Equivalent Finite Difference Scheme

We have

$$v_{lm}(n+1) = \frac{1}{2}\left[ v_{l,m+1}^{-\textsc{s}}(n) + v_{... ...textsc{w}}(n) + v_{l,m-1}^{-\textsc{n}}(n) + v_{l-1,m}^{-\textsc{e}}(n)\right]$$

$$v_{lm}(n-1) = \frac{1}{2}\left[ v_{l,m+1}^{+\textsc{s}}(n) + v_{... ...textsc{w}}(n) + v_{l,m-1}^{+\textsc{n}}(n) + v_{l-1,m}^{+\textsc{e}}(n)\right]$$

Adding gives a finite difference equation satisfied by the mesh

$$\zbox{v_{lm}(n+1) + v_{lm}(n-1) = \frac{v_{l,m+1} + v_{l+1,m} + v_{l,m-1} + v_{l-1,m}}{2}}$$

• Physical variables only (no traveling-wave components)
• Omitted time arguments are all '$(n)$'

Subtracting $2v_{lm}(n)$ from both sides yields

$$\begin{eqnarray*} \lefteqn{v_{lm}(n+1) - 2 v_{lm}(n) + v_{lm}(n-1)} \\ &=& \fra... .... \left[v_{l+1,m}(n) - 2 v_{lm}(n) + v_{l-1,m}(n)\right]\right\} \end{eqnarray*}$$

or, assuming $X=Y$ (``square hole'' case),

$$\begin{eqnarray*} \lefteqn{\frac{v_{lm}(n+1) - 2 v_{lm}(n) + v_{lm}(n-1)}{T^2}} ... ...ft.\frac{v_{l+1,m}(n) - 2 v_{lm}(n) + v_{l-1,m}(n)}{X^2}\right]. \end{eqnarray*}$$

In the limit,

$$\frac{\partial^2 v(x,y,t)}{\partial t^2} = \frac{X^2}{2T^2} \left... ...l^2 v(x,y,t)}{\partial x^2} + \frac{\partial^2 v(x,y,t)}{\partial y^2} \right]$$

i.e., the ideal 2D wave equation

$$\frac{\partial^2 v}{\partial t^2} = c^2 \left[ \frac{\partial^2 ... ...v}{\partial y^2} \right] \mathrel{\stackrel{\mathrm{\Delta}}{=}}c^2\nabla ^2 v$$

where $\nabla ^2$ denotes the Laplacian, and

$$c = \frac{1}{\sqrt{2}}\frac{X}{T}$$