Fourier transforming the wave equation into frequency space yields, for 10 points, what differential equation, which states that the Laplacian of phi plus k-squared phi equals zero?
Typical multigrid methods tend to work poorly when solving the "indefinite" form of this equation. Arnold Sommerfeld developed a set of boundary conditions for which solutions of this equation are considered "radiating." By substituting a generalized solution into this time-independent equation, ignoring the imaginary part, and taking the limit as frequency becomes infinite, one can derive the eikonal equation. This equation can be rewritten in a special form by expanding one operator into longitudinal and transverse components and expressing the solution in terms of a slowly-varying complex envelope. That form of this equation, which is used to describe Gaussian beams and Fresnel diffraction, is its "paraxial" form. Fourier transforming the wave equation into frequency space yields, for 10 points, what differential equation, which states that the Laplacian of phi plus k-squared phi equals zero?
Typical multigrid methods tend to work poorly when solving the "indefinite" form of this equation.
Helmholtz equation