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An analysis of electromagnetic wave propagation in waveguides (in TH and TE modes) as well as in TEM transmission lines (e.g., coaxial) at frequencies in the extremely high frequency band (30 to 300 kmc) and higher is carried out. It is shown that in the case of metallic waveguides excited in TE0 modes, the attenuation at sufficiently high frequencies is always proportional to (frequency)-3/2. On the other hand with TH waves (including TEM waves) although the attenuation at moderately high frequencies is proportional to (frequency)1/2, yet at sufficiently high frequencies the attenuation drops, becoming proportional to (frequency)-5/2. In other words, with TH waves the attenuation becomes a more rapidly decreasing funcion than it is with TE waves. The analysis is concluded with numerical examples.