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The purpose of this work is twofold. First, we present a new simple model of photonic crystal structures that can be treated analytically. Second, from the rigorous analysis of propagation and resonance of the models, we point out two novel properties of waves in the structure. The first is that there is a waveguide in which a leakage-free guided mode can have the same propagation constant (wavenumber) as that of continuum waves. The second novel property is that there is a resonator in which the wave can be localized, even in the absence of a "full bandgap." These facts disprove some "common beliefs" about photonic crystal structures: many people believe that (1) in a photonic crystal waveguide, a radiation-free guided mode cannot have the same wavenumber as that of continuum modes and (2) in a photonic crystal resonator, lossless localization can take place only if the host photonic crystal has an absolute bandgap. Our examples show that such beliefs are overstatements.