What is the resultant amplitude when two equal-frequency sinusoids with an arbitrary phase difference are superposed?

Conditions

• The two waves have the same frequency.
• The two waves have an arbitrary phase difference $\phi$.

Reasoning, step by step

1. Identify the amplitudes $A_1$ and $A_2$ of the two individual waves.
2. Determine the phase difference $\phi$ between the two waves.
3. Apply the formula $A=\sqrt{A_1^2+A_2^2+2A_1A_2\cos\phi}$ to calculate the resultant amplitude.

Example

The bottom coordinate system demonstrates superposition with an arbitrary phase difference. The blue wave ($A=2$) combines with a phase-shifted red wave ($A=0.8$), producing a green wave that exhibits changes in both amplitude and phase position.

Common misconceptions

• Believing that the resultant amplitude is always the sum of the individual amplitudes.
• Assuming that the phase difference only affects the position of the wave, not its amplitude.

Watch the explanation

Explore next

• How does the phase difference $\phi$ affect the resultant amplitude?
• What happens to the resultant amplitude when the phase difference is zero or $\pi$?