Many of the sounds we hear are periodic. When these sounds occur, vibrations repeat the same motion over and over again. One repetition of the vibration is called a cycle. Each cycle occurs over a consistent length of time called a period. Conceptually, a period is the time it takes for a signal to complete one cycle. To represent a signal’s period, we will use the greek letter tau, $\tau$.

$\tau = \frac{time}{1 cycle}$

A related concept to period is a signal’s frequency, which is the number of cycles per second. Therefore, frequency is the inverse of period. To represent a signal’s frequency, we will use the letter, $f$. The unit of frequency is called Hertz $(Hz)$.

$f = \frac{cycles}{1 second} Hz = \frac{1}{\tau}$

Below is a demonstration of different types of signals created from cyclical motion. In each case, the time it takes to complete a cycle (frequency) is the same. However, variations of the path for each cycle create different waveforms.

Several mathematical functions can be used to synthesize different periodic signals. Let’s begin by looking at the sine function which will lead us to synthesizing a sine wave.

Photo credit