Questions: Phase and Phase Relationships in Waves

Question 1

Two speakers emit sound of wavelength 2 m. Speaker A is 5 m from a listener; Speaker B is 6 m from the listener. What does the listener hear?

• A. Constructive interference — the waves reinforce because they originated from the same source type
• B. Destructive interference — the path difference of 1 m equals λ/2, producing a phase difference of π
• C. No interference — interference only occurs when the two waves travel exactly the same distance
• D. Partial constructive interference — the 1 m path difference is too small to matter

Answer: (B)

Path difference = 6 − 5 = 1 m = λ/2 (since λ = 2 m). Phase difference = (2π/λ) × path difference = (2π/2) × 1 = π radians. A phase difference of π means crest meets trough — complete destructive interference. Option A reflects the misconception that same-type sources automatically reinforce; what matters is not the source type but the phase relationship at the listening point, which depends on path geometry.

Question 2

Two waves of the same frequency start in phase at their source. Wave A travels 3 m to a detector; Wave B travels 5 m. The wavelength is 2 m. The phase difference at the detector is:

• A. π radians — path difference 2 m is one wavelength, so there must be a phase shift
• B. 2π radians — path difference 2 m equals one full wavelength, meaning the waves arrive in phase
• C. π/2 radians — half the path difference divided by the wavelength
• D. 0 radians — they started in phase so they remain in phase regardless of path

Answer: (B)

Phase difference = (2π/λ) × Δd = (2π/2) × 2 = 2π. A phase difference of 2π is equivalent to 0 — the wave that traveled the extra wavelength has completed one full extra cycle and arrives back in phase with the other. The result is constructive interference. Option A is the classic trap: students see '2 m path difference and 2 m wavelength' and assume that means a full π phase shift, forgetting that one full wavelength corresponds to 2π (not π) of phase — a complete cycle, not a half cycle.

Question 3

A path difference of exactly one wavelength between two otherwise identical waves produces constructive interference.

• T. True
• F. False

Answer: True

A path difference of λ corresponds to a phase difference of (2π/λ) × λ = 2π, which is a complete cycle — the wave arrives as if no extra path had been traveled at all. The two waves are back in phase, and their amplitudes add. Constructive interference occurs whenever the path difference is any integer multiple of λ: 0, λ, 2λ, etc.

Question 4

Two waves with a phase difference of π/2 (90°) interfere destructively.

• T. True
• F. False

Answer: False

Destructive interference requires a phase difference of π (180°), so crest aligns with trough and the two amplitudes cancel. A phase difference of π/2 (90°) produces partial interference: the waves are neither in phase (which would give maximum constructive addition) nor fully opposed (which would give complete cancellation). The resultant amplitude is √2 times the individual amplitude — less than constructive but more than zero.

Question 5

How does path difference relate to phase difference, and what path difference produces complete destructive interference?

Model answer: Phase difference (in radians) = (2π/λ) × path difference. Complete destructive interference requires a phase difference of π radians (180°), which occurs when the path difference equals λ/2 — one half-wavelength. More generally, destructive interference occurs at path differences of λ/2, 3λ/2, 5λ/2, etc. (odd multiples of λ/2).

The factor 2π/λ converts spatial distance into angular phase: one full wavelength corresponds to one full cycle (2π radians), so half a wavelength corresponds to π radians. This conversion is the bridge between the geometry of path lengths and the physics of interference, and it underlies every interference calculation from double-slit experiments to thin-film optics.