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
Atmospheric tide The latitudinal structure of the tides is described by the "horizontal structure equation" which is also called "Laplace's tidal equation": with "Laplace operator" using formula_30, formula_31 and "eigenvalue" Hence, atmospheric tides are eigenoscillations (eigenmodes)of Earth's atmosphere with eigenfunctions formula_33, called Hough functions, and eigenvalues formula_34. The latter define the "equivalent depth" formula_35 which couples the latitudinal structure of the tides with their vertical structure. Longuet-Higgins has completely solved Laplace's equations and has discovered tidal modes with negative eigenvalues ε (Figure 2). There exist two kinds of waves: class 1 waves, (sometimes called gravity waves), labelled by positive n, and class