Perfectly matched layer

Remarks

Check that a plane wave passes into the layer without reflection. Let $u = e^{i(k_x x + k_y y + k_z z)}$ solve the equation for $x < L$, with $k_x > 0$, so that it travels towards the layer. The function $e^{i(k_x \tilde{x} + k_y y + k_z z)}$ solves the stretched equation, and it equals $u$ for $x \le L$. For $x > L$ it equals $u \exp\big(-\frac{k_x}{\omega} \int_L^x \sigma(s) \, ds\big)$, which decays. So the incoming wave continues into the layer without any reflected wave, whatever $k_y$, $k_z$ and $\omega$ are.

After the wave reflects at the end of the layer and comes back, its amplitude is multiplied by $R = \exp\big(-\frac{2 k_x}{\omega} \int_L^{L+\delta} \sigma(s) \, ds\big)$. In vacuum, with $k_x = (\omega / c) \cos \theta$ at incidence angle $\theta$, this is $R = \exp\big(-\frac{2 \cos \theta}{c} \int_L^{L+\delta} \sigma \, ds\big)$, which is small except near grazing incidence.

Non-example: adding a real conductivity to the material in the layer, without the matching magnetic loss, is not a PML. Its wave impedance differs from that of the domain, so part of the wave reflects at $x = L$.

After discretization the layer is no longer exactly reflectionless, so $\sigma$ is made to grow gradually, for example as a power of $x - L$.