Questions: Path Difference and Phase Difference in Waves

Question 1

Two coherent wave sources reach point P: Wave 1 travels 3.0 m and Wave 2 travels 3.5 m. The wavelength is 0.5 m. What type of interference occurs at P?

• A. Destructive — the path difference of 0.5 m produces a phase difference of π
• B. Constructive — the path difference equals exactly one wavelength, giving a phase difference of 2π
• C. Neither — a phase difference of 2π means the waves are out of phase
• D. Partially constructive — 0.5 m path difference never produces full constructive interference

Answer: (B)

Path difference Δx = 3.5 − 3.0 = 0.5 m = 1 wavelength (since λ = 0.5 m). Phase difference Δφ = 2πΔx/λ = 2π(1) = 2π. A phase difference of 2π means the waves are exactly in phase — their peaks and troughs align — producing constructive interference. The most common error is confusing a phase difference of 2π (full cycle, in phase) with π (half cycle, out of phase).

Question 2

Two coherent sources each produce waves with a path difference of 6 cm to point P. Source A has wavelength 2 cm; Source B has wavelength 4 cm. At which source does the path difference produce constructive interference at P?

• A. Source A only — Δx/λ = 3 (integer), giving full constructive interference; Source B has Δx/λ = 1.5 (half-integer), giving destructive interference
• B. Source B only — the longer wavelength is less sensitive to path differences
• C. Both sources — the same path difference always produces the same type of interference
• D. Neither source — 6 cm is too large a path difference for either wavelength

Answer: (A)

For Source A: Δx/λ = 6/2 = 3 (integer multiple of λ) → constructive. For Source B: Δx/λ = 6/4 = 1.5 (half-integer multiple) → destructive. The same path difference produces different interference outcomes for different wavelengths. This is why 'path difference' and 'type of interference' cannot be connected without knowing the wavelength — the conversion Δφ = 2πΔx/λ is the essential link.

Question 3

A path difference of exactly one wavelength always produces constructive interference, regardless of the wavelength's actual value.

• T. True
• F. False

Answer: True

If Δx = λ, then Δφ = 2πΔx/λ = 2π(λ/λ) = 2π. A phase difference of 2π is a full cycle — the two waves arrive perfectly in phase. This is true for any wavelength: what matters is the ratio Δx/λ, not the absolute values. A path difference of one wavelength always means full constructive interference.

Question 4

Path difference and phase difference are the same quantity measured in different units — one in meters, one in radians.

• T. True
• F. False

Answer: False

Path difference (Δx, in meters) and phase difference (Δφ, in radians) measure fundamentally different things, and the conversion between them depends on wavelength: Δφ = 2πΔx/λ. The same path difference of 1 meter produces a phase difference of 2π for λ = 1 m, but a phase difference of 4π for λ = 0.5 m. They are not simply different units for the same quantity — wavelength is a necessary factor in the conversion.

Question 5

Point P has a path difference of 450 nm from two coherent sources with wavelength λ = 300 nm. Determine whether P is a bright or dark fringe and explain how the path difference–wavelength relationship leads to that conclusion.

Model answer: Δx/λ = 450/300 = 1.5, which is a half-integer (n + ½ for n = 1). Therefore Δφ = 2π(1.5) = 3π, an odd multiple of π. This means the waves arrive exactly out of phase — a peak of one aligns with a trough of the other — producing destructive interference. Point P is a dark fringe. The key step is computing Δx/λ: if the result is an integer, it's a bright fringe; if it's a half-integer, it's a dark fringe.

The ratio Δx/λ is the central quantity in interference analysis. It tells you how many wavelengths of path difference exist, which directly determines whether the waves arrive in phase or out of phase. Students who memorize the conditions (nλ for constructive, (n+½)λ for destructive) without computing this ratio often make sign errors or apply the wrong condition.