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Everyone has some experience of how radio turns into static when one is driving into a tunnel. It just does not work (unless there is a special radiating cable installed inside the tunnel). Once upon a night, I could not get sleep pondering on why does this happen. From my experience, mobile phone signals seem to work better. The mobile phone signals seem to work. Someone might have told me that FM radio wavelength is just so large that the wave “does not fit the tunnel”. But as intuitively acceptable as this explanation could be (I don't think it is even that), it is not a satisfying explanation as such.
After some research, I found that the explanation lies in the behaviour of Maxwell equations (well, what a surprise). All backs down to the boundary conditions of the electromagnetic field at the tunnel wall: the tangential component of the electric field component of the electromagnetic wave has to be near to zero in the tunnel interface. This leads to a situation where the wave has, in a sense, no room the oscillate inside the tunnel.
Intuitive or not, this is what the equations tell us. To demonstrate this fact I solved, using FEM, the Helmholtz equation for a plane-wave in a two-dimensional setting mimicking a tunnel and its entrance. The electromagnetic wave's polarization is so that the electrical component is pointing at the right angle w.r.t. the plane (either towards or against the reader of this page). Boundary conditions inside the 2D tunnel are zero. Solution colouring, shown in the figures following, represents the magnitude of the electrical component.
We compare the behaviour of two different wavelengths:
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