Phasor diagram

Parametric equations

A phasor diagram writes the superposition of waves as rotating vectors on the complex plane. Each harmonic wave is a vector centered at the origin, turning counterclockwise at constant angular speed. When two waves meet, the combined displacement is their vector sum, with the two vectors joined head to tail.

• A and B are the lengths of the two rotating vectors, so they set each component’s amplitude. In this figure ${\omega }_A=1$, and the second amplitude is fixed at $B=0.8$.
• ω_A and ω_B are the angular speeds, the frequencies. When ${\omega }_A:{\omega }_B$ is an integer ratio, for example $1:3$, the endpoint often traces a closed, symmetric curve.
• δ is the initial phase difference of the two vectors. Changing $\delta$ twists the combined curve continuously and does not change the frequency ratio.

Interaction

• Amplitude A: drag the slider to change the length of the first rotating vector. The combined trace scales with it, and the view follows the largest radius $A+B$
• Frequency B: switch the integer ${\omega }_B$ and watch the endpoint trace change from a simple closed figure to a more knotted one
• Phase δ: change the initial phase difference continuously and watch the combined curve twist and its symmetry shift

What to notice

• A harmonic wave can be read as a vector rotating at constant angular speed on the complex plane. Superposition is vector addition, the same diagonal as the parallelogram.
• When ${\omega }_A:{\omega }_B$ is a simple integer ratio, the endpoint trace closes more readily. The phase $\delta$ mainly changes how rotated the figure looks.
• The faint history marks the recent stretch of the combined path. The vector chain at the current time shows both components and the combined endpoint together.

Related

• Lissajous curve
• Euler’s formula, rotating
• Geometric significance of complex arithmetic
• Superposition of trigonometric functions and wave interference

Further reading

• Phasor (Wikipedia)