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This letter utilizes a simple locally resonant cavity cell (LRCC) model to provide an insight into the physical mechanism and the relationship between locally resonant modes and surface-wave suppression band-gaps of the proposed meandering slotted electromagnetic band-gap (MSEBG) structure. The MSEBG structures have two surface-wave suppression band-gaps that are separately yielded by cross slots and meandering slots. It was proved that the cross slots and the meandering slots produced two different groups of monopolarized (MP) degenerate modes whose corresponding frequencies located within the two surface-wave suppression band-gaps. The dual-band MSEBG structure was fabricated, and an experimental verification was conducted by surface-wave band-gap measurement using a pair of shield loops. By adjusting the dimension of meandering slot, we can design a dual-band-gap EBG structure effectively.