Worked example
Trigonometric Interference: sin(x) + cos(2x)
When two waves travel through the same medium, their displacements add together point by point — a phenomenon called superposition. Graphing sin(x), cos(2x), and their sum on the same axes makes this addition visible: at every x, the height of the combined curve is exactly the sum of the heights of the two component curves.
Notice how the sum is not simply a bigger sine wave. The cos(2x) term oscillates twice as fast, so it alternately reinforces and cancels the sin(x) wave. Where both waves peak together, the sum reaches its highest points; where one is at a crest and the other at a trough, they partially cancel. This is the same mathematics behind beats in sound and interference patterns in light.
Plotted expressions
- y = sin(x)
- y = cos(2*x)
- y = sin(x) + cos(2*x)
The link preloads these exact expressions into the calculator — no typing needed.
What to notice
- The combined wave sin(x) + cos(2x) is periodic, but its shape is more complex than either component alone.
- Toggle each expression on and off in the calculator to isolate the contribution of each wave.
- Try changing cos(2*x) to cos(3*x) and watch how a faster second wave changes the interference pattern.