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If we consider the Helmholtz equation for electromagnetic wave propagation and let $\partial \Omega$ be the surface of a perfect conductor with Dirichlet boundary conditions, ie. $u(x) = 0$ for $x \in \partial \Omega$, then any incoming waves will be completely reflected.

If we change to Neumann boundary conditions, ie. $\frac{\partial u(x)}{\partial \nu} = 0$ for $x \in \partial \Omega$, these conditions imply no flux through the surface of the perfect conductor. But as it is a perfect conductor it is impossible to have flux through the surface anyway! So the waves reflect completely just like with the Dirichlet boundary conditions.

So it seems using Dirichlet or Neumann boundary conditions on the surface of a perfect conductor are equivalent? Am I missing something here, what exactly do these conditions mean for the field when they are placed on the surface of a perfect conductor? Is there any difference between them?

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You need to choose particular type of boundary conditions for particular components of the EM wave. Namely, normal component of $\vec E$ must have Neumann condition at the perfect uncharged conductor-vacuum interface, while tangential component must have Dirichlet condition.

Indeed, both conditions don't allow the field to flow through the interface, resulting in reflection. But the difference does exist. It will be apparent when you consider two parallel perfectly conducting planes and calculate the normal modes of EM field between them. You'll find that when you have Neumann conditions, they allow a constant solution, so lowest-frequency mode is actually constant $\vec E$-field. And when you have Dirichlet conditions, your lowest frequency is the higher the smaller the gap between the surfaces.

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