Obstacle-insensitive eigenfields due to boundary condition–symmetry compatibility

It is known that the eigenspectra and eigenfields of wave systems are sensitive to boundary conditions. We show a counterintuitive result that two hard-boundary sound cavities of different boundary shapes and of different sizes have a common set of eigenfrequencies. Moreover, their eigenfields at shared frequencies are the same in some of their common subregions. The reason is that both cavities can be created by cutting along locally mirror-symmetric lines in a honeycomb lattice, with their boundaries compatible with the hexagonal cavity's eigenfields at those common eigenfrequencies. Based on this observation, we have discovered a phenomenon where cavity eigenfields remain unaffected by obstacles of certain shapes within the cavity. The key lies in aligning the obstacles' boundaries with the antinodal lines of the cavity eigenfields, which are easy to find when the cavity contains some local mirror symmetries. All the results can be generalized to electromagnetic waves and three-dimensional cavities.