The dense spectrum of periodic plasma-filled waveguide

The dispersion relation for low-frequenncy plasma waves in a rippled-wall waveguide ($w<wp$) is derived under the nonlocalized condition that the contour integral from the electric field along the rippled boundary is equal the new branches of a spectrum, which are produced by the scattering of waves at finite amplitude ripple. These new modes satisfy the integral from of Maxwell's equations under the above nonlocalized boundary condition.