The assignment was simple enough. All I had to do was choose an everyday phenomenon to model graphically using a trigonometric function. My topic could have been any number of things. I could, for example, have modeled the motion of a swing or a point on a tricycle wheel. I decided to make this project into a challenge for myself because I enjoy making sense of the world. A shortcut would have been to do an Internet search on a topic that interests me, but there was much more to gain from trying to figure things out on my own. Even if I failed, I knew I would learn a great deal from exploring incorrect theories.

After weighing my options for several days, hoping that the perfect topic would come, it did, in the form of the Doppler effect. It was everyday, it was elegant, and I wanted to quantify it. In case you are unfamiliar with the Doppler effect, it is the effect of movement on the perceived frequency of a continuously emitted sound wave (or anything that has a tempo). A person can hear this effect in the change in pitch of train whistles or car horns when one quickly passes by. The sound begins at a high pitch as the vehicle approaches, the pitch rapidly drops as the vehicle passes, and as the vehicle speeds into the distance, the pitch is much lower than before. Specifically, I wanted to graph the sound wave perceived by the hearer of this phenomenon in the case of an emitted sound traveling in a straight line as it passed in front of a stationary listener.

Several factors influenced my decision to study the Doppler effect. For one, my father introduced me to the concept at a young age. I have distinct memories of shooting hoops on our street, and as we waited for a car to pass, he explained the Doppler effect by focusing my attention on the noise created by the passing vehicle. Also, sound has always intrigued me, and at the time, sound was something about which I often thought. I imagined sound waves on the molecular level as many molecules bunched up in some areas (the crest of a sound wave) alternating between areas with molecules spaced farther apart (the trough of a sound wave). Knowing what sound waves were like inspired me to imagine how the waves are created on the molecular level. And what about waves mixing together to make chords? I also thought about how the brain can take input from the ear and pick out words and voices from the violent sea of sound waves in a room of conversation or how a single radio speaker can create the music of an orchestra. Having grasped these concepts and with precursory knowledge of what the graph of a sound wave distorted by the Doppler effect should look like, I thought I had a chance to quantify the Doppler effect for myself.

The project soon took on a new pace that none of its predecessors had ever realized. I would work tirelessly, armed with a pencil, paper, and an indispensable computer graphing program to test my theoretical equations. I began to summon up any equations that seemed relevant to my quest and proceeded to combine them in an attempt to create a mathematically precise relationship between time and the correlating point of a sound wave distorted by the Doppler effect. The experience was very much like trying to solve a mathematical jigsaw puzzle. The equations that I had accumulated throughout my schooling were the pieces and my final equation would be the completed puzzle. Naturally, from time to time I would not be able to find a “piece” or see where they could fit together in any productive capacity. I would be frustrated that an aspect of the Doppler effect proved particularly difficult to “fit together” mathematically, but my frustration only made me want to know its secrets even more. At times like these, I could hardly maintain focus enough to follow the plot of a movie, read a paragraph to the end without restarting, or actively participate in a conversation with friends or family.

When the due date arrived, I had all but conquered the Doppler effect. I fell short of absolute victory because the problem required math beyond my level at the time (specifically limits and derivatives for instantaneous velocity), but my final equation had an insignificant margin of error. My graph was a tad off as well (due to the result of a changing frequency value in a sine function), but the final result looked very much as it should have. I explained these deficiencies on my final project. I was not disappointed with my imperfection because I had reached a new level of mathematical and scientific understanding of the Doppler effect and had made so many discoveries over the course of the project. For example, the problem brought me very close to conceptually understanding derivatives through my independent efforts.

If anything, I was disappointed with my peers’ responses to my project. Of course, my teacher recognized my investment of time and effort, and brought it to the attention of my classmates, but even so, none of them were excited about my discovery. None of them were willing to appreciate the caliber of my solution. I showed my finished project to those I considered intellectual equals, but they only glanced briefly over it, said, “wow,” and then moved on to something else. Somehow, they were unmoved to know the mysteries of math and the Doppler effect that so motivated me.

This project has great personal significance for me because it validates my aspiration that I might one day make discoveries in a math and science based career with the same sensations of joy, success, and accomplishment that I experienced while working on this project. I yearn to rediscover the secrets already uncovered in the world’s history of scientific and mathematical progress. I wish to be guided to the edge of society’s knowledge in these areas by the leadership of my college teachers and with the aid of the many skills with which they will equip me. Then I will be able to venture into the unknown and hopefully map new trails through the conceptual wilderness, using and developing technology to solve problems that frustrate and mystify society today.