IB Mathematics: Diffraction Analysis of a Uniformly Radiated Strip | IB数学：均匀辐射条的衍射分析

📚 IB Mathematics: Diffraction Analysis of a Uniformly Radiated Strip | IB数学：均匀辐射条的衍射分析

Diffraction is one of the most beautiful and practical phenomena in wave physics. It occurs whenever a wave encounters an obstacle or an aperture, causing the wave to bend and spread into regions that simple ray geometry would predict to be dark. For a uniformly radiated strip — a thin rectangular slit that emits waves of constant amplitude across its width — the resulting intensity pattern can be derived entirely with tools from the IB Mathematics syllabus: definite integrals, complex numbers, Euler’s formula, and trigonometric limits.

衍射是波物理学中最优美也最实用的现象之一。当波遇到障碍物或孔径时就会发生衍射，使波弯曲并扩展到简单光线几何所预测的暗区。对于均匀辐射条——即宽度上振幅恒定的细长矩形狭缝——其强度分布可以完全用 IB 数学大纲中的工具推导：定积分、复数、欧拉公式和三角极限。

1. Physical Model of the Strip | 辐射条的物理模型

Imagine a long, narrow strip of width a lying in a plane screen. The strip radiates monochromatic waves of wavelength λ, and every point on the strip emits a secondary wavelet with the same amplitude A per unit length. This is the Huygens–Fresnel picture: an extended source is treated as a continuous collection of infinitesimal point sources.

设想一个位于平面屏内、宽度为 a 的细长条。该条辐射波长为 λ 的单色波，条上每一点都发出振幅为每单位长度 A 的次波。这就是惠更斯-菲涅耳图像：将扩展源视为无穷多个微小点源的连续集合。

We observe the combined radiation at a distant point P located at an angle θ from the normal to the strip. In far-field (Fraunhofer) diffraction, the rays arriving at P from different parts of the strip are effectively parallel, so the

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