Interactive physics simulator
Wave Diffraction
Explore how mechanical waves bend and spread. Study single-slit apertures, double-slit interference envelopes, obstacle shadows, and Huygens' wavelet superposition in a 2D wave tank.
2D Wave Diffraction Ripple Tank
Modify the wavelength, slit opening, or frequency. Analyze the live intensity distribution graph in real time.
Live Wave Diffraction Telemetry
- Wavelength (λ)
- 3.0 cm
- Aperture Size
- Slit: 3.0 cm
- Frequency (f)
- 3.0 Hz
- Dimension Ratio
- λ / w = 1.00
- Diffraction Level
- Dominant (waves spread very wide)
Understanding Wave Diffraction
In physics, diffraction refers to the bending, spreading, and interference of waves when they pass through narrow openings (apertures), around obstacles, or past sharp edges. Unlike reflection or refraction, diffraction occurs within a single, uniform medium without waves bouncing off boundaries or changing speeds due to depth or density changes.
According to Huygens' Principle, every point on a propagating wavefront behaves as a source of secondary spherical wavelets. When the wavefront is unobstructed, these wavelets combine to form a flat plane wave or expanding circle. However, when a barrier blocks the wavefront, the wavelets at the opening propagate unimpeded, expanding outwards in circular paths that bend deep into the geometrical shadow zone.
Key Principles
For diffraction to be noticeable, the wave properties must match the boundary dimensions:
- Wavelength to Width Ratio (λ / w): Diffraction is strongest when the wavelength is comparable to or larger than the slit opening size (λ ≥ w).
- Circular Spreading: When waves pass through a slit much smaller than their wavelength (λ >> w), they spread out in semi-circular wavefronts as if originating from a single point.
- Straight Propagation: When the wavelength is much smaller than the opening (λ ≪ w), the wave travels straight through with negligible bending at the corners.
Single Slit Diffraction
When wavefronts pass through a single slit of width w, wavelets from different points across the slit interfere with each other, producing dark and bright bands on a distant screen:
Where:
- w is the width of the slit (meters)
- θ is the angle of the minimum from the central axis (degrees)
- λ is the wave's wavelength (meters)
- m is the order of the minimum (non-zero integer, m = ±1, ±2...)
Solved Examples
Plane water waves of wavelength λ = 3.0 cm approach a slit of width w = 5.0 cm in a ripple tank. Calculate: (a) the angle of the first diffraction minimum, and (b) the total angular width of the central maximum.
- Identify the given values: wavelength λ = 3.0 cm and slit width w = 5.0 cm.
- Part (a): Solve for the angle of the first diffraction minimum. The formula for single-slit minima is w · sin(θ) = m · λ. For the first minimum, set m = 1:
sin(θ) = λ / w = 3.0 cm / 5.0 cm = 0.60. - Compute the angle θ using the inverse sine function: θ = arcsin(0.60) ≈ 36.9°.
- Part (b): The central maximum extends from the first minimum on the left (-θ) to the first minimum on the right (+θ). Therefore, the total angular width is 2θ:
2θ = 2 × 36.9° = 73.8°. - The waves spread out past the opening at a total angle of 73.8°, creating a wide central peak.
Answer: θ = 36.9°, Central Width 2θ = 73.8°
A monochromatic light wave of wavelength λ = 600 nm passes through a narrow slit, producing a diffraction pattern on a screen located D = 2.0 m away. If the width of the central maximum on the screen is measured to be y = 4.8 cm, calculate the width w of the slit.
- Convert all given measurements to standard SI units: wavelength λ = 6.0 × 10-7 m, screen distance D = 2.0 m, and central maximum width y = 0.048 m.
- Find the distance from the center of the pattern to the first diffraction minimum on the screen: x = y / 2 = 0.048 m / 2 = 0.024 m.
- Apply the small-angle approximation, since the screen is far away and the pattern is small: sin(θ) ≈ tan(θ) = x / D = 0.024 m / 2.0 m = 0.012 rad.
- Recall the single-slit minimum condition for m = 1: w · sin(θ) = λ.
- Solve for the slit width w:
w = λ / sin(θ) = (6.0 × 10-7 m) / 0.012 = 5.0 × 10-5 m (or 0.05 mm).
Answer: w = 5.0 × 10-5 m (0.05 mm)
A doorway of width w = 0.9 m acts as a wave aperture. (a) Calculate the ratio λ / w for a sound wave of frequency f = 440 Hz (speed of sound v = 343 m/s). (b) Calculate the ratio for yellow light of wavelength λ = 580 nm. (c) Explain why we can hear someone talking around a corner but cannot see them.
- Part (a): For sound, calculate the wavelength using the wave equation v = f · λ:
λs = v / f = 343 m/s / 440 Hz ≈ 0.78 m.
Now, calculate the ratio of wavelength to aperture width:
λs / w = 0.78 m / 0.9 m ≈ 0.87. - Since the wavelength of sound is comparable to the doorway width (λ ≈ w), sound diffracts strongly, bending around the door frame into the adjacent room.
- Part (b): For light, convert wavelength to meters: λL = 5.8 × 10-7 m. Calculate the ratio:
λL / w = (5.8 × 10-7 m) / 0.9 m ≈ 6.4 × 10-7. - Since this ratio is extremely small (λ ≪ w), light does not diffract noticeably at the door frame. Instead, it travels in straight lines (geometric rays) and creates sharp shadows.
- Part (c): Sound waves are large enough to diffract around everyday obstacles like walls and doors, while light waves are far too small to bend around macro-sized corners, creating a straight line of sight.
Answer: (a) λs/w ≈ 0.87, (b) λL/w ≈ 6.4 × 10-7, (c) Sound waves bend around corners while light travels in straight rays.
Common Mistakes
- Effort vs. Wave bending: Thinking waves bend due to friction with the slit edges. Wave bending is entirely a wave superposition phenomenon (Huygens' wavelets) and occurs in completely friction-free environments.
- Confusing diffraction with refraction: Diffraction does not require a change in wave speed or medium, whereas refraction is caused exclusively by a change in speed.
- Assuming no diffraction for large slits: Diffraction always occurs at boundaries. However, when the slit is large, the central maximum is extremely narrow and bright, making the spreading difficult to observe.
Babinet's Principle
Babinet's Principle states that the diffraction pattern of a solid obstacle is identical to that of a complementary opening of the same shape and size. For instance, a thin wire of thickness b generates the same single-slit diffraction fringes on a screen as a slit of width w = b.
Practice Questions
1. What is wave diffraction, and what determines how much a wave spreads when passing through an opening?
Wave diffraction is the bending and spreading of waves as they pass around obstacles or through narrow openings (apertures). The extent of diffraction is determined by the ratio of the wave's wavelength to the size of the opening (λ / w). When the wavelength is comparable to or larger than the opening (λ ≥ w), the wave behaves like a point source and spreads out widely in a semicircular pattern. When the wavelength is much smaller than the opening (λ ≪ w), the wave travels straight through with minimal bending at the edges.
2. Why does a smaller slit width lead to a wider central maximum in a diffraction pattern?
According to the single-slit diffraction minimum formula, sin(θ) = mλ / w. As the slit width (w) decreases, the value of sin(θ) increases for a constant wavelength (λ). This means the first diffraction minimum occurs at a larger angle (θ) relative to the center. Because the central maximum is bounded on both sides by the first minima, reducing the slit width pushes these boundaries outward, widening the central maximum and spreading the wave energy over a larger area.
3. State Huygens' Principle and explain how it accounts for wave diffraction past a barrier.
Huygens' Principle states that every point on a wavefront behaves as a source of tiny, secondary spherical waves called wavelets. These wavelets propagate outward in all directions at the speed of the wave, and the new wavefront is the tangent line (envelope) of all these wavelets combined. When a wavefront hits a barrier with a slit, the barrier blocks most wavelets. However, the wavelets at the open slit pass through and expand in all directions—including sideways into the geometrical shadow zone—producing the circular diffracted wave pattern.
4. Explain Babinet's Principle and its implications for the diffraction patterns of a slit versus a solid wire.
Babinet's Principle states that the diffraction pattern from an obstacle (like a solid wire) is identical to the diffraction pattern from a complementary opening (like a slit of the exact same width), except for the intensity at the very center of the pattern. This occurs because the wave field behind a barrier is the difference between an unobstructed wave field and the wave field of the complementary opening. Consequently, a thin hair or wire placed in front of a laser pointer will produce the exact same single-slit interference fringes as a slit of the same thickness.
FAQ
Frequently Asked Questions
What is wave diffraction?
Wave diffraction is the bending, spreading, and interference of waves when they encounter an obstacle, corner, or opening.
What is the condition for strong wave diffraction?
Diffraction is most noticeable when the wavelength (λ) of the wave is comparable to or larger than the size of the opening (w) or obstacle (d), meaning λ ≥ w.
Does wave speed change during diffraction?
No. Wave speed, frequency, and wavelength do not change during diffraction, because the wave remains in the same medium.
How does wavelength affect diffraction?
Longer wavelengths diffract (bend) much more easily than shorter wavelengths. This is why long-wavelength radio signals travel around hills while short-wavelength light is blocked.
What is the difference between diffraction and refraction?
Refraction is the bending of waves due to a change in speed when crossing into a new medium. Diffraction is the bending of waves around obstacles or openings within the same medium.
What is Huygens' Principle?
Huygens' Principle states that every point on a wavefront acts as a source of secondary spherical wavelets. The new wavefront is the sum of these wavelets.
Why can you hear sound around a corner but not see light?
Sound waves have long wavelengths (meters) comparable to doorways and walls, allowing them to bend. Light waves have extremely tiny wavelengths (nanometers) and travel in straight lines.
What is single-slit diffraction?
It is the diffraction pattern produced when a wave passes through a single slit, creating a wide, bright central maximum flanked by narrower, dimmer side peaks.
What is the formula for single-slit diffraction minima?
The formula is w · sin(θ) = m · λ, where w is the slit width, θ is the diffraction angle, λ is the wavelength, and m is a non-zero integer (1, 2, 3...).
What is Babinet's Principle?
It is a theorem stating that complementary screen patterns (like a slit and a wire of equal width) produce identical diffraction patterns.
How does double-slit diffraction differ from single-slit?
Double-slit diffraction combines the spreading of waves from each slit (diffraction envelope) with the constructive and destructive interference between the two slits (fine fringes).
Can longitudinal waves undergo diffraction?
Yes. All waves, including longitudinal waves (like sound) and transverse waves (like light and water ripples), experience diffraction.