How does the amplitude of the resultant wave behave when two equal-frequency sinusoids with arbitrary phase difference superpose?

Conditions

• Two sinusoids have equal frequency.
• There is an arbitrary phase difference $\phi$ between them.
• Amplitudes are non-zero ($A_1=2, A_2=0.8$ in the demo).

Reasoning, step by step

1. Observe the bottom coordinate system demonstrating superposition with an arbitrary phase difference.
2. Note that the blue wave ($A=2$) combines with a phase-shifted red wave ($A=0.8$).
3. Analyze the resulting green wave, which exhibits changes in both amplitude and phase position.
4. Recall the general formula for the resultant amplitude: $A=\sqrt{A_1^2+A_2^2+2A_1A_2\cos\phi}$.
5. Conclude that the amplitude varies continuously depending on $\phi$, staying within the bounds defined by complete constructive and destructive interference.

Example

From 00:14 to 00:21, the bottom coordinate system shows a blue wave ($A=2$) combining with a phase-shifted red wave ($A=0.8$). The script notes the production of a green wave exhibiting changes in both amplitude and phase position. The card provides the formula $A=\sqrt{A_1^2+A_2^2+2A_1A_2\cos\phi}$.

Common misconceptions

• Believing that the equilibrium position shifts during superposition (it does not; only amplitude and phase change).
• Assuming the resultant amplitude is simply the arithmetic mean of the two amplitudes regardless of phase.

Watch the explanation

Explore next

• What specific value of the phase difference $\phi$ maximizes the resultant amplitude?
• Why does complete cancellation require equal amplitudes in addition to anti-phase alignment?