a nature setting, we used Iterated Function System or IFS fractals. IFS fractals are created on a basis of simple plane transformations. These transformations include: scaling, translations, rotations, dislocations, etc. To build an IDS fractals we have to follow 5 steps:
1. Defining a set of plane transformations,
2. Drawing an initial pattern on the plane (any pattern),
3. Transforming the initial pattern using the transformations defined in first step,
4. Transforming the new picture (combination of initial and transformed patterns) using the same set of transformations,
5. Repeating the fourth step as many times as possible (in theory, this procedure can be repeated an infinite number of times).
The ocean wave fractal was created using JavaScript code including a p5 library and was built as an IFS fractal.
In the first step in which we have to define a set of plane transformations, we use the rotation and translation transformations.
In the second step in which we draw an initial pattern on the plane, we use a simple line of 150px for the low tide waves and 200px for the high tide waves. This line is drawn blue to represent the color of water. The code illustrated bellow demonstrates how these parameters can be changed to represent different sizes of an initial pattern of an IFS fractal.
In the third step in which we transform the initial line pattern using the rotation and translation transformations, we create a line identical to the initial pattern. We then translate this second line to the tip of the first one and rotate it at any specified angle. The code bellow illustrates how these parameters can be changed to represent different angles of rotation and translation coordinates.
In the fourth and fifth step in which we apply this IFS fractal transformation at multiple iterations, we end up forming a fractal that resembles an ocean wave in nature as illustrated bellow.
Furthermore, we can easily determine the amount of fractal iterations to complete a tidal crash at any rotation angle independent of the tidal length. For a tide to crash in the fractal simulation, the last iteration should end at an angle of 180 degrees, ending at the initial y position. This means that the formula would be:
n = 180/θ
WHERE
n = number of iterations
θ = angle of rotation
In the end, if we complete a full fractal representation of a general ocean wave by designing the base of the wave collapsing at a single point, we can end up with a representation as such:
Simulated Waves in a Water Tank
For this part of the research we decided to make a real-life wave simulation and compare it to the fractal waves simulation. To do so, we used plywood as the walls and the bottom of the tank and wanted to use plexy-glass as the transparent wall, so I can see the wave. Plexy-glass was to expensive, so we decided to use transparent plastic bag instead. After filming our waves we saw that the video was not that good of a quality so instead, we used a YouTube video of someone doing the same simulation and analyzed his video.
Tracker Video Analysis
In the following video we analyzed waves generated in a small water tank. We observed that from it’s initial point, the waves gain in height as it advances in the water tank. In addition we observed, with our simulations, that as you increase the angle of the incline plane, the waves gain in height and are breaking faster.
Comparison with Ocean Waves Fractal Simulation
As seen in the following pictures, our wave simulation matches the fractal wave generated by the code. This further proves that waves in nature hold a certain linear fractal pattern guided by an angle of rotation.
Simulation of a Wave as a Particle
Waves can be well represented by water particles following certain paths. As demonstrated in the video showing multiple waves created by using this method, an offset between every circle is necessary to create the peaks and troughs of a wave. This offset also dictates the direction in which the wave is travelling. An offset is created by changing the x and y components of a water particle along a predetermined path. If the y component of a water particle is at its highest, the particle is at the peak of the wave. Inversely, if the y component of a particle is at its lowest, the water particle is at the trough of the wave. This is in fact a sine or cosine graph, depending on its y component at the origin, in relation with the horizontal position.
A simple equation can be built to illustrate a water wave with its vertical height as well as its movement on the horizontal axis.
The amplitude (A) of a wave is obtained by the difference between the height of the peaks of a wave and the normal water level if there were no waves. In the previously shown video, the middle of the circles, or the point where the water particles are rotating about, is the normal water level with no wave. Since sine and cosine “reset” after 2π, we need to multiply 2π by the x-position divided by the wavelength of the wave minus a point in time divided by the period of the wave. The wavelength of a wave is easily known by calculating the distance between two peaks or two troughs. In other words, to make a connection with the video, the wave length is the distance on the x-axis for a water particle (white dot) to complete a full rotation. Finally, to simulate a wave moving along the horizontal axis, the period (T) of a wave has to be included in the equation. The period defined by the time it takes for a water particle to complete a rotation.
Conclusion
In conclusion, if you currently ask graphic designers if they have profound understanding in math, they would probably answer “no”. This is because they don’t understand all the mathematical relations that are present in their drawings such as simply having symmetry in their patterns. Also, popular designs such as the apple logo uses mathematical notions, in this case the golden ratio. However, many designers simply use particles or shapes to draw figures such as clouds, trees, waves and many more. As Mandelbrot said: “…clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line…” (Mandelbrot 1982). In fact, these objects in nature can not simply be described by a shape since they have a fractal-like pattern associated to them.
That is why we decided to compare our own research into two categories: fractals and particles. To do so, we decided to create ocean waves with the help of these two notions and compare them. When comparing both, we can see that when we simply use particles we simply get a plain wave that looks like a sinus function, whereas when using fractals, we can see some of the patterns of the wave when crashing on the land. In addition, the fractal wave respects the properties of a real wave, whereas the particle wave simply follows a continuous trend. Finally, to further this research we could investigate other figures in nature such as snowflakes, trees, lightning … Otherwise, we can explore fractals in other domains such as architecture (Eiffel Tower, Cathedral towers), in art (Salvador Dali, Jackson Pollock), in movie effects (“Titanic”, “Star Trek II”), video games design and many others.
Bibliography
Edyta Patrzalek , “General Introduction to Fractal Geometry”, Stan Ackermans Institute. 2016
Fairclough, Caty. “Currents, Waves, and Tides: The Ocean in Motion.” Ocean Portal | Smithsonian, Smithsonian’s National Museum of Natural History. 2017
Jones,H. “Fractals Before Mandelbrot A Selective History” Fractals and Chaos. pp 7–33. 1991
Mandelbrot, B. “Fractals and the Geometry of Nature” W.H. Freeman and Company. 1982
Science class is great. “Ocean Waves (Part 1): Wave Structure & Formation.”, YouTube. 2012
UT Dallas Education, “Ocean waves”. 2015
Zaleski, A. Fractals And The Weierstrass-Mandelbrot Function. Rose-Hulman Mathematics Journal. Art. 7, Vol.13, Iss.2. 2012