Superposition and interference
Add displacements and use phase to explain reinforcement or cancellation.
What you will learn
- Apply the principle of superposition.
- State the conditions for complete cancellation.
Waves overlap by adding displacement
When waves overlap in a linear medium, the resultant displacement at a point is the sum of the individual displacements there. Displacements in the same direction reinforce; opposite displacements reduce one another.
Coherent waves have the same frequency and a constant phase difference. This permits a stable interference pattern. Coherence does not require that the waves are in phase.
Amplitude and phase both matter
Equal-amplitude waves exactly in phase can produce a resultant amplitude twice either individual amplitude. Equal-amplitude waves exactly in antiphase can cancel at that point. If the amplitudes differ, the minimum resultant amplitude is not zero.
For initially in-phase coherent sources, a path difference of an integer number of wavelengths gives constructive interference; an odd half-wavelength gives destructive interference.
Worked example
Two waves have displacements +3 mm and −2 mm at the same point and time. Find the resultant.
Show the worked solution
- Use signed displacements along the same axis.
- Add +3 and −2.
- The positive direction remains because the first displacement is larger.
Answer +1 mm
Common mistakes
- Add displacements, not frequencies.
- Antiphase waves only cancel completely when their amplitudes are equal.
What can you explain now?
Two equal-amplitude coherent waves arrive with phase difference π radians. What is the resultant amplitude?
Compare with the explanation
Zero
Their displacements are equal and opposite at that point, so the sum is zero.
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