Helmholtz equation ($\Delta u+k^{2}u=0$)

The Helmholtz equation is the constant-coefficient elliptic equationOn a bounded domain, nonzero Dirichlet solutions occur when $k^2$ is an eigenvalue of the negative Laplacian.

A phase screen is a thin inhomogeneous layer idealized as changing only a wave's phase. A random phase screen converts a coherent incident wave into coherent and diffuse components whose moments are determined by the phase covariance.

The parabolic wave equation is a one-way, slowly varying envelope approximation to the Helmholtz equation. For carrier wavenumber $k$ and transverse coordinate $z$, a common form is $2ik{E}_x+E_{zz}=0$.

The coherent field is the ensemble mean $⟨E⟩$ of a randomly scattered wave. The zero-mean remainder $E-⟨E⟩$ is the diffuse or incoherent field.