Radio atmospheric 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. Resonant waves of this zeroth mode can be excited in the Earth–ionosphere waveguide cavity, mainly by the continuing current components of lightning flowing between two return strokes. Their wavelengths are integral fractions of the Earth's circumference, and their resonance frequencies can thus be approximately determined by "f" ≃ "mc"/(2π"a") ≃ 7.5 "m" Hz (with "m" = 1, 2, ...; "a" the Earth's radius and "c" the
Spherical Earth below the horizon). Because of their longer wavelengths, radio waves are even more susceptible to atmospheric refraction and reflection, which can cause radio and television signals to be received from towers thousands of miles away which cannot be seen with visible light. When the atmosphere is relatively well-mixed, the visual effects generally expected of a spherical Earth can be observed. For example, ships travelling on large bodies of water (such as the ocean) disappear over the horizon progressively, such that the highest part of the ship can still be seen even when lower parts cannot, proportional to distance from the