The Wave Equations node is the main node for the Electromagnetic Waves, Time Explicit interface. The governing transient equations can be written in the form

with the constitutive relations B = μ0μrH and D = ε0εrE, which reads

The default Relative permittivity εr (dimensionless), Relative permeability μr (dimensionless), and Electric conductivity σ (SI unit: S/m) take values From material. For User defined select Isotropic, Diagonal, Symmetric, or Full and enter values or expressions in the field or matrix.

Select the Flux type — Lax–Friedrichs or Upwind flux.

Enter a Scaling factor αupwind, the default is 1 (dimensionless). When it is set to 0, the flux type becomes the central flux.

To display this section, click the Show More Options button () and select Advanced Physics Options in the Show More Options dialog.

where V is a Vandermonde matrix induced by the node points, and Λ is a diagonal matrix with the exponential damping factors on the diagonal:

and Np is the basis function and im the polynomial order for coefficient m. Furthermore, α, ηc, and s (for default values, see below) are the filter parameters that you specify in the corresponding text fields. The damping is derived from a spatial dissipation operator of order 2s. For s = 1, you obtain a damping that is related to the classical 2nd-order Laplacian. Higher order (larger s) gives less damping for the lower-order polynomial coefficients (a more pronounced low-pass filter), while keeping the damping property for the highest values of η, which is controlled by α. Maximal damping is obtained for η = 1. It is important to realize that the effect of the filter is influenced by how much of the solution (energy) is represented by the higher-order polynomial coefficients. For a well resolved solution this is a smaller part than for a poorly resolved solution. The effect is stronger for poorly resolved solutions than for well resolved ones. This is one of the reasons why this filter is useful in an absorbing layer where the energy is transferred to the higher-order coefficients through a coordinate transformation. See Ref. 1 (Chapter 5) for more information.

α must be positive; α = 0 means no dissipation, and the maximum value is related to the machine precision, −log(ε), which is approximately 36. ηc should be between 0 and 1, where ηc = 0 means maximum filtering, and ηc = 1 means no filtering, even if filtering is active.

Set Filter settings to Automatic (the default), Only use filter on absorbing layers, or User defined. When Automatic is selected, α = 0.5 in Absorbing Layer domains and α = 0.05 in all other domains, and ηc = 0.01 and s = 4 in all domains. When Only use filter on absorbing layers is selected, the filter parameters in Absorbing Layer domains are taken from the settings in the Filter Parameters for Absorbing Layers section for The Electromagnetic Waves, Time Explicit Interface, whereas there is no filtering in non-Absorbing Layer domains. Finally, when User defined is selected, set values for α (default value: 0.05), ηc (default value: 0.1), and s (default value: 4) for non-Absorbing Layer domains, whereas the filter parameters in Absorbing Layer domains are taken from the settings in the Filter Parameters for Absorbing Layers section.