The Wiener–Hopf perspective on embedding formula: reusing solutions of boundary value problems

Embedding formula allows solutions to be reused within a family of boundary value problems by expressing a family of solutions in terms of a small number of solutions. Such formulas have been previously derived in the context of diffraction by applying a cleverly chosen operator to the solution and the construction of edge Green’s functions, which are introduced in an elaborate manner specific for each problem. We demonstrate that embedding formula naturally appears from a matrix Wiener–Hopf equation, and the embedding formula is derived from the normal solution to this matrix Wiener–Hopf problem. This allows us to drive the embedding formula in any context where the problem can be formulated as a Wiener–Hopf equation. We illustrate the effectiveness of this approach by revisiting known problems, such as the problem of diffraction by a half-plane, a strip and the problem of diffraction by a wedge (for which a new method for deriving a matrix Wiener–Hopf formulation is presented).