A small lab for coherent waves
Two waves.
One signal.
Turn phase into a picture. Add two rotating arrows, then watch their real projections combine into a single cosine.
Add the arrows
01 / PHASORSThe arrows rotate together at the time cursor.
The faint shifted copy of B completes head-to-tail addition. The sum's horizontal projection is its real signal value.
Read the real signal
02 / TIMEThe same calculation, followed through two cycles.
- COMBINED PEAK Common signal units
- RESULTANT PHASE At t = 0 · [−180°, 180°)
- CYCLE MEAN SQUARE Squared signal units
- ROOT MEAN SQUARE Common signal units
Inspect all 129 model values
Uniform evaluations across two cycles include both endpoints. Displayed numbers use up to six decimal places, with scientific notation for small nonzero values. Downloads retain numerical precision.
| Index | Cycle | Time (s) | Wave A | Wave B | Sum |
|---|
Keep the current experiment
Save the settings and calculated results, or a table of the two-cycle trace.
Settings stay in this tab only. Refreshing returns to the starting example.
Practice the connections
16 original questions connect complex amplitudes, addition, relative phase and the model's boundaries.
In RecallWeave, use Bring your own lesson, choose this JSON file, review the preview, then select Start this deck.
What the picture assumes
Both signals are real cosines with the same positive frequency, constant amplitudes, fixed relative phase and common scalar units. We project onto the real axis; positive phase advances a cosine in time.
ZA = A eiφ · ZB = B ei(φ+δ)
R² = A² + B² + 2AB cos δ
For a nonzero resultant, the sum is one cosine of peak amplitude R. A zero resultant has no defined phase. Its cycle mean square is R²/2; this is a mathematical signal average, not a calibrated intensity. Arrow angle encodes phase, without specifying a direction in physical space.
The curves join evaluations of the stated model. These are calculated values, not measurements; this lab does not model different-frequency beats or a physical propagation medium.
Three things to try
- Rotate together. Keep 3 and 4 a quarter-cycle apart. Change common phase. Which numbers stay the same, and what shifts?
- Cancel, then disturb it. Choose the cancellation preset. Change B to 2.1. Is the remaining signal identically zero?
- Change the clock. Hold amplitudes and phases fixed, then double frequency. Compare the seconds axis with the unchanged phase relationships.
The worked guide develops the calculations and explains the course's answer choices.