Wave speed:
frequency times wavelength, and what stays fixed

Three quantities that must agree

Any regular wave can be described by three numbers: how often it repeats, how long each repeat is in space, and how fast the pattern travels. Frequency is the count of crests passing a point each second. Wavelength is the distance from one crest to the next. Speed is how fast any crest moves.

Because each second brings f crests, each of length λ, the pattern advances f × λ metres in that second. That is the whole derivation, and it holds for water ripples, sound, light and waves on a string.

Picture a long rope shaken at the end. Each shake sends one crest down the rope. Shake faster and the crests are closer together; shake slower and they spread out. The rope itself sets how fast each crest travels, so the spacing is the only thing you can change by changing your rhythm.

The wave equation and its rearrangements

Use SI units throughout. Frequency in kilohertz or megahertz should be converted to hertz, and wavelengths in centimetres or millimetres to metres, before multiplying.

Because the period is T = 1 ÷ f, the equation can also be written v = λ ÷ T: one wavelength per period.

Worked example: 20 Hz and a 3 m wavelength

Suppose the source is retuned to 40 Hz but the medium is unchanged, so the speed stays at 60 m/s. The new wavelength is 60 ÷ 40 = 1.5 m. Doubling frequency halved the wavelength, which is the central behaviour of the wave equation.

Going the other way, a 2 m wavelength at 60 m/s needs a frequency of 30 Hz.

It also helps to visualise. At 20 Hz, 20 crests pass any fixed point every second, each separated by 3 m. In one second the leading crest has moved through 20 wavelengths, which is 60 m. That picture is the same as the formula, only slower to say.

The medium fixes the speed, the source fixes the frequency

In a given medium under given conditions, the wave speed is a property of the medium: tension and mass per length for a string, temperature for sound in air, depth for shallow water. Changing the frequency does not change that speed, so the wavelength has to adjust.

When a wave crosses from one medium into another, the frequency stays the same because the source sets it, while the speed and wavelength both change. A light wave entering glass slows down and its wavelength shortens, but its colour, which is tied to frequency, does not shift.

Typical speeds and the wavelengths that result

The radio row shows how a wave with an enormous frequency still has a human-scale wavelength because light is so fast. Antennas are sized on this basis, with common designs using a half or quarter wavelength.

Sound in air varies with temperature, rising by about 0.6 m/s for each degree Celsius, so a figure of 343 m/s applies near 20 °C only.

Sound in other media is much faster than in air: roughly 1,480 m/s in water and several thousand metres per second in steel. At the same 440 Hz, the wavelength in water is therefore about 3.4 m. The frequency is the same, but the pattern is stretched out because the wave moves faster.

A routine for wave problems

1. 1Write down which two of the three quantities you know.
2. 2Convert each to SI: hertz for frequency, metres for wavelength, metres per second for speed.
3. 3Choose the arrangement: multiply for speed, divide speed by frequency for wavelength.
4. 4Check the result with the other form, using period if it helps.

A general rate calculator can handle simple quantity-per-time conversions that sit next to these problems, though it is not specific to the wave equation itself.

As a final check, estimate the order of magnitude. If your sound wavelength comes out in kilometres or millimetres for a note you can hear, a unit has slipped. Audible sound in air has wavelengths from a couple of centimetres to about 17 m, which makes a handy sanity range.

Where the simple rule stops working

• Dispersive media, where speed depends on frequency, such as light in glass or water waves in deep water. Then v = f × λ still holds for each frequency, but v is not a single constant.
• Standing waves, where the pattern does not travel, though the equation still links the frequency and wavelength of the underlying travelling waves.
• Very large waves in shallow water, where amplitude and depth also change the speed.
• Moving sources or observers, which shift the observed frequency through the Doppler effect.

For introductory problems and most engineering checks, the formula as written is exact, and the main task is keeping the units straight.

A related check involves energy. Frequency and wavelength alone say nothing about how strong a wave is; amplitude does that. Two waves with the same f and λ travel at the same speed whether they are gentle ripples or large swells, which is why the wave equation never needs the height of the wave.