Abstract

Using symbolic calculations, a method of local asymptotics is developed, which describes the diffraction focusing of electromagnetic and acoustic fields for edge and corner catastrophes. Explicit expressions are found for the unfolding coefficients, the functional module, and the phase of the traveling wave, when the family of primary (geometrical-optical) and secondary (edge) rays form focuses on the cuspoid \({\mathrm{\boldsymbol A}}_{\mathrm{2}}\) type (simple edge catastrophe \({\mathrm{\boldsymbol F}}_{\mathrm{4}}\) ), when the family of primary and secondary rays form focuses on the \({\mathrm{\boldsymbol A}}_{\mathrm{3}}\) type (unimodal edge catastrophe \({\mathrm{\boldsymbol K}}_{\mathrm{4,2}}\) ), as well as for a corner catastrophe \({\mathrm{\boldsymbol A}_{1}}^{4}\).

中文翻译：

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边缘和角落灾难发生的局部渐近性

摘要

使用符号计算，开发了一种局部渐近方法，该方法描述了边缘和拐角突变的电磁场和声场的衍射聚焦。当主要（几何光学）和次要（边缘）射线形式的族群集中在尖齿\（{\ mathrm {\ boldsymbol）上时，可以找到展开系数，功能模块和行波相位的明确表达式。 A}} _ {\ mathrm {2}} \）类型（简单边缘突变\（{\ mathrm {\ boldsymbol F}} _ {\ mathrm {4}} \）），当一次和二次射线族形成时专注于\（{\ mathrm {\ boldsymbol A}} _ {\ mathrm {3}} \）类型（单峰边缘突变\（{\ mathrm {\ boldsymbol K}} _ {\ mathrm {4,2}} \）），以及发生角落灾难\（{\ mathrm {\ boldsymbol A} _ {1}} ^ {4} \）时。