Poster Presentation
Fourier Series
Josh Davis
Faculty Mentor: Dr. Marko Kranjc
Mathematics
We explore the vibration of a string. The simplest possible stationary vibration of a string that is attached on both ends looks like a simple sine curve with the nodes at the endpoints. The frequency of the vibration in this case is called the fundamental frequency.
In general, there could be several stationary waves such that the corresponding nodes divide the string into intervals of equal length. The corresponding frequencies are called the harmonic frequencies and are integer multiples of the fundamental frequency. The waves in these cases again look like sine curves.
Therefore it is natural to try to describe vibrations of a string as combinations of trigonometric functions. Combinations of trigonometric functions lead to the notion of
Fourier series. We show how Fourier series lead to solutions of the string equation.
Since every solution of the string equation can be written as a Fourier series, the natural question to ask is if every “nice enough” function can also be written as a Fourier series.
We define the Fourier series of a function as the orthogonal projection of a function onto the subspace spanned by appropriate trigonometric functions. In order to do that, we use the notion of inner product. We show how simple geometric idea of projection leads to the dot product in the usual two and three-dimensional spaces and how this motivates the definition of the inner product in a general setting.