We consider the solution of the Helmholtz equation−u(x)−n(x)2ω2u(x)=f(x),x=(x,y),in a domain which is infinite in x and bounded in y.We assume that f(x)is supported in 0:={x∈|a−<x<a+}and that n(x)is x-periodi...We consider the solution of the Helmholtz equation−u(x)−n(x)2ω2u(x)=f(x),x=(x,y),in a domain which is infinite in x and bounded in y.We assume that f(x)is supported in 0:={x∈|a−<x<a+}and that n(x)is x-periodic in\0.We show how to obtain exact boundary conditions on the vertical segments,−:={x∈|x=a−}and+:={x∈|x=a+},that will enable us to find the solution on 0∪+∪−.Then the solution can be extended in in a straightforward manner from the values on−and+.The exact boundary conditions as well as the extension operators are computed by solving local problems on a single periodicity cell.展开更多
文摘We consider the solution of the Helmholtz equation−u(x)−n(x)2ω2u(x)=f(x),x=(x,y),in a domain which is infinite in x and bounded in y.We assume that f(x)is supported in 0:={x∈|a−<x<a+}and that n(x)is x-periodic in\0.We show how to obtain exact boundary conditions on the vertical segments,−:={x∈|x=a−}and+:={x∈|x=a+},that will enable us to find the solution on 0∪+∪−.Then the solution can be extended in in a straightforward manner from the values on−and+.The exact boundary conditions as well as the extension operators are computed by solving local problems on a single periodicity cell.
