How are wavelength, frequency and speed related?
Speed equals frequency times wavelength — v = fλ. For a fixed wave speed, frequency and wavelength are inversely proportional: double the frequency and the wavelength halves. Light in a vacuum travels at 299,792,458 m/s, so a 100 MHz FM signal has a wavelength of about 3 metres. Sound in air travels at roughly 343 m/s, so a 343 Hz note has a wavelength of exactly 1 metre.
Does wavelength change when a wave enters a different medium?
Yes, and this is the key insight most people miss. Frequency is set by the source and does not change; speed changes with the medium, so wavelength must change to match. Light entering glass slows to about two-thirds of its vacuum speed, so its wavelength shortens by the same factor while its colour — determined by frequency — stays the same. That change in speed is what causes refraction.
What are typical wavelengths across the spectrum?
Visible light runs from about 700 nm (red) to 400 nm (violet). Beyond red is infrared, then microwaves at centimetres and radio at metres to kilometres. Below violet is ultraviolet, then X-rays at around 0.1 nm and gamma rays smaller still. Audible sound spans roughly 17 metres at 20 Hz down to 17 mm at 20 kHz — far longer than light, which is why sound bends round corners and light does not.
Why do lower frequencies travel further?
Longer wavelengths diffract more readily around obstacles and are absorbed less by the atmosphere and by materials. It is why AM radio reaches beyond the horizon while FM needs line of sight, why bass passes through walls from a neighbour's party while treble does not, and why submarines use extremely low frequency for communication despite the tiny data rates it allows.
How does this relate to photon energy?
Energy equals Planck's constant times frequency, so energy is inversely proportional to wavelength. Short-wavelength radiation carries far more energy per photon — which is why ultraviolet, X-rays and gamma rays are ionising and can damage DNA, while infrared and radio simply heat things. It is the reason a sunbed harms skin and a radio transmitter does not.
What is the relationship to antenna length?
Antennas are sized as fractions of the wavelength they handle — commonly a quarter or a half. A quarter-wave antenna for 100 MHz FM is about 75 cm, which is why car aerials are that length. It is also why 2.4 GHz Wi-Fi antennas are small: the wavelength is only about 12.5 cm, so a quarter-wave element is roughly 3 cm.