Wiener-Hopf method

The Wiener-Hopf method solves a boundary value problem with complementary half-line traces by Wiener-Hopf factorization and additive splitting into upper- and lower-half-plane analytic functions. Analytic continuation gives a common entire function, whose growth and edge conditions determine its admissible value.

For a scalar wave above a line with a Dirichlet half and a Neumann half, the boundary transform factors into ${\beta }_{+}(\alpha )=\sqrt{k+\alpha }$ and ${\beta }_{-}(\alpha )=\sqrt{k-\alpha }$. For $e^{-i\omega t}$, the limiting absorption principle puts $-k$ below and $k$ above the inversion contour. With Dirichlet conditions on the negative half-line and incident tangential wavenumber $a$, pole subtraction gives the diffracted trace $U_{+}=2i\sqrt{k-a}/[(\alpha +a)\sqrt{k+\alpha }]$. Its large-transform decay gives a bounded field with a square-root edge trace. The normal derivative has an inverse-square-root singularity on the Dirichlet half; it is identically zero on the Neumann half.

The plus factor is analytic and nonzero above the transform contour, and the minus factor below it. Their continuations inherit singularities and zeros in the opposite half-planes. Allocation of branch cuts and modal poles must agree with the physical outgoing continuation; no factor may have a zero inside its designated analytic domain.

A scalar Wiener-Hopf equation couples two Half-range Fourier transforms analytic in complementary half-planes. After dividing by a kernel factor, additive splitting separates the known forcing. Source poles, boundary terms and contour prescriptions are part of the problem data.

The scalar coefficient multiplying an unknown half-line transform in a Wiener-Hopf equation is its kernel. This is a transform-domain function, distinct from the null space of a linear map. Its analytic zeros, poles and winding number govern Wiener-Hopf factorization. For mixed scalar Helmholtz half-plane diffraction a typical kernel is the outgoing vertical wavenumber $\sqrt{k^2-{\alpha }^2}$.

A forcing pole can be split between analytic half-planes by subtracting a factor's value at the pole. For instance,The second quotient has a removable pole and belongs to the plus expression; the first keeps the prescribed minus-side forcing pole. After Wiener-Hopf factorization, analytic continuation identifies an entire remainder, whose value is fixed by growth and edge conditions. The pole prescription must be stated together with the Fourier transform convention.