The Earth–ionosphere waveguide is dispersive. Its propagation characteristics are described by a transfer function T(ρ, f) depending mainly on distance ρ and frequency f. In the VLF range, only mode one is important at distances larger than about 1000 km. Least attenuation of this mode occurs at about 15 kHz. Therefore, the Earth–ionosphere waveguide behaves like a bandpass filter, selecting this band out of a broadband signal. The 15 kHz signal dominates at distances greater than about 5000 km. For ELF waves (< 3 kHz), ray theory becomes invalid, and only mode theory is appropriate. Here, the zeroth mode begins to dominate and is responsible for the second window at greater distances.