Describe how wavelength affects diffraction at an edge

Cambridge IGCSE Physics 0625 – Topic 3.1: General Properties of Waves

Learning Objective

Describe how the wavelength of a wave influences the amount of diffraction that occurs when the wave encounters an edge or a narrow opening, and relate this to other wave phenomena required by the syllabus (reflection, refraction, interference).

Key Terminology (AO1)

• Wavefront: an imaginary surface joining points of equal phase (e.g., all crests). In a plane wave the wavefronts are straight lines; in a spherical wave they are concentric circles.
• Wavelength (λ): distance between two successive points of the same phase (crest‑to‑crest, trough‑to‑trough, or successive compressions).
• Frequency (f): number of wave cycles that pass a given point each second (unit Hz).
• Period (T): time for one complete cycle; T = 1/f.
• Amplitude: maximum displacement of the medium from its equilibrium position.
• Transverse wave: particle motion is perpendicular to the direction of propagation (e.g., water ripples, light).
• Longitudinal wave: particle motion is parallel to the direction of propagation (e.g., sound in air).

Wave Speed Relationship

The speed v of a wave is the product of its frequency and wavelength:

\$\$

v = f\lambda

\$\$

• For a given medium, v is constant → if λ increases, f must decrease, and vice‑versa.
• Example: In air, sound travels at ≈ 340 m s⁻¹.

  • f = 500 Hz → λ = v/f = 340 / 500 = 0.68 m.
  • f = 1000 Hz → λ = 0.34 m (half the previous wavelength).

Transverse vs. Longitudinal Waves

Reflection of Waves (AO2)

• The incident wave bounces off a surface such that the angle of incidence i equals the angle of reflection r (measured from the normal).
• Law of reflection: i = r.
• Applies to all types of waves (light, water, sound).

Refraction of Waves (AO2)

• When a wave passes from one medium to another its speed changes, causing a change in direction.
• Snell’s law (for light and any wave with a well‑defined speed):

  \$\$

  n1 \sin\theta1 = n2 \sin\theta2

  \$\$

  where n is the refractive index (ratio of wave speed in vacuum to that in the medium).

• Critical angle and total internal reflection occur when the wave moves from a higher‑index to a lower‑index medium.

Diffraction – Bending and Spreading of Waves

Diffraction becomes noticeable when the size of an obstacle or aperture (a) is comparable to the wavelength (λ). The amount of spreading is governed by the ratio λ/a.

Quantitative condition for a single slit (or a straight edge)

Minima in the diffraction pattern satisfy

\$\$

a\sin\theta = m\lambda \qquad (m = 1,2,3,\dots)

\$\$

• a – width of the slit (or effective width of the edge).
• θ – angle measured from the original direction of propagation.
• m – order of the minimum (first order = 1, second order = 2, …).
• A larger λ gives a larger sin θ, i.e. a wider spread of the diffracted beam.

Qualitative relationship

• λ ≈ a – strong diffraction; the wave spreads over a wide angular region.
• λ < a – weak diffraction; most of the wave proceeds straight ahead.
• λ > a – the obstacle blocks most of the wave, but the small portion that passes diffracts very broadly.

Diagram – single‑slit diffraction

Ripple‑Tank Demonstration (AO3)

1. Generate a plane wave using a vibrator.
2. Place a straight barrier with a narrow slit (or a sharp edge) in the path.
3. Illuminate the water surface and view the pattern on a screen below.

Observations:

• If slit width ≈ ripple wavelength, a semicircular pattern appears – clear diffraction.
• If the slit is much wider than the wavelength, the outgoing wavefront remains almost straight (weak diffraction).
• Simultaneous reflection from the barrier and refraction when the water depth changes can be seen, linking the three phenomena.

Practical Applications (AO2)

• Radio communication: Long‑wave (λ ≈ 1 m) signals diffract around buildings and hills, allowing reliable long‑distance transmission.
• Optical instruments: Apertures are made many times larger than visible‑light wavelengths (λ ≈ 500 nm) to minimise diffraction blur and produce sharp images.
• Acoustic engineering: Low‑frequency sounds (long λ) can be heard around corners, whereas high‑frequency sounds are more directional – important for speaker placement and noise control.
• Medical imaging: Ultrasound (λ ≈ 0.5 mm) experiences limited diffraction, giving fine resolution in tissue scanning.
• Laser cutting & microscopy: Short wavelengths (e.g., 0.4 µm) give minimal diffraction, enabling precise material processing and high‑resolution imaging.

Summary Table – Effect of Wavelength on Diffraction at an Edge or Slit

Worked Example – Diffraction Angle for a Radio Wave

Given: λ = 2 m (medium‑frequency radio), slit width a = 1 m.

Find the first‑order minimum angle (m = 1).

1. Use the condition a sinθ = mλ → 1 sinθ = 1 × 2 → sinθ = 2.
2. Since sinθ cannot exceed 1, the first minimum does not exist; the wave is strongly diffracted and the pattern is very broad.
3. Conclusion: For λ > a the diffraction pattern spreads over essentially the whole forward hemisphere.

Practice Questions (AO3)

1. A water ripple has λ = 5 cm. It encounters a vertical slit 1 cm wide. Predict the appearance of the diffracted pattern and calculate the angle to the first minimum.
2. Sound of frequency 250 Hz travels in air (v ≈ 340 m s⁻¹). Calculate its wavelength and comment on how well it will diffract around a doorway 0.5 m wide.
3. Light of wavelength 600 nm passes through a slit 0.03 mm wide. Find the angular position of the second-order minimum (m = 2). What does this tell you about the need for larger apertures in telescopes?

Suggested Composite Diagram – Wavelength vs. Diffraction

Link to Other Wave Phenomena (AO2)

• Interference: Diffraction provides the coherent wavefronts needed for constructive and destructive interference (e.g., double‑slit experiment).
• Resolution of Instruments: The Rayleigh criterion (θ ≈ 1.22 λ/D) shows directly how diffraction limits the resolving power of microscopes and telescopes.
• Standing Waves: In a pipe, the ends act as edges; the degree of diffraction at the open end influences the formation of nodes and antinodes.