Wave Speed Formula v = fλ: Worked Examples Explained

The wave speed formula is v = fλ: the speed of a wave (v, in metres per second) equals its frequency (f, in hertz) multiplied by its wavelength (λ, in metres). It works for every wave, from ripples on a rope to sound and light. If you know any two of the three quantities, you can find the third.

What does each part of v = fλ mean?

A wave repeats a pattern in space and in time, and the formula links the two repeats.

• Wavelength (λ) is the length of one complete wave, for example crest to the next crest. It is measured in metres.
• Frequency (f) is how many complete waves pass a point each second. One wave per second is 1 hertz (Hz).
• Wave speed (v) is how fast the pattern travels, in m/s.
• Time period (T) is the time for one wave, T = 1/f. So v = λ/T says the same thing: one wavelength travels in one period.

The formula can be rearranged: f = v/λ and λ = v/f. Always convert to metres, hertz and m/s first (1 kHz = 1000 Hz, 1 cm = 0.01 m).

How do you use the wave speed formula? A method

1. Write down what you know, with units, and what you need.
2. Convert everything to m, Hz and m/s.
3. Rearrange: v = fλ, f = v/λ or λ = v/f.
4. Substitute, calculate, and give the unit.

Worked example 1: find the speed

A rope wave has a frequency of 5 Hz and a wavelength of 0.40 m. v = fλ = 5 × 0.40 = 2.0 m/s.

Worked example 2: find the wavelength

Middle A on a piano is 440 Hz. Sound travels at about 343 m/s in air at 20 °C. λ = v/f = 343 ÷ 440 = 0.78 m.

Worked example 3: find the frequency

A radio wave has a wavelength of 3.0 m. Radio waves travel at the speed of light, 3.00 × 10⁸ m/s. f = v/λ = 3.00 × 10⁸ ÷ 3.0 = 1.0 × 10⁸ Hz = 100 MHz, which is in the FM radio band.

What decides the speed of a wave?

The speed is set by the medium, not by the source. Change the frequency of a vibrating source and the wavelength changes so that v stays the same.

Wave and medium	Approximate speed
Sound in air at 20 °C	343 m/s
Sound in water	1480 m/s
Sound in steel	5960 m/s
Light in a vacuum	3.00 × 10⁸ m/s

Sound travels faster in solids than in gases because the particles are closer together and pass the vibration on more quickly. Sound needs a medium; light does not.

What is the relationship between frequency and wavelength?

At a fixed speed, frequency and wavelength are inversely proportional: double the frequency and the wavelength halves. This is why a high-pitched sound has a short wavelength and a low-pitched sound a long one. A 20 Hz bass note in air has λ = 343 ÷ 20 ≈ 17 m, while a 20 kHz squeak has λ ≈ 1.7 cm.

How does this explain a guitar string?

A plucked string is held at both ends, so the longest wave that fits is half a wavelength: λ = 2L. The fundamental frequency is then f = v/2L, where v is the wave speed along the string. Shorten the string by pressing a fret and L falls, so f rises. A thicker string carries waves more slowly, so it plays a lower note.

Common mistakes

• Mixing units. Using kHz or cm in the formula gives answers that are out by factors of 1000 or 100.
• Confusing amplitude with wavelength. Amplitude is the height of the wave, measured from the rest line; it does not appear in v = fλ.
• Thinking a higher frequency makes a wave faster. In the same medium it only shortens the wavelength.
• Measuring wavelength between a crest and a trough. That is half a wavelength.

Frequently asked questions

What is the formula for wave speed?

Wave speed is v = fλ, where v is speed in m/s, f is frequency in Hz and λ is wavelength in m. It is the same for sound, light and water waves. Rearranged, f = v/λ and λ = v/f. You can also write v = λ/T, because the time period T equals 1/f.

How do you find wavelength from frequency?

Divide the wave speed by the frequency: λ = v/f. For sound in air, use 343 m/s at 20 °C. A 686 Hz note has a wavelength of 343 ÷ 686 = 0.50 m. For light and radio waves, use 3.00 × 10⁸ m/s.

Does wave speed change with frequency?

Not for most waves in a given medium. The medium sets the speed, so changing the frequency changes the wavelength instead. When a wave moves into a new medium, such as sound passing from air into water, the speed changes and the frequency stays the same, so the wavelength changes.

What is the speed of sound in air?

Sound travels at about 343 m/s in air at 20 °C. It is a little slower in colder air (about 331 m/s at 0 °C) and faster in warmer air, because the air particles move faster and pass vibrations on sooner.

Try it yourself

Open the Waves & Sound tool and drag the frequency slider to make waves on the rope. Watch the wavelength shrink as the frequency rises while the speed stays constant, then check your own v = fλ answer against the readout. Next, pluck the 3D guitar and shorten a string to hear f = v/2L. For the related idea of repeating motion, read simple harmonic motion explained.