Wavelength and Frequency, Converted
After reading this you can convert between wavelength and frequency for light, sound or any wave, know which wave speed to use, and avoid the unit slips that produce answers off by a factor of a thousand or a billion.
What the converter does and one example to anchor it
A wave carries a repeating disturbance through space. Two numbers describe it. The frequency counts how many full cycles pass a fixed point each second, measured in hertz (Hz). The wavelength measures the distance between two consecutive peaks, in metres. These two are not independent. Once you fix the speed of the wave, one determines the other.
Here is the anchor. Tune a radio to an FM station at 100 MHz. That signal is an electromagnetic wave travelling at the speed of light, about 3 \times 10^8 m/s. Its wavelength comes out to 3 metres, roughly the length of a small car. The antenna length inside your radio is not arbitrary: it is sized to that 3 metre wave. This tool does that arithmetic for light in vacuum, sound in air or water, or any custom speed you type in.
When to use it, and when the simple formula fails
Use the direct conversion whenever the wave travels through a single uniform medium at a known, constant speed. Radio and light in vacuum, sound in still air at a fixed temperature, ultrasound in water: all fine.
The formula quietly assumes the speed is one fixed number. That breaks in three common cases. First, dispersion: in glass or water the speed of light depends on wavelength, so blue light travels slower than red and a prism splits white light. Second, temperature and medium changes: the speed of sound in air is 343 m/s at 20 °C but about 331 m/s at 0 °C, a 3.5% shift that moves every wavelength by the same fraction. Third, light in a material: inside glass with refractive index 1.5, light slows to 2 \times 10^8 m/s, so a 550 nm wave in vacuum becomes about 367 nm inside the glass while its frequency stays fixed. When speed changes, frequency is the quantity that stays constant; wavelength is the one that stretches or shrinks.
Frequency is the invariant when a wave crosses into a new medium. A green photon does not become blue when it enters water. Its wavelength shortens because the speed dropped, but the frequency, and therefore the colour your eye reports, is unchanged.
The formula and why it is just speed equals rate times spacing
Here \lambda is the wavelength in metres, v is the wave speed in metres per second, and f is the frequency in hertz (cycles per second). To go the other way, rearrange to f = v / \lambda.
The intuition is a marching band. Suppose players cross a line at 2 per second (that is the frequency) and the whole line advances at 1 m/s (the speed). Then the gap between players must be 0.5 m, because in the half second between two crossings the line moves half a metre. Wavelength is that gap. Faster marching with the same crossing rate spreads the players out; a higher crossing rate at the same speed packs them tighter. So wavelength grows with speed and shrinks with frequency, exactly as v / f says.
One useful product falls out immediately: \lambda f = v. Wavelength times frequency always equals the wave speed. If someone quotes you a wavelength and a frequency, multiply them. You should get the speed of the medium, and if you do not, one of the numbers is wrong or the units are mismatched.
A worked example: the demo FM station
100 MHz radio in vacuum
These are the tool's default values: mode frequency to wavelength, medium light in vacuum, frequency 100 MHz.
- Convert the frequency to hertz.
100 MHzis 100 \times 10^6 = 1 \times 10^8 Hz. - Take the speed of light in vacuum, v \approx 2.998 \times 10^8 m/s. The tool uses this value; rounding to 3 \times 10^8 changes the answer by 0.07%.
- Divide: \lambda = \frac{2.998 \times 10^8}{1 \times 10^8} = 2.998 m.
- Report as roughly
3.00 m. With the rounded speed you get exactly 3 m, which is why the footnote quotes 3 m.
Check it backwards: 2.998 \times 1 \times 10^8 = 2.998 \times 10^8 m/s, the speed of light. The numbers close.
How frequency and wavelength trade off
Because \lambda = v / f is a reciprocal, doubling the frequency halves the wavelength. On a log-log plot this is a straight line with slope -1. The table below fixes the speed of light and steps frequency up by factors of ten. Watch the wavelength drop by the same factor each row.
| Frequency | Wavelength | Everyday label |
|---|---|---|
| 1 MHz | 300 m | AM radio |
| 100 MHz | 3 m | FM radio |
| 2.4 GHz | 0.125 m | Wi-Fi, microwave oven |
| 545 THz | 550 nm | green light |
| 30 EHz | 0.01 nm | X-ray |
Explore the trade-off yourself
Reading the result and the electromagnetic band
For light, the tool also names the band. The boundaries are conventions, not sharp physical walls, but the common divisions are worth memorising in wavelength terms.
- Radio
- longer than about 1 mm (frequencies below 300 GHz). FM at 3 m, AM at hundreds of metres.
- Microwave
- 1 mm to 1 m overlaps the radio range by convention; 2.4 GHz Wi-Fi at 12.5 cm lives here.
- Infrared
- about 700 nm to 1 mm. Thermal radiation from warm objects.
- Visible
- roughly 380 nm (violet) to 700 nm (red). The tool's default 550 nm is green.
- Ultraviolet
- 10 nm to 380 nm. Sunburn wavelengths.
- X-ray and gamma
- shorter than 10 nm, frequencies above 3×10^16 Hz.
When you enter 550 nm in wavelength-to-frequency mode, expect f = \frac{3 \times 10^8}{550 \times 10^{-9}} = 5.45 \times 10^{14} Hz, or 545 THz, squarely in the visible band.
Common mistakes
The failures here are almost always unit slips, and they hit by round factors, so they are easy to spot once you know the pattern.
- Forgetting the prefix on frequency. Entering 100 for a 100 MHz station while the unit still reads Hz gives a 3 million metre wavelength instead of 3 m. Wrong by 10^6.
- Mixing wavelength units. A 550 nm wave is 550 \times 10^{-9} m, not 550 m. Type the number and set the unit; do not pre-convert in your head and then also pick nm.
- Using light speed for sound. A 1 kHz tone in air is 34.3 cm. If you accidentally use 3 \times 10^8 m/s you get 300 km, the radio-wave answer. The two speeds differ by a factor near 874,000.
- Treating the air speed as universal. The 343 m/s value is for 20 °C air. In water sound runs at about 1481 m/s, over four times faster, so the same frequency gives a wavelength four times longer.
Sanity-check any conversion by multiplying your answer back: wavelength times frequency should return the wave speed. If \lambda f is not close to 3e8 for light or 343 for air, a unit is wrong.
Related tools
If you came here from a mechanics problem rather than optics, these calculators cover neighbouring ground. For motion under gravity, see the Projectile Motion Calculator and the Free Fall Calculator. For structures, the Beam Deflection Calculator handles the four standard load cases, and the Gear Ratio Calculator works out output speed and torque. For atmospheric conditions that set the speed of sound, the Wind Chill, Heat Index and Dew Point tool covers the temperature side.
Frequently asked questions
Why does the same 1 kHz mean 34 cm for sound but 300 km for radio?
Because wavelength scales with speed. Sound in air travels at 343 m/s, radio at 3 \times 10^8 m/s. The ratio is about 874,000. At the same frequency the radio wavelength is 874,000 times longer: 0.343 m becomes about 300,000 m.
Which speed should I use for light?
In vacuum, 2.998 \times 10^8 m/s. In air the speed is only 0.03% slower, so vacuum is fine for everyday work. Inside glass or water, divide by the refractive index (about 1.5 for glass, 1.33 for water) and use the custom speed.
Does frequency or wavelength change when light enters glass?
Wavelength changes, frequency stays fixed. The speed drops by the refractive index, and since \lambda = v / f with f held constant, the wavelength shrinks by the same factor. A 600 nm wave becomes 400 nm inside glass of index 1.5.
What speed of sound should I use underwater?
About 1481 m/s in fresh water at 20 °C. It rises with temperature, salinity and pressure, so sonar work uses measured profiles rather than a single number. For a rough estimate 1500 m/s is the common textbook figure.
How do I convert an energy or wavenumber instead?
Photon energy is E = hf with Planck's constant h = 6.626 \times 10^{-34} J·s, so once you have frequency you have energy. Wavenumber, common in spectroscopy, is just 1/\lambda, usually quoted per centimetre.
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