(d) constructive interference and destructive interference in terms of…

The idea

When two waves meet, their displacements add. Constructive interference gives maximum amplitude when the waves arrive in phase: phase difference is zero or a whole number of cycles (0, 2π, 4π, ...), and path difference is zero or a whole number of wavelengths (0, λ, 2λ, ...). Destructive interference gives minimum amplitude when the waves arrive in antiphase: phase difference is an odd number of π (π, 3π, 5π, ...), and path difference is an odd number of half-wavelengths (λ/2, 3λ/2, 5λ/2, ...). Phase difference and path difference are linked by Δφ = 2πΔx/λ, where Δx is path difference and λ is wavelength.

For example: If λ = 0.60 m and Δx = 0.90 m, then Δx = 1.5λ, so Δφ = 3π rad and the interference is destructive.

What earns the marks

• Constructive interference occurs when waves arrive in phase, with phase difference 0, 2π, 4π, ... rad.
• Constructive interference occurs when path difference is 0, λ, 2λ, ... (a whole number of wavelengths).
• Destructive interference occurs when waves arrive in antiphase, with phase difference π, 3π, 5π, ... rad.
• Destructive interference occurs when path difference is λ/2, 3λ/2, 5λ/2, ... (an odd number of half-wavelengths).
• Phase difference and path difference are related by Δφ = 2πΔx/λ.

Common mistakes and how to put them right

• Thinking that destructive interference means the waves cancel completely everywhere.

  Put it right: At a point of destructive interference the resultant displacement is zero at that instant, but energy is redistributed and the waves still exist.

• Confusing path difference with phase difference.

  Put it right: Path difference is a distance in metres, while phase difference is an angle in radians, linked by Δφ = 2πΔx/λ.

• Assuming any path difference produces constructive interference.

  Put it right: Only whole-number multiples of λ give constructive interference, while odd multiples of λ/2 give destructive interference.

• Forgetting that phase difference can be expressed in radians or degrees.

  Put it right: 2π rad equals 360°, so π rad equals 180° and represents antiphase.

Exam tips

• Read the question carefully to see whether it asks about path difference or phase difference, and give the answer in the correct unit.
• When calculating, convert path difference to a multiple of wavelength first; this makes it easy to classify as constructive or destructive.
• Remember that a path difference of exactly λ/2 gives destructive interference, not constructive, because it corresponds to a phase difference of π rad.
• If a question gives phase difference in degrees, convert to radians before using Δφ = 2πΔx/λ, or work directly with the fraction of a cycle.