For the sake of brevity, consider the 1D Equation 2-48. In order to derive the PML formulation in the time domain, the following steps are taken (Ref. 39). First, consider a special form of the mapping Equation 2-47:

Then, taking Equation 2-49 into account, multiply Equation 2-48 by 1+σ(x)/iω. Equation 2-48 transforms to the following form:

The transformation to the time domain is performed according to the rule . Its direct application to Equation 2-50 would result in a time integral of p. To avoid this, an auxiliary variable u is introduced:

Equation 2-50 and Equation 2-51 yield a system of partial differential equations in the time domain equivalent to the frequency domain Equation 2-48:

The derivation of the model in 3D space, where more auxiliary variables are required, is given in Ref. 39.