The Schumann resonances are a set of natural electromagnetic resonances of the Earth's ionosphere. These resonances occur in the extremely low-frequency (ELF) range of the electromagnetic spectrum. The fundamental Schumann resonance, known as the first mode, has a frequency of approximately 7.83 Hz.
To understand why the Schumann frequency wavelength is shorter than the smallest circumference of the Earth, we need to consider the relationship between frequency, wavelength, and the speed of light. The speed of light in a vacuum is a constant value, approximately 299,792 kilometers per second (km/s).
The wavelength (λ) of a wave is inversely proportional to its frequency (f) according to the equation:
λ = c / f
where c is the speed of light. Since the Schumann resonance has a relatively low frequency of 7.83 Hz, its corresponding wavelength is larger compared to higher-frequency waves.
In the case of the fundamental Schumann resonance with a frequency of 7.83 Hz, its wavelength can be calculated using the speed of light:
λ = 299,792 km/s / 7.83 Hz
This calculation gives us a wavelength of approximately 38,247 kilometers.
The smallest circumference of the Earth is the equatorial circumference, which is roughly 40,075 kilometers. As you correctly inferred, the Earth's rotation plays a significant role in the formation of the Schumann resonances. The resonances are created by the global electrical activity in the Earth-ionosphere cavity, with lightning discharges acting as a primary source. The Earth's surface, combined with the ionosphere, forms a waveguide that supports these resonant frequencies. The circumference of the Earth influences the way these waves propagate and create standing waves within this waveguide.
Regarding longer wavelengths, higher-order modes of the Schumann resonances do exist, such as the second, third, and fourth modes. These modes have longer wavelengths and correspondingly higher frequencies. The second mode, for example, has a frequency of around 14.3 Hz and a wavelength of approximately twice that of the fundamental mode.
In summary, the Schumann frequency wavelength is shorter than the smallest circumference of the Earth due to the inverse relationship between wavelength and frequency. The Earth's rotation and the Earth-ionosphere cavity create a waveguide that supports these resonant frequencies, forming the Schumann resonances. Additionally, higher-order modes with longer wavelengths and higher frequencies do exist in the Schumann resonance spectrum.