Introduction: A study done in 2019 by the U.S. Environmental Protection Agency found that "more than half of the Nation's stream miles have ecosystems in poor conditions," ("Water Quality in the Nation's"). As a result of our actions, ecosystems across the globe are facing the impacts of climate change and pollution. Turbidity and pH levels are measures that determine a river's overall water quality. In turn, these measures are affected by different variables including temperature, changes in weather, and flow rate. With these factors at play, actions must be taken to remediate the negative impacts they can have on overall water quality. Through education and awareness, water quality issues can be easily prevented if we know the variables that are causing them. This paper aims to explain how turbidity and pH levels, specifically focusing on a local river in my area, fluctuate in correspondence to seasonal change. This concept can be mathematically demonstrated through the trigonometric function of a sine wave. In classes I have taken at my school, I have listened to various discussions surrounding the numerous applications of sinusoidal waves. However, with my interest in water quality and conservation, I wanted to know if the concept could be applicable to the study of climate and environmental restoration. After doing research, I realized that sine waves potentially have a connection with pH and turbidity levels, both of which fluctuate in response to other variables. Through my research, I wanted to know if fluctuations in pH and turbidity could be graphed through a sinusoidal function, and if this function could help predict changes in water quality. This is the point in my research where I knew I needed to collect data to confirm my hypothesis, so I decided to collect water samples from a local river.

Background: As defined by Merriam-Webster, a sine function is "a waveform that represents periodic oscillations in which the amplitude of displacement at each point is proportional to the sine of the phase angle of the displacement and that is visualized as a sine curve," ("Sine Wave"). However, the strict definition of a sine function was once unknown. Famous Greek mathematician Hipparchus was the first person to develop a table of values for a trigonometric function (Barnard and Maor). However, other major contributions to the branch came from India, dating back to as early as 550 c, as Aryabhata was the first mathematician to develop a table of sine values (Barnard and Maor). But it was not until John Napier's fundamental work in 1614 that the branch was able to develop sine into a logarithmic function. Napier was a Scottish mathematician and theological writer who was able to create tables where "roots, products, and quotients could be quickly determined from tables showing powers of a fixed number used as a base," (Struik). As a result of Napier's contributions, later, French mathematician Joseph Fourier was able to use the functions of sine and cosine to describe and compute any periodic wave function. This mathematical equation became what is now known as the Fourier series. Whilst being introduced to the Internal Assessment for math, I knew that I wanted to do something that relates to my interest in climate and environmental restoration. Within the past few years, there have been numerous examples where scientists and climate activists have used math to help us predict the impact human activity has on the world around us. For instance, one specific example was in 2018, when a group of 90 climate scientists who served on the Intergovernmental Panel on Climate Change found that "if humans don't take immediate, collective action to limit global warming to 1.5 degrees Celsius by 2040, the consequences will effectively be basked into the natural systems of the planet," (Turrentine). Using current and past changes in temperature, scientists were able to predict how long we have until we begin facing extreme backlashes from climate change. However, given that this environmental topic has already been extensively researched, I wanted to shift my focus toward something that has significantly less research: water quality. As I mentioned in my introduction, pH and turbidity are measures that determine a river, lake, or ocean's overall water quality. pH is, "a measure of how acidic/basic water is," and is determined through a scale that ranges from 0 to 14, with 14 indicating that the water is alkaline and zero indicating that the water is highly acidic. With pH being a measure of acidity, turbidity in comparison is "the measure of relative clarity of a liquid. It is an optical characteristic of water and is a measure of the amount of light that is scattered by material in the water when a light is shined through the water sample," (Water Science School). Turbidity determines the number of particles that can be found in a water sample and is measured through nephelometric turbidity units, or NTU. Nephelometric units determine, "the concentration or particle size of suspensions by means of transmitted or reflected light," ("Nephelometer"). Both elements are of importance when determining a river's overall water quality. For reference, I needed to have base numbers regarding pH levels and turbidity levels with river quality. It is important that when determining the quality and health of the river, I have comparative numbers I can observe my data with. For river water, NASA found that "the optimum pH ... is around 7.4," as elevated levels in pH "can make a river inhospitable to life," ("Water Quality"). Regarding normal turbidity levels, during a period of low flow turbidity should be "less than 10 NTU," (Water Science School). During periods of high flow, it can be found that a river is going to have higher turbidity levels, as increased flow leads to an increased amount of debris. Sinusoidal Waves: For my project, I wanted to focus on the concept of sinusoidal waves, as they mirror a continuous cycle. As seen in the picture below, a sinusoidal wave is easily identifiable with its s-like shape, amplitude of one, and period of 2π. It is defined as a continuous function, denoting that it has no specific beginning or end. Furthermore, it is a trigonometric function, meaning it is a "function...of an arc or angle most simply expressed in terms of the ratios of pairs of sides of a right-angled triangle," ("Trigonometric Function"). Simply put, it is a function of an angle. There are six trigonometric functions, being: sine, cosine, tangent, secant, cosecant, and cotangent. For the intents and purposes of this paper, I will be focusing on the sine function, as this is the most applicable in the context of water quality. Any sinusoidal function can be modeled through the following base equation: y=asin⁡(((x-h))/b)+k In the equation, "a" models the amplitude of the function, "h" models the horizontal shift, "b" models the frequency, and "k" models the vertical shift. For example, theoretically, let us say that I wanted to create a sinusoidal function with a period of 4π, and that has been vertically shifted down by 5. We would be able to represent that sinusoidal function with the following equation: y=sin⁡�-(x/2)-5�--.

In a theoretical sense, I knew that there were thousands of different variations of sinusoidal waves. However, I was curious as to how it could be applied to real-life applications, specifically focusing on water. That's when my research came to the following question: can a sinusoidal function be used to demonstrate cyclical occurrences in real life?

Real-life Application: To create consistency within my data collection, I first needed to know what outside variables might influence my data results. After conducting research, I found that the outside temperature could play an impact on my data. As a result, I wanted to account for the outside temperature for every water sample I collected. Subsequently, I created a data table using a Microsoft Excel spreadsheet where I could record my data. In my data table, I created columns where I could record the date, time, flow rate, turbidity level, pH level, and temperature. To measure the turbidity and pH levels, I used Vernier Software and Technology sensors. For my research purposes, I chose to collect my water sample from the same spot on the river every time. I collected all my data at the spot seen in my thumbnail image. I tried to collect a water sample once a week and measured the pH and turbidity levels using my tools and calculator. I collected samples for 8 months starting from April through December, with my first beginning date being the third of April.

The Relationship Between Temperature and Turbidity: After forming the table, I started analyzing my data points to find trends/patterns across different fields. Upon closer look, I found a strong correlation between temperature and turbidity levels. The trend I found is that as temperature increased, the turbidity levels decreased. While there are a few data points that were outliers, for example, 64 degrees Fahrenheit, 39.488 NTU, it can be found that the points continually decrease with increasing temperatures. Therefore, it can be inferred that turbidity patterns can be determined throughout the seasons. As an example, it can be predicted that turbidity levels would increase during the fall/winter seasons as the temperature decreases.

Modeling an Equation for Temperature and Turbidity Levels: To determine a sinusoidal relationship with my data, I first began by graphing all the points that I had collected throughout the nine months of my research. I did this through an online graphing calculator platform titled Desmos, as seen in the image below. On the x-axis, I graphed the month in which the data was taken, given that this was the independent variable in my research. For the y-axis, I graphed the turbidity level in NTU. Furthermore, given that I took multiple samples each month, I separated my points using decimal places depending on the week in which they were taken. I used decimal places of 0.2 between each point, given that there are four weeks in each month. For both my x and y-axis on the graph, I used intervals of five. After graphing my data, I found that it modeled, as I predicted, the periodic function of a sine wave. As we can see in the graph. The data taken in the spring had significantly higher turbidity levels in comparison to that of the one taken in the summer and fall months. This models the graph seen in Figure 3.2, where turbidity peaks during the winter and spring months and decreases during the summer and fall months.

I knew that I wanted to create an equation using the sine function that predicts how turbidity levels will increase/decrease throughout the course of the year. To do this, I started with the basic equation for the function, and this can be seen in the equation below: y=sin⁡x

As previously mentioned, I knew that there were four variables that affected the function: amplitude, frequency, horizontal shift, and vertical shift. Given that the highest turbidity level recorded was 41.181 NTU in April, I knew that I needed to vertically shift my function upwards to match my data. The lowest turbidity level I recorded was 21.555 NTU in September, so I knew that my function needed to fall between 20 and 45. Given that my first data points were not taken during the peak of the winter and spring season, I knew that the amplitude of the function needed to be a bit higher than these points, to indicate that turbidity levels were then decreasing after reaching a peak level. Thus, I decided to vertically shift the function upwards by 33, changing the "k" variable in my equation, to be in line with the gradual decrease in turbidity levels. This is seen below: y=sin⁡x+33

After shifting my function upwards, I knew that I needed to change the horizontal shift so that the function fit more in line with the curve of my data. Thus, I changed the "h" variable of my equation to three to shift the graph over to the left so that it modeled the curve of my data points. This can be seen in the equation depicted below: y=sin⁡(x+3)+33

Given that the current amplitude of my function was one, I knew that I needed to change the "a" variable so that the amplitude modeled the range between my points. The highest turbidity level recorded throughout my research period was 41.181 NTU taken in April and the lowest was 21.555 NTU taken in September. Thus, I knew that I needed to model the amplitude based on the difference between the crest and trough of my data. I chose to change the amplitude of the function to 12 so that both the maximum and minimum points would be included. This is shown below: y=12 sin⁡(x+3)+33

The last step that I took to get the function to match the curve of my data was to change the frequency or the "b" variable. Given that my data was spread out between the months of April through December (a range of 8 months), I knew that the function needed to oscillate one full-time. This would show how turbidity levels decrease during the spring and summer months and increase during the fall and winter months. Thus, I decided to change the frequency in my equation to 2.6 to match the curve of my data. This can be seen in the equation below: y=12 sin⁡((x+3)/2.6)+33

Evaporation and Turbidity: After analyzing my data and finding a connection, I asked myself the following question: Why is there a relationship between temperature and turbidity? When I first thought about this question, I was confused as to why there was a relationship between the two factors. In theory, they should be completely unrelated. However, I started thinking about what turbidity was: the measure of the number of particles in each sample. At that point, I came to the following conclusion: evaporation. It has been seen that, "evaporation rates are higher at higher temperatures because as temperature increases, the amount of energy necessary for evaporation decreases," ("Evaporation and Climate"). As a result, I predicted that turbidity rates decrease as temperature increases because the evaporation rate increases proportionally to the temperature. With increased evaporation, there will be less particle matter left in the water.

Water Treatment Applications: Water treatment and care are vital to maintaining healthy ecosystems for aquatic life. There are certain standards for chemical composition, pH, turbidity, etc. that must be met when maintaining the quality of a river. If these standards are not met, there are adverse effects on the aquatic life that inhabits the river. As a result, it is of the utmost importance that water treatment is looked at closely so that the perfect ratio of chemicals can be put into a river to maintain overall health. However, it can be hard to predict what treatment is necessary considering ongoing changes in weather patterns, temperatures, flow rates, etc. But, as we have seen earlier in this paper, it is possible to monitor certain factors of river quality with mathematics. Using sinusoidal waves, we can see the pattern of temperature and turbidity throughout the year. Knowing the relationship between temperature and turbidity allows us to determine what treatment is going to be needed for the river. Therefore, mathematics allows us to predict necessary treatment before the quality of a river is adversely affected. The relationship between temperature and turbidity not only applies to aquatic life, but also to humans. Turbidity is a measure that determines if water is safe for us to drink. It is monitored closely to ensure that we do not get sick from drinking contaminated water. With that, it is possible that the mathematical relationship between temperature and turbidity can be applied to the overall quality of water for drinking purposes. Drinking water can become more accessible when we are able to predict what treatment is necessary to maintain its overall quality. Considering ongoing changes, we can predict the necessary treatment and prevent people from drinking unsafe/contaminated water samples. This mathematical relationship can alleviate many of the worries people around the world have regarding safe drinking water.

Conclusion: Through this paper, I found how seasonal changes can affect water quality, and how these fluctuations in turbidity can be monitored through the trigonometric function of sine waves. Throughout this paper, I have demonstrated how mathematics can be applied to river quality and treatment. In my real-life application, I measured pH and turbidity levels in a local river for eight months. As a result of this study, I found a strong correlation between temperature and turbidity. I found that as temperature increases, turbidity decreases. Because of this finding, I learned how the relationship between the two factors can be applied to water treatment, for both aquatic life and drinking water. Through sinusoidal waves, we can predict changes in turbidity levels to provide the most effective water treatment.