Questions: Wave Interference: Constructive and Destructive

Question 1

Two speakers 3 meters apart emit identical 440 Hz tones. A listener walking between them finds a spot of near-silence midway between the speakers. What best explains this?

• A. The two sound waves cancel each other's energy, converting it to heat at that point
• B. The path difference at that point is approximately λ/2, so crests from one speaker arrive simultaneously with troughs from the other, producing destructive interference
• C. The speakers are out of phase with each other because they are different brands
• D. Sound waves cannot interfere in air — only light waves produce interference patterns

Answer: (B)

A quiet node occurs where the path difference equals a half-integer multiple of the wavelength (λ/2, 3λ/2, etc.) — a crest from one speaker arrives simultaneously with a trough from the other, and the displacements cancel. The energy is not destroyed; it has been redistributed to the loud antinodes nearby. Option A is the most common misconception — destructive interference does not convert energy to heat, it redistributes energy spatially. Option D is false; sound waves absolutely interfere.

Question 2

Two coherent wave sources produce waves with wavelength 0.5 m. At a given point, wave A travels 1.75 m and wave B travels 1.00 m to reach that point. What kind of interference occurs there?

• A. Constructive — the path difference of 0.75 m is close to a whole wavelength
• B. Destructive — the path difference of 0.75 m equals 1.5λ, a half-integer multiple of the wavelength
• C. Constructive — the path difference is less than one full wavelength, so waves reinforce
• D. Neither — interference only occurs when path differences are exact whole numbers

Answer: (B)

Path difference = 1.75 − 1.00 = 0.75 m. With λ = 0.5 m, this equals 0.75/0.5 = 1.5 wavelengths — a half-integer multiple (n + ½)λ with n = 1. This is the condition for destructive interference: a crest from one source arrives with a trough from the other. Option A makes the error of checking whether 0.75 is 'close to' a whole wavelength in meters, ignoring that the criterion is measured in units of λ.

Question 3

When two waves undergo complete destructive interference at a point, the energy carried by those waves is permanently destroyed at that location.

• T. True
• F. False

Answer: False

Energy is conserved in wave interference. When energy is 'missing' from a dark fringe or node, it has been redistributed to the bright fringes or antinodes nearby. The bright fringes in a double-slit pattern are brighter than either source alone precisely because energy from the destructive regions accumulates there. This redistribution is a fundamental consequence of wave behavior and does not violate energy conservation — it merely concentrates energy in space rather than distributing it uniformly.

Question 4

Two identical lasers aimed at the same spot on a screen from the same direction will produce a visible interference pattern with bright and dark fringes.

• T. True
• F. False

Answer: False

Interference requires coherence — a constant phase relationship between sources over time. Two independent lasers, even if identical in wavelength, have independently fluctuating phases. Their phase difference changes randomly on timescales shorter than any detector can resolve, so the interference pattern averages away to uniform intensity. Stable, visible interference requires a coherent source: a single laser split into two beams (Young's double slit), or two sources locked in phase. This is why everyday light sources (bulbs, separate lasers) don't produce observable interference.

Question 5

Explain why the bright fringes in a two-source interference pattern are brighter than either source alone, even though dark fringes appear nearby where the intensity is zero.

Model answer: Energy is conserved but redistributed by interference. At destructive nodes, the wave displacements cancel and the local intensity is zero. That energy does not vanish — it is transferred to the constructive antinodes. The total energy across the entire pattern equals the sum of what both sources emit. Because energy is concentrated into fewer bright regions (with dark regions in between), those bright spots are more intense than either source alone would produce uniformly. Interference is a spatial redistribution of energy, not creation or destruction of energy.

This is also why increasing the number of coherent sources (as in a diffraction grating) makes the bright fringes even sharper and more intense — more sources means more complete destructive interference everywhere except at the principal maxima, concentrating all the energy into increasingly narrow bright lines.