Wave Speed, Frequency and Wavelength
Whether it is a ripple crossing a pond, a musical note, or a beam of light, every wave can be described by just three numbers: how fast it travels, how often it wiggles, and how long each wiggle is. A single elegant equation links them, and it is one of the most useful relationships in all of physics.
Meet the three quantities
Before the equation, it helps to picture exactly what each quantity means:
- Wavelength (λ) — the distance between two identical points on the wave, such as crest to crest. Measured in metres.
- Frequency (f) — how many complete waves pass a fixed point each second. Measured in hertz (Hz), where 1 Hz is one cycle per second.
- Wave speed (v) — how fast a crest travels through the medium. Measured in metres per second.
A related quantity is the period (T), the time for one full wave to pass. Period and frequency are simply reciprocals: T = 1/f. A high-frequency wave has a short period and vice versa.
The wave equation
The three quantities are bound together by the wave equation, one of the simplest and most far-reaching formulas in physics:
The logic is direct. In one second, f complete waves pass by, and each is λ long, so the total length of wave that streams past is f × λ — and that length per second is exactly the speed. Memorise this single relationship and you can find any one of the three quantities given the other two.
For a given medium, the wave speed is usually fixed. That means frequency and wavelength are inversely linked: raise the frequency and the wavelength must shrink to keep their product constant. Higher pitch means shorter wavelength.
Speed is set by the medium
A crucial subtlety: it is usually the medium, not the source, that determines wave speed. Sound travels at roughly 343 m/s in air at room temperature regardless of whether you whisper or shout. If you sing a higher note (higher f), the wavelength automatically adjusts so that v stays the same.
Different media give different speeds. Sound moves faster in water than in air, and faster still in steel, because the medium’s stiffness and density control how quickly the disturbance is passed along. Light, by contrast, travels fastest in a vacuum at about 3 × 10⁸ m/s and slows down in glass or water.
A worked example
Suppose a radio station broadcasts at a frequency of 100 MHz, which is 100 × 10⁶ Hz. Radio waves are light, travelling at v = 3 × 10⁸ m/s. What is their wavelength?
So the waves are three metres long — which is why FM radio aerials are sized in metres. The same calculation, run backward, lets engineers choose antenna lengths to match the wavelengths they want to receive. Try variations with our wave speed calculator.
From sound to the whole spectrum
The same equation spans wildly different waves:
- Audible sound ranges from about 20 Hz to 20,000 Hz, giving wavelengths from many metres down to centimetres.
- Visible light has frequencies around 10¹⁴ Hz, with tiny wavelengths of a few hundred nanometres — red being longer, violet shorter.
- Radio, microwaves, infrared, ultraviolet, X-rays and gamma rays are all light, differing only in frequency and wavelength, all obeying v = fλ.
This unity is one of physics’ great themes: a single equation governs everything from a double-bass string to a gamma-ray burst. To go deeper into the physical motion behind these numbers, see transverse and longitudinal waves.
Why pitch and colour are really frequency
Our senses respond to frequency. A high musical pitch is a high-frequency sound wave; a low rumble is a low-frequency one. For light, colour is frequency: red light is lower frequency than blue. Because speed in a given medium is fixed, you could equally describe these by wavelength, which is why scientists quote light colours in nanometres and musicians sometimes think in terms of wave size. Both descriptions are two sides of the same v = fλ coin, and the link to energy is explored in the electromagnetic spectrum.
Frequently asked questions
If I shout louder, does the sound travel faster?
No. Loudness is the wave’s amplitude, which carries more energy but does not change the speed. In a given medium at a given temperature, sound travels at a fixed speed regardless of how loud it is.
Why does a wave’s wavelength change when it enters a new medium?
When a wave crosses into a new medium its speed changes, but its frequency stays the same because the source still oscillates at the same rate. To keep v = fλ consistent, the wavelength must change. This is what bends light as it enters glass.
What is the difference between frequency and period?
Frequency counts cycles per second; period is the time for one cycle. They are reciprocals: T = 1/f. A 50 Hz wave has a period of 1/50 = 0.02 seconds.
