The Wave Equation: v = fλ
Every wave, from ocean swell to a radio broadcast, obeys the same short equation: wave speed equals frequency multiplied by wavelength, written v = fλ. Frequency, measured in hertz, counts how many wave crests pass a point each second. Wavelength, measured in metres, is the distance between one crest and the next. Multiply them and you get the speed at which the wave pattern travels. If a sound wave vibrates four hundred and forty times per second and each wave is zero point seven eight metres long, its speed is four hundred and forty times zero point seven eight, which is three hundred and forty-three point two metres per second — the speed of sound in air at room temperature.
The equation's power lies in its flexibility. Know any two quantities and you can find the third. A radio station broadcasting at one hundred megahertz has a wavelength of about three metres, because the speed of light divided by the frequency gives the wavelength. A guitar string vibrating at one hundred and ten hertz with waves travelling along it at two hundred and twenty metres per second supports a wavelength of two metres, which is why the string's length and tension together determine its pitch. The calculator on this page solves all three arrangements of the equation, so you never need to rearrange it by hand.
Frequency, Wavelength, and Pitch
For sound, frequency is perceived as pitch. Higher frequency means a higher note: a piccolo plays at thousands of hertz, while a bass guitar lives below one hundred hertz. Because the speed of sound in air is roughly constant at a given temperature, higher frequency automatically means shorter wavelength. This inverse relationship is why a small tweeter speaker can reproduce high notes while a large woofer is needed for bass — the speaker cone must move a distance comparable to the wavelength it produces, and bass wavelengths are metres long.
For light, frequency determines colour, though we more often speak of wavelength. Visible light spans wavelengths from about three hundred and eighty nanometres for violet to about seven hundred and fifty nanometres for red, with frequencies around four hundred to seven hundred and ninety terahertz. Beyond the visible range, the same equation continues to work: radio waves have wavelengths of metres to kilometres, microwaves centimetres, X-rays nanometres, and gamma rays shorter still. The electromagnetic spectrum is one continuous family, sorted by a single equation.
Wave Speed Depends on the Medium
A wave's speed is set by the medium it travels through, not by whoever created it. Sound travels at about three hundred and forty-three metres per second in air at twenty degrees Celsius, about one thousand four hundred and eighty metres per second in water, and about five thousand metres per second in steel. The denser and stiffer the medium, the faster sound generally travels, because the particles pass the disturbance along more efficiently. Temperature matters too: sound in air speeds up by roughly zero point six metres per second for each degree Celsius of warming, which is why concert pitches and sonar calculations need temperature corrections.
Light behaves differently: it travels fastest in a vacuum, at two hundred and ninety-nine million seven hundred and ninety-two thousand four hundred and fifty-eight metres per second, and slows down in any material. In water it travels at about seventy-five percent of its vacuum speed, and in glass at about sixty-seven percent. This slowing is what bends light at boundaries, the phenomenon of refraction that makes a straw look broken in a glass of water and allows lenses to focus images. When a wave enters a new medium, its frequency stays the same — it is set by the source — but its speed and wavelength change together to satisfy the equation.
Period and Other Wave Quantities
Period, written T, is the time for one complete wave cycle, and it is simply the reciprocal of frequency: T = 1/f. A four hundred and forty hertz concert A has a period of about two point two seven milliseconds. Amplitude is a separate quantity: it measures how tall the wave is, which for sound corresponds to loudness and for light to brightness. Amplitude does not appear in the wave equation at all, which is a useful reminder that a loud sound and a quiet sound of the same pitch travel at exactly the same speed.
Two more quantities appear in advanced work. Wave number counts how many wavelengths fit into a fixed distance, and angular frequency measures the same idea as frequency but in radians per second rather than cycles per second. Both are rescalings of the same ideas, and neither changes the underlying relationship. For most practical problems — homework, audio work, radio planning, or simple curiosity — speed, frequency, and wavelength are all you need.
Real-World Uses of the Wave Equation
Musicians use the equation constantly, whether they know it or not. Tuning a string instrument means adjusting tension until the wave speed on the string gives the desired frequency for its length. Organ builders choose pipe lengths from the wavelength of the note, with open pipes sounding a wavelength twice their length and stopped pipes four times. Audio engineers use it to place acoustic treatment: absorbing a bass frequency requires material thick enough to interact with a wavelength of several metres, which is why bass traps are large.
Radio engineers use it to size antennas, which work best when their length is a simple fraction of the wavelength — a quarter-wave antenna for a one hundred megahertz station is about seventy-five centimetres long. Medical ultrasound uses high frequencies and therefore short wavelengths, giving fine resolution inside the body, at the cost of shallower penetration. Seismologists use it to locate earthquakes, comparing the arrival times of waves that travel at different speeds through the Earth. One equation, an extraordinary range of applications.
Common Mistakes
The most common mistake is mixing units: frequency in kilohertz with wavelength in centimetres gives a speed that is off by a factor of one hundred thousand unless the units are converted first. Always convert to hertz and metres before multiplying, or use the unit conversions in this tool and check the result. Another mistake is confusing wavelength with amplitude, or frequency with speed — a high note is not a fast wave in the sense of travel speed, since all sounds in the same air travel at the same speed regardless of pitch. Finally, remember that the equation describes the wave pattern's speed, not the speed of the medium itself: air molecules in a sound wave jiggle back and forth but do not travel with the sound.
Use this calculator whenever two wave quantities are known and the third is needed, and use the medium presets to check your intuition: sound in air, sound in water, and light in vacuum are the three speeds worth memorising, because nearly every other wave problem is a variation on one of them.
Measuring Waves Yourself
You can verify the wave equation at home with surprisingly simple equipment. For sound, a phone app that displays frequency can measure the note from a guitar string or a tuning fork, and a tape measure gives you the wavelength once you know the speed of sound in air at your room temperature. For water waves, a shallow tray of water and a ruler let you time waves travelling a known distance, and dividing distance by time gives speed directly. Comparing your measured speed with the value predicted by frequency times wavelength closes the loop between theory and experiment, and any discrepancy usually points to a measurement worth repeating — a mistimed stopwatch, a misread ruler, or a temperature you forgot to account for.