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The properties of surface waves guided by an uncoated cylinder of finite conductivity (Sommerfeld wave) or by a perfectly conducting cylinder with a dielectric coating (Goubau wave) have been analyzed theoretically for frequencies in the millimeter and submillimeter wavelength regions. Previous analysis by Goubau for lower frequencies provided the primary basis for this investigation. For wire sizes which provide low loss propagation above about 100 kMc, the approximations used by earlier workers are not sufficiently accurate to provide useful results. More accurate solutions for some of the properties have been obtained for wires of circular cross section by the use of fewer or different approximations. Numerical values have been calculated for attenuation, power-handling ability, and radial extent of the region of principal power flow. The results show that above 70 kMc the significant field extent is reasonable and the attenuation may be orders of magnitude smaller than for rectangular waveguide. Consideration was also given to the problem of surface-wave propagation on wires of elliptical cross section, but in this case the solutions for the field components must normally be expressed in the form of infinite series of products of the radial and angular Mathieu functions. Numerical results are therefore extremely difficult to obtain for elliptical wires.