Plate, 1-D, Steady-Periodic GF

X11 Plate, with $G=0$ (Dirichlet) at x=0 and at x=L.

$$G_{X11}(x,x^{\prime},\omega)=\frac{1}{2 \alpha \sigma} \left... ...e^{- \sigma [2L-\vert x+x'\vert]}}{1-e^{-2 \sigma L}} \right ]$$

$$\sigma=(1+\it {i})\sqrt{\frac{\omega}{2\alpha}}$$

X12 Plate, with $G=0$ (Dirichlet) at x=0 and $\partial{G}/\partial{x}=0$ (Neumann) at x=L.

$$G_{X12}(x,x^{\prime},\omega)=\frac{1}{2 \alpha \sigma} \left... ...-e^{-\sigma [2L-\vert x+x'\vert]}}{1+e^{-2 \sigma L}} \right ]$$

X13 Plate, with $G=0$ (Dirichlet) at x=0 and $\it {k}\partial{G}/\partial{x}+\it {h}G=0$ (Convection) at x=L.

$$G_{X13}(x,x^{\prime},\omega)=\frac{1}{2 \alpha \sigma} \left... ...ert x-x'\vert]} \right \}}{1+R \cdot e^{-2 \sigma L}} \right ]$$

$$R~=~ \frac{k \sigma - h}{k \sigma + h}$$

X21 Plate, with $\partial{G}/\partial{x}=0$ (Neumann) at x=0 and $G=0$ (Dirichlet) at x=L.

$$G_{X21}(x,x^{\prime},\omega)=\frac{1}{2 \alpha \sigma} \left... ...-e^{-\sigma [2L-\vert x+x'\vert]}}{1+e^{-2 \sigma L}} \right ]$$

X22 Plate, with $\partial{G}/\partial{x}=0$ (Neumann) at x=0 and at x=L.

$$G_{X22}(x,x^{\prime},\omega)=\frac{1}{2 \alpha \sigma} \left... ...e^{- \sigma [2L-\vert x+x'\vert]}}{1-e^{-2 \sigma L}} \right ]$$

X23 Plate, with $\partial{G}/\partial{x}=0$ (Neumann) at x=0 and $\it {k}\partial{G}/\partial{x}+\it {h}G=0$ (Convection) at x=L.

$$G_{X23}(x,x^{\prime},\omega)=\frac{1}{2 \alpha \sigma} \left... ...ert x+x'\vert]} \right \}}{1-R \cdot e^{-2 \sigma L}} \right ]$$

X31 Plate, with $\it {k}\partial{G}/\partial{x}+\it {h}G=0$ (Convection) at x=0 and $G=0$ (Dirichlet) at x=L.

$$G_{X31}(x,x^{\prime},\omega)=\frac{1}{2 \alpha \sigma} \left... ...ert x-x'\vert]} \right \}}{1+R \cdot e^{-2 \sigma L}} \right ]$$

X32 Plate, with $\it {k}\partial{G}/\partial{x}+\it {h}G=0$ (Convection) at x=0 and $\partial{G}/\partial{x}=0$ (Neumann) at x=L.

$$G_{X32}(x,x^{\prime},\omega)=\frac{1}{2 \alpha \sigma} \left... ...ert x-x'\vert]} \right \}}{1-R \cdot e^{-2 \sigma L}} \right ]$$

X33 Plate, with $\it {k}\partial{G}/\partial{x}+\it {h_1}G=0$ (Convection) at x=0 and $\it {k}\partial{G}/\partial{x}+\it {h_2}G=0$ (Convection) at x=L.

$$G_{X33}(x,x^{\prime},\omega)=\frac{1}{2 \alpha \sigma} \left... ...vert x+x'\vert]}}{1-R_{1}R_{2} \cdot e^{-2 \sigma L}} \right ]$$

$$R_{1}~=~\frac{k \sigma - h_{1}}{k \sigma + h_{1}}$$

$$R_{2}~=~\frac{k \sigma - h_{2}}{k \sigma + h_{2}}$$