Superposition, coherence and interference

Key words

• Wavefront: a line or surface joining points that are in phase
• Phase: the stage a point has reached in its cycle; phase difference is measured in degrees or radians (one cycle = 360° = 2π rad)
• Coherent sources: same frequency and a constant phase difference
• Path difference: the difference in distance travelled by two waves to a point

Superposition

When two or more waves meet, the resultant displacement at any point is the vector sum of the individual displacements. Interference is the result of superposing coherent waves: a stable pattern of maxima and minima.

Constructive and destructive interference

For two coherent sources in phase:

• path difference = nλ (0, λ, 2λ...): waves arrive in phase, constructive interference, maximum amplitude
• path difference = (n + 1/2)λ: waves arrive in antiphase (180° out of phase), destructive interference, minimum amplitude

Phase and path difference

phase difference (in radians) = 2π x path difference / λ

phase difference (in degrees) = 360° x path difference / λ

Worked example

Two loudspeakers driven in phase emit sound of wavelength 0.25 m. Point P is 3.0 m from one speaker and 3.5 m from the other. Is P loud or quiet?

1. Path difference = 3.5 - 3.0 = 0.50 m
2. 0.50 / 0.25 = 2, so the path difference is 2λ
3. The waves arrive in phase: constructive interference

Answer: P is loud (a maximum)

Check yourself

1. What does it mean for two sources to be coherent?
   • They have the same amplitude
   • They have the same frequency and a constant phase difference
   • They are the same distance from a point
   • They emit only one wavelength of light
   Show answer

   They have the same frequency and a constant phase difference. Coherence requires a fixed phase relationship, which needs equal frequencies.

2. Two waves of wavelength 0.20 m have a path difference of 0.050 m. Find their phase difference in degrees.
   Show answer

   90 °. Phase difference = 360° x 0.050 / 0.20 = 90°.

3. Find the phase difference in radians for a path difference of half a wavelength, to 3 s.f.
   Show answer

   3.14 rad. 2π x 0.5 = π rad = 3.14 rad.

4. Two in-phase coherent sources give a path difference of 1.5λ at a point. What happens there?
   • Constructive interference (a maximum)
   • The waves do not superpose
   • Destructive interference (a minimum)
   • The frequency doubles
   Show answer

   Destructive interference (a minimum). 1.5λ = (1 + 1/2)λ, so the waves arrive in antiphase.

5. Waves of amplitude 3.0 mm and 2.0 mm meet exactly in antiphase. Find the resultant amplitude in mm.
   Show answer

   1 mm. In antiphase the displacements subtract: 3.0 - 2.0 = 1.0 mm.

6. What is a wavefront?
   • The front edge of the first wave
   • The direction in which the wave travels
   • A line joining points that are in phase
   • The point of maximum amplitude
   Show answer

   A line joining points that are in phase. For example, a line along a crest.

7. What is the principle of superposition?
   • The resultant displacement is the vector sum of the individual displacements
   • Waves cancel out whenever they meet
   • Two waves always produce a standing wave
   • The resultant amplitude is always the sum of the amplitudes
   Show answer

   The resultant displacement is the vector sum of the individual displacements. Displacements add (with signs) at every point and every instant.

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Sources

• Pearson Edexcel International Advanced Subsidiary/Advanced Level in Physics specification (XPH11/YPH11), Issue 3
• Pearson qualifications: International Advanced Levels Physics (2018)

Last updated 7 November 2026. Spotted a mistake? Tell us.