Parabolic Equation of Diffraction Theory: Why It Works Better than Expected?

It is shown for the classical half-plane diffraction problem why parabolic equation of diffraction theory provides a good approximations to the exact solution of Helmholtz equation. Particularly, a boundary integral equation is formulated in the parabolic approximation. This equation is analogous to the corresponding integral equation for the initial (Helmholtz) problem. Then it is found out that the kernel of the “parabolic” equation is one of the factors of the factorization (in the Wiener-Hopf sense) of the “Helmholtz” equation kernel. That is why the parabolic method gives an almost exact solution even near the edge of the half-plane, where, seemingly, it should work poorly.