zbMATH — the first resource for mathematics

Diffraction of a wave at a corner. (Diffusion d’une onde par un coin.) (French) Zbl 0779.35063
The authors consider the boundary-value problem given by $(\partial^ 2_ t -\Delta)u=0, \qquad u|_{\partial\Omega}=0, \qquad u|_{t<0}= u_ i| _{t<0},$ where $$u_ i\equiv u_ i(t,x,y)\in H_{\text{loc}}'(\mathbb{R}\times \mathbb{R}^ 2)$$ is a certain given function satisfying $$(\partial^ 2_ t-\Delta)u_ i=0$$ and $$\Omega\subset \mathbb{R}\times \mathbb{R}^ 2$$ is an open domain defined by $$\Omega= \mathbb{R}\times(\mathbb{R}^ 2 \setminus F)$$ with $$F=\{(x,y)\in\mathbb{R}^ 2$$, $$x\geq 0$$, $$b(x)\leq y\leq a(x)\}$$, and study the singularities of the solution for $$t>0$$. Here $$a(x)\in C^ \infty$$ and $$b(x)\in C^ \infty$$ are two given functions, analytic near the origin, such that $$a(x)>0>b(x)$$ when $$x>0$$ while $$a(0)=b(0)=0$$ and $$a'(0)>0>b'(0)$$. This problem corresponds to the diffraction of a cylindrical wave (namely: $$u_ 1$$) by the curved wedge $$W=\{(x,y,z)\mid z\in(-\infty,\infty)$$, $$(x,y)\in F\}$$.
The paper is rather long and includes three theorems, eleven propositions and eighteen lemmas. It is worthwile to notice that the title of the paper is misleading in physical point of view because the diffraction does not occur here at a cone tip but rather at an edge.

MSC:
 35L05 Wave equation 35S15 Boundary value problems for PDEs with pseudodifferential operators 35A20 Analyticity in context of PDEs 78A45 Diffraction, scattering
Full Text: