Wave Equation (c = λ·f)

Multiply wavelength and frequency to get the propagation speed. Usually, however, c is known and one of the other two quantities is wanted: λ = c/f or f = c/λ. Example: concert pitch a¹ with f = 440 Hz propagates in air at c = 343 m/s, so its wavelength is λ = 343/440 ≈ 0.78 m. For light insert c = 3×10⁸ m/s: green light with λ = 500 nm = 5×10⁻⁷ m has f = 6×10¹⁴ Hz. Be strict about units; nanometres, centimetres and kilohertz must be converted to metres and hertz before substituting.

The frequency. It is set by the source; the wave in the new medium is driven by the incoming wave and necessarily oscillates in the same rhythm. The propagation speed, however, changes with the medium, and the wavelength adapts with it: λ = c/f. Light entering glass from air slows down (c/n with n ≈ 1.5), its wavelength compresses accordingly, while the frequency and hence the colour stay the same. Exactly this change of speed is the cause of refraction. The common exam mistake is to claim the frequency changes; remember: the rhythm comes from the source, the speed from the medium.

Because light and sound have vastly different propagation speeds. The flash reaches you at about 3×10⁸ m/s, practically instantly; the sound crawls after it at about 343 m/s. Each second of delay therefore means the storm is about 340 m away, giving the familiar rule of thumb: count the seconds between flash and thunder and divide by 3 to get the distance in kilometres. If you count 6 s, the storm is about 2 km away. Echo sounding and ultrasonic ranging use the same principle: from travel time and known speed follows the distance s = c·t (halved for an echo, since the sound travels there and back).

Because antennas are tuned to the wavelength of the signal; they radiate and receive efficiently at lengths of λ/2 or λ/4. The wavelength follows from λ = c/f with c = 3×10⁸ m/s. An FM station at 100 MHz has λ = 3×10⁸/10⁸ = 3 m, so the λ/4 antenna is only 75 cm long, the classic car antenna rod. A long-wave station at 150 kHz, by contrast, has λ = 2,000 m; such stations need enormous masts. Wi-Fi at 2.4 GHz manages with λ ≈ 12.5 cm, which is why the antennas fit inside routers and smartphones. The formula c = λ·f is thus the basis of all radio antenna sizing.

Yes. The relation follows purely from the definitions of wavelength and period and holds equally for sound, water waves, waves on a rope, seismic waves and electromagnetic waves. What differs is the speed c itself: it is a property of the medium and the type of wave. Sound travels at 343 m/s in air, about 1,480 m/s in water and roughly 5,900 m/s in steel; light needs no medium at all and reaches 3×10⁸ m/s in vacuum. In dispersive media there is a subtlety: c additionally depends on frequency, which is why a prism splits white light into colours and water waves of different lengths travel at different speeds. The equation itself nevertheless remains valid for each individual frequency.