Sum of Sines

Lesson notes

Overview

Adding waves of different frequency creates complicated shapes — the seed of Fourier analysis. This topic is rated intermediate and is number 53 of 245 in Mathematics. Builds on the basics with relationships, methods and trade-offs.

What you are seeing

The panel above shows a graph of the function. Each slider changes one parameter, so you can see exactly how the curve bends, shifts or stretches. The animation is drawn live from the model, so everything you see responds to the sliders. Press Space to pause, use Restart to replay from the start, and change the speed to slow down quick events.

In depth

Adding sine waves of different frequencies produces a more complicated periodic shape. The waves reinforce each other where their peaks line up and cancel where a peak meets a trough. This is the principle of superposition. Fourier showed that nearly any periodic signal can be built from sines in this way, so sums of sines underlie sound synthesis, signal processing and image compression.

Reading the relationship

The expression "sin(x) + a·sin(bx)" is a compact summary of the idea. It uses the symbols x, a, b; name what each one stands for before you start. Notice that it contains a trigonometric function, so the result repeats as the angle goes round. Then use the sliders to see how changing one quantity changes the others.

A guided exploration

1. Set every slider to its middle value and note the shape of the curve. 2. Move one slider at a time and describe whether the curve shifts, stretches or turns. 3. Find the value where the behaviour changes sharply, then connect it to the formula.

Try this

Predict the outcome first, then test extreme slider values to find where the pattern changes or breaks.

Common pitfalls

A frequent slip is memorising the rule without the picture. If a result surprises you, sketch the graph or test a simple number by hand before trusting the algebra.

How it connects

It builds on “Damped Oscillations”, the topic before it in Mathematics. Once you are comfortable, continue with “Law of Sines and Cosines”.

Why it matters

Mathematical ideas like this one are the language that physics, engineering, computing and statistics are written in.