Difference between revisions of "Template:Energy contour and preliminaries"
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Based on the method used in [[Evans and Davies 1968]], a check can be made to ensure the solutions energy balance. | Based on the method used in [[Evans and Davies 1968]], a check can be made to ensure the solutions energy balance. | ||
The energy balance equation is derived by applying Green's theorem to <math>\phi</math> and its conjugate. | The energy balance equation is derived by applying Green's theorem to <math>\phi</math> and its conjugate. | ||
− | The domain of integration is shown in the figure on the right. | + | The domain of integration is shown in the figure on the right. We assume that the angle is sufficiently small that we do not get total reflection. |
− | [[Image:energy_schematic.jpg|thumb|right| | + | [[Image:energy_schematic.jpg|thumb|right|410px|A diagram depicting the area <math>\Omega</math> which is bounded by the rectangle <math>\partial\Omega</math>. The rectangle <math>\partial\Omega</math> is bounded by <math> -h\leq z \leq0</math> and <math>-\infty\leq x \leq \infty</math>]] |
− | |||
− | The rectangle <math>\ | ||
Applying Green's theorem to <math>\phi</math> and its conjugate <math>\phi^*</math> gives | Applying Green's theorem to <math>\phi</math> and its conjugate <math>\phi^*</math> gives | ||
<center><math> | <center><math> | ||
− | { \ | + | { \iint_\Omega\left(\phi^*\nabla^2\phi - \phi\nabla^2\phi^* \right)\mathrm{d}x\mathrm{d}z |
− | = \int_\ | + | = \int_{\partial\Omega}\left(\phi^*\frac{\partial\phi}{\partial n} - \phi\frac{\partial\phi^*}{\partial n} \right)\mathrm{d}l }, |
</math></center> | </math></center> | ||
where <math>n</math> denotes the outward plane normal to the boundary and <math>l</math> denotes the plane parallel to the boundary. | where <math>n</math> denotes the outward plane normal to the boundary and <math>l</math> denotes the plane parallel to the boundary. | ||
As <math>\phi</math> and <math>\phi^*</math> satisfy the Laplace's equation, the left hand side of the Green theorem equation vanishes so that it reduces to | As <math>\phi</math> and <math>\phi^*</math> satisfy the Laplace's equation, the left hand side of the Green theorem equation vanishes so that it reduces to | ||
<center><math> | <center><math> | ||
− | \Im\int_\mathcal{S}\phi\frac{\partial\phi | + | \Im\int_\mathcal{S}\phi^*\frac{\partial\phi}{\partial n} \mathrm{d}l = 0, |
</math></center> | </math></center> |
Latest revision as of 08:59, 11 March 2009
Based on the method used in Evans and Davies 1968, a check can be made to ensure the solutions energy balance. The energy balance equation is derived by applying Green's theorem to [math]\displaystyle{ \phi }[/math] and its conjugate. The domain of integration is shown in the figure on the right. We assume that the angle is sufficiently small that we do not get total reflection.
![](/files/thumb/6/61/Energy_schematic.jpg/410px-Energy_schematic.jpg)
Applying Green's theorem to [math]\displaystyle{ \phi }[/math] and its conjugate [math]\displaystyle{ \phi^* }[/math] gives
where [math]\displaystyle{ n }[/math] denotes the outward plane normal to the boundary and [math]\displaystyle{ l }[/math] denotes the plane parallel to the boundary. As [math]\displaystyle{ \phi }[/math] and [math]\displaystyle{ \phi^* }[/math] satisfy the Laplace's equation, the left hand side of the Green theorem equation vanishes so that it reduces to