  For a spinless quantum particle in a one-dimensional box or an
electromagnetic wave in a one-dimensional cavity, the respective Dirichlet and
Neumann boundary conditions both lead to non-degenerate wave functions.
However, in two spatial dimensions, the symmetry of the box or microstrip
antenna is an important feature that has often been overlooked in the
literature. In the high-symmetry cases of a disk, square, or equilateral
triangle, the wave functions for each of those two boundary conditions are
grouped into two distinct classes, which are one- and two-dimensional
representations of the respective point groups, $C_{\infty v}$, $C_{4v}$, and
$C_{3v}$. Here we present visualizations of representative wave functions for
both boundary conditions and both one- and two-dimensional representations of
those point groups. For the one-dimensional representations, color contour
plots of the wave functions are presented. For the two-dimensional
representations, the infinite degeneracies are presented as common nodal points
and/or lines, the patterns of which are invariant under all operations of the
respective point group. The wave functions with the Neumann boundary conditions
have important consequences for the coherent terahertz emission from the
intrinsic Josephson junctions in the high-temperature superconductor
Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$: the enhancement of the output power from
electromagnetic cavity resonances is only strong for wave functions that are
not degenerate.
