s(t) = s1(t) + s2(t)
Hover over or drag a slider to see what it changes. Drag across the graphs to move through time, or use the arrow keys.
t = 0.00 s:
0.00 + 0.00 = 0.00
Start from:
Wave 1
Wave 2
Adding the arms
Each wave is an arm turning once per cycle. Placed tip to tail, they make the dark arm. Its height, read off the vertical line, is the oscillating part of the sum.
Phase difference now:
0° lined up
Mystery sum challenge
A green band appears on the sum graph. Build two waves that add up to it.
Things to try
- Make the two waves identical. Predict the amplitude of the sum before you look.
- Slowly slide φ₂ from 0° to 180°. Watch the arms. When does the sum vanish completely, and what has to be true about A₁ and A₂ for that to happen?
- Pause at a moment when the sum is close to zero. What do the two stacked arrows look like?
- Choose “Beats”. How long is it between the big swings? Compare with 1/|f₁ − f₂|.
- Predict first: if you move f₂ closer to f₁, do the beats come faster or slower? Then try it.
- Change C₁ and C₂. Does the baseline of the sum do what you expect?
- Choose “Double frequency”. Is the sum still a simple cosine? What repeats, and how often?
- Turn on damping and give one wave much more damping than the other. Which wave dominates the sum at late times?
- Try a mystery sum. Is there more than one pair of waves that matches it?
Superposition: when two waves reach the same point in a linear medium, the signal there is the sum of the two signals at every instant. Nothing about either wave changes; only the total does. The arms ignore C₁ and C₂ because a baseline doesn’t oscillate. It just shifts the sum up or down.