The 2D scalar wave equation I am trying to solve is,

∇ · (μ∇u) − \frac{1}{c_{0}^{2}} \frac{∂^{2}}{∂t^{2}} u = 0 , x ≥ 0

with boundary force

\frac{∂u}{∂x} = f , at x = 0

In weak form, the augmented equation on the region with PML attached at one end along the x direction looks like

∫_{Ω} (\frac{1}{c_{0}} \frac{∂}{∂t} + σ(x))^{2} \frac{1}{c_{0}^{2}} \frac{∂^{2}}{∂t^{2}} u w dV  (1)
− ∫_{Γ_{f}} μ (\frac{1}{c_{0}} \frac{∂}{∂t} + σ(x)) \frac{1}{c_{0}} \frac{∂f}{∂t} wdS  (2)

+ ∫_{Ω} μ ( \begin{matrix}\frac{1}{c_{0}^{2}} \frac{∂^{2}}{∂t^{2}} & 0 \\ 0 & (\frac{1}{c_{0}} \frac{∂}{∂t} + σ(x))^{2} \end{matrix} ) ∇u · ∇w dV = 0  (3)

where σ ≡ 0 on the regular region but is nonzero on PML. Notice that when σ = 0, the equation degenerates to the original wave equation after integrating twice in time.