Oscillation: Definition, Theory & Equation

Instructor: Aaron Miller

Aaron teaches physics and holds a doctorate in physics.

In this article you will learn about mathematical functions that are used to model oscillations. An oscillation is any motion that repeats itself, and the function describing a particular oscillation is an example of a periodic function. Sine and cosine are the most common periodic functions.

Examples of Oscillations

What do the pendulum of a grandfather clock, your heart as you sit and read this article, and the Earth in orbit around the Sun have in common? Hint: the answer is related to a characteristic of the motion of these three very different systems. All three scenarios involve repeating motion -- the pendulum swings back and forth, the muscular walls of your heart's chambers contract during each beat, and the Earth follows the same path around the Sun each year. More specifically, these are examples of oscillations. An oscillation is defined as a motion that repeats itself.

What Do Graphs of Oscillations Look Like?

One of the easiest ways to identify an oscillation is by examining a graph of a variable characterizing the location or state of system as it changes with time. Here is an example of the graph of a function that describes a simple oscillation:

Figure 1: A graph of a simple oscillation.
Figure 1: A graph of a simple oscillation.

The independent variable on the horizontal axis of this graph is time, and the dependent variable on the vertical axis is a variable characterizing the system, like its position. Accordingly, this graph shows that as time advances, the variable quantity on the vertical axis (which might be, for example, the position of the pendulum in a grandfather clock) cycles across the same values in a predictable, repeating pattern. This is how you can identify, from looking at the graph, that this is an oscillation.

Not all oscillations are simple, like the example in Figure 1. In fact, most oscillations in the real world are not. Here is an example of the graph of another oscillation. Can you spot the repeating pattern in the motion?

Figure 2: A graph of a non-simple oscillation.
Figure 2: A graph of a non-simple oscillation.

In Figure 2, even though the system is not characterized by a simple 'back-and-forth-and-back,' it is still an oscillation because the variation over time of the dependent variable repeats after a predictable amount of time. One repetition is labeled with a horizontal red line and a T. We will give T a name and definition in the next section.

Finally, it is important to discuss what sorts of variables describing a system might oscillate. A common example is the position coordinate of the center of the system. The position is a variable that oscillates in a pendulum system or in the orbital motion of Earth around the Sun; however, in cases like the heart, where the oscillating object is stationary, blood pressure in a particular chamber would likely be the variable you could graph to observe the characteristic repeating pattern of the oscillation. The important point here is that position of an object is not always the variable that is oscillating in a real system.

Periodic Functions

We just learned we could identify an oscillation by looking for a repeating pattern in the graph of a variable capturing the state of a system under consideration. There is a special class of functions whose graphs exhibit a repeating pattern, and therefore, are the best functions for modeling the behavior of an oscillating system. These are called periodic functions. The mathematical definition of a periodic function basically translates the visual procedure for identifying a periodic function, which we discussed in the previous section, into a symbolic definition. Specifically, a function f(t) is a periodic function if and only if for some number T called its period, the function obeys the condition, shown in Figure 3, for all t in the domain of the function.

Figure 3: General condition that defines a periodic function.
periodic_condition

What this definition means in practice is that periodic functions repeat their pattern over a time interval equal to the period T. Furthermore, it doesn't matter at what value of t in the domain of the function you start looking for the pattern, the period is the same. This definition gives major significance to the quantity T: the period is the time required for a motion to repeat itself. Take a moment to look back at Figures 1 and 2 and see if you can identify the period visually. What is the period in time units along the horizontal axis? No matter where you start your visual measurement of the interval required for the pattern to repeat, you should see that one repetition happens over 2 units of time.

Simple Harmonic Functions

There are many examples of periodic functions. In fact, because the period of a function can be any positive number, we can define an infinite number of periodic functions. On top of that, the pattern in the graph that repeats can take on any shape. For example, this is a periodic function called a 'square wave':

Figure 4: A graph of a square wave.
square wave

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