Diffraction

🎯 Learning Objectives

By the end of this chapter, you should be able to:

Introduction to Diffraction:

Diffraction occurs when a sound wave encounters an obstacle with a free edge, corners, or boundaries between materials with differing impedances. The diffracted wave appears to be radiated from the edges or perimeters of these obstacles.

• If the obstacle is small compared with the wavelength, the incident wave remains largely unaffected.
• As the obstacle size increases relative to the wavelength, a shadow region appears, becoming sharper as the diffraction effect grows. This shadow is a result of interference between the incident wave and the diffracted wave, leading to partial or total cancellation.

Diffraction is not merely an academic curiosity; it influences:

• Binaural hearing and sound localization.
• Transmission of sound through doors or windows with small gaps.
• Orchestra sound projection from pits in opera houses or concert halls.

Historical and Theoretical Foundation:

The mathematical description of diffraction dates back centuries:

• Huygens (1690) proposed that every point on a wavefront acts as a secondary source, radiating spherical wavelets.
• Fresnel (1815–1821) expanded this concept to include superposition and interference of wavelets, yielding diffraction patterns.
• Kirchhoff formulated diffraction integrals for apertures and obstacles, extending Huygens–Fresnel principles.
• Sommerfeld (1896) addressed diffraction around obstacles comparable to the wavelength, providing a wave-theoretic approach.
• Keller (1962) formulated the Geometrical Theory of Diffraction (GTD), linking diffraction to ray splitting at edges.
• Biot and Tolstoy (1957) developed a frequency-dependent transfer path model, describing diffraction in closed form.
• Medwin (1981) refined this into the Biot-Tolstoy-Medwin (BTM) approach, enabling accurate calculations for finite-sized obstacles.

The BTM method allowed closed-form evaluation of diffraction around finite screens, edges, and barriers, paving the way for engineering applications.

Practical Diffraction Models:

Maekawa’s Model (1968)

• Maekawa conducted experimental studies on diffraction by an infinite half-plane, measuring sound reduction as a function of frequency and geometry.
• The model computes the detour distance $d$ of the wave around a vertical screen to the receiver.
• Insertion loss approximation:

• Widely used in urban noise propagation and open-plan office acoustics.
• Forms the basis for ISO 9613-2 standard calculations.

Unified Theory of Diffraction (UTD)

• UTD extends GTD by treating edges as lines of secondary point sources, integrating contributions along the edge line.
• Based on Biot-Tolstoy-Medwin but adapted for finite edges, producing time-discrete impulse responses.
• Key features:
  • Accounts for multiple paths over the edge.
  • Computes energy and delay of each diffracted path.
  • Supports continuous, decaying impulse response for practical simulation.

Mathematical Description:

For a single edge or slit in an infinite screen, diffraction can be described by Huygens-Fresnel integral:

Where:

• $A(s)$ is the amplitude contribution from each secondary source along the edge.
• $r(s)$ is the distance from the secondary source to the observation point $P$"
• The integral sums contributions along the edge line.

For finite edges, time-discrete impulse response accounts for detours and phase delays:

Where:

• $A_i$ is the amplitude along path $i$ over the edge.
• $\tau_i$ is the travel time along that path.

Frequency and Geometrical dependence:

• Low-frequency waves $\lambda \gtrsim$ obstacle size: strong diffraction, shadow region blurred.
• High-frequency waves $\lambda \ll$ obstacle size: sharper shadow, weaker bending.
• Diffraction effects are pronounced for apertures and gaps comparable to the wavelength.

Practical engineering: frequency-dependent diffraction coefficients are used in simulations of noise barriers, urban sound propagation, and room acoustics.

Applications in Acoustics:

• Noise Barriers: insertion loss estimation using Maekawa’s model.
• Architectural Acoustics: sound propagation from orchestra pits or around columns.
• Urban Planning: modeling sound detours over buildings, edges, and fences.
• Transmission through small openings: doors, windows, and ventilation.

Animated Examples: Animation of Diffraction over a Barrier

Barrier Height $= h=λ$

Barrier Height $= h=2λ$

Barrier Height $= h=4λ$

🧪 Interactive Examples

📝 Key Takeaways

🧠 Quick Quiz