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Diffraction is the deviation of waves from straight-line propagation due to an obstacle or through an aperture. The diffracting object or aperture effectively becomes a secondary source of the propagating wave.

Green's second identity links the behavior of a field (e.g., sound pressure) inside a volume V to its values on the enclosing surface S.

1. Applying Green's identity to sound waves:

   • Assume the waves satisfy the Helmholtz equation, which governs wave propagation in the frequency domain.

   • Use a free-space Green's function.

2. The integral formulation derived from Green's identity allows calculating the field U(x) at any point inside the volume based on the field values and their derivatives on the surface S.

Simplified: The Kirchhoff-Helmholtz integral allows predicting the sound field behind an aperture based on the field and its normal derivative on the aperture's surface.

The Sommerfeld radiation condition is used to discard S2​.This is satisfied if the wave attenuates atleast as fast as a spherical wave.

Fresnel–Kirchhoff diffraction formula approximates light intensity and phase in optical diffraction: light fields in the boundary regions of shadows. The approximation can be used to model light propagation in a wide range of configurations, either analytically or using numerical modelling. It gives an expression for the wave disturbance when a monochromatic spherical wave is the incoming wave of a situation under consideration.

Kirchhoff Boundary Conditions:

1. Within the aperture Σ

   • The field U and its normal derivative are the same as if the screen wasn’t there.

2. Outside the aperture:

   • Both U and its normal derivative are zero.

Key Difference: Introduces "mirror sources" to avoid the inconsistencies in Kirchhoff’s approach.

Two formulations are possible depending on how the mirror source interacts with the primary source.

1. The Green’s function is zero on the screen (S1), eliminating the field value constraint.

2. The normal derivative of the Green’s function is zero on S1​, eliminating the derivative constraint.

1. Accounts for the directional effect of the aperture’s normal relative to the wave direction.

2. Obliquity factor calculation varies with model.

Note that planar waves seem to be necessary fpor both approaches.

1. For small angles (where cos⁡θ≈1)

   • All models behave similarly.

   • The obliquity factor has minimal effect.

2. For large angles:

   • The obliquity factor significantly alters the results.

3. Plane Wave Insonification:

   • For a plane wave approaching the aperture, the obliquity factors simplify as the incoming wave direction aligns with the normal to the aperture.

1. Fresnel-Kirchhoff Model: Uses average direction cosines and is mathematically simpler but has inconsistencies.

2. Rayleigh-Sommerfeld Models: More rigorous, avoiding inconsistencies by choosing Green’s functions that simplify the constraints.

3. Obliquity Factor: Governs the directional influence of diffraction, with minimal impact at small angles but growing significance at larger angles.