Physics

Waves: wavelength, frequency and speed

A wave is a disturbance that carries energy from place to place without carrying the material along with it. Its wavelength λ\lambda is the distance between two crests, its frequency ff is how many waves pass each second, and its speed is v=fλv = f\lambda. Transverse waves, like waves on a string, shake across the direction they travel. Longitudinal waves, like sound, shake along it.

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Key ideas

v=fλT=1fv = f \lambda \qquad T = \frac{1}{f}
  • vv is the wave speed, in meters per second (m/s).
  • ff is the frequency, in hertz (Hz). 1 Hz is one wave per second.
  • λ\lambda (lambda) is the wavelength, in meters (m): crest to crest, or compression to compression.
  • TT is the period, the time for one full wave to pass, in seconds (s).
  • Amplitude is how far the medium moves from its rest position. Bigger amplitude means more energy (louder sound, brighter light), but it does not change the speed.

The speed of a wave is set by the medium it travels through, not by how fast you shake it. Shake a rope faster and you get a higher frequency with a shorter wavelength, at the same speed. Sound in air at 20 °C travels about 343 m/s. Light in a vacuum travels 3.00×1083.00 \times 10^8 m/s.

Transverse vs longitudinal
TransverseLongitudinal
Medium movesAt right angles to the wave's travelBack and forth along the wave's travel
PartsCrests and troughsCompressions and rarefactions
ExamplesWaves on a string, light, the S waves of earthquakesSound, a pushed Slinky, the P waves of earthquakes

Worked examples

Example 1: wavelength of a musical note

Problem The note A above middle C has a frequency of 440 Hz. In air where sound travels at 343 m/s, what is its wavelength?

  1. Solve v=fλv = f\lambda for λ\lambda.
    λ=vf=343 m/s440 Hz=0.7795 m\lambda = \frac{v}{f} = \frac{343\ \text{m/s}}{440\ \text{Hz}} = 0.7795\ \text{m}
  2. Round to 3 significant figures. That is about 78 cm, a bit less than a meter stick.

Answer 0.780 m

Example 2: an FM radio wave

Problem An FM station broadcasts at 98.1 MHz. Radio waves travel at 3.00×1083.00 \times 10^8 m/s. What is the wavelength?

  1. Convert megahertz to hertz: 1 MHz = 10610^6 Hz.
    f=98.1×106 Hzf = 98.1 \times 10^6\ \text{Hz}
  2. Divide.
    λ=3.00×108 m/s98.1×106 Hz=3.058 m\lambda = \frac{3.00 \times 10^8\ \text{m/s}}{98.1 \times 10^6\ \text{Hz}} = 3.058\ \text{m}

Answer 3.06 m

Example 3: water waves at a buoy

Problem A buoy bobs up and down 12 times in 30. s. The crests are 2.5 m apart. Find the frequency, the period and the wave speed.

  1. Frequency is waves per second.
    f=1230. s=0.40 Hzf = \frac{12}{30.\ \text{s}} = 0.40\ \text{Hz}
  2. Period is the flip of frequency.
    T=10.40 Hz=2.5 sT = \frac{1}{0.40\ \text{Hz}} = 2.5\ \text{s}
  3. Speed.
    v=fλ=(0.40 Hz)(2.5 m)=1.0 m/sv = f\lambda = (0.40\ \text{Hz})(2.5\ \text{m}) = 1.0\ \text{m/s}

Answer 0.40 Hz, period 2.5 s, speed 1.0 m/s

Common mistakes and how to fix them

  • Measuring wavelength from crest to trough. That is only half a wavelength. Fix: crest to the next crest.
  • Measuring amplitude from trough to crest. That is twice the amplitude. Fix: measure from the rest line to a crest.
  • Thinking a higher frequency travels faster. In one medium, all frequencies of sound travel at the same speed. Fix: if ff goes up, λ\lambda goes down.
  • Forgetting metric prefixes. Fix: kHz is 10310^3 Hz, MHz is 10610^6 Hz, nm is 10−910^{-9} m.

Practice problems

  1. A wave has a frequency of 5.0 Hz and a wavelength of 0.60 m. What is its speed?

    1. 3.0 m/s
    2. 8.3 m/s
    3. 0.12 m/s
    4. 5.6 m/s
    Show answer

    Answer: 3.0 m/s

    v=fλ=5.0×0.60=3.0v = f\lambda = 5.0 \times 0.60 = 3.0 m/s.

  2. What is the period of a wave with a frequency of 50. Hz?

    Show answer

    Answer: 0.020 s

    T=1/f=1÷50.=0.020T = 1/f = 1 \div 50. = 0.020 s.

  3. What kind of wave is sound in air?

    1. Transverse
    2. Longitudinal
    Show answer

    Answer: Longitudinal

    Air molecules move back and forth along the direction the sound travels, making compressions and rarefactions.

  4. In the same medium, a wave's frequency doubles. What happens to its wavelength?

    1. It doubles
    2. It halves
    3. It stays the same
    4. It becomes 4 times as long
    Show answer

    Answer: It halves

    The medium fixes the speed. With vv constant, λ=v/f\lambda = v/f, so doubling ff halves λ\lambda.

  5. A tuning fork vibrates at 256 Hz. Sound travels at 343 m/s. What is the wavelength?

    Show answer

    Answer: 1.34 m

    λ=v/f=343÷256=1.340\lambda = v/f = 343 \div 256 = 1.340 m, which is 1.34 m.

Frequently asked questions

Does a wave carry matter along with it?

No. Each bit of the medium moves back and forth around one spot, while the wave's energy moves forward. A duck on a pond bobs up and down as waves pass but stays in about the same place.

Why does sound travel faster in water than in air?

Sound speed depends on how stiff the medium is and how dense it is. Water is far harder to squeeze than air, which more than makes up for its higher density. Sound travels about 1,480 m/s in water and roughly 343 m/s in air at room temperature.

Can light travel without a medium?

Yes. Light is an electromagnetic wave: changing electric and magnetic fields that keep each other going. It needs no material, which is why sunlight crosses empty space. Sound is a mechanical wave and needs a medium, so there is no sound in space.

Sources

  1. OpenStax College Physics 2e, 16.9 Waves, accessed October 1, 2026
  2. OpenStax College Physics 2e, 17.2 Speed of Sound, Frequency, and Wavelength, accessed October 1, 2026

Try asking Ducky

  • “I measured the wavelength on this diagram. Did I go crest to crest or crest to trough?”
  • “Check my unit conversion from MHz to Hz on number 2.”
  • “Quiz me on transverse versus longitudinal with real examples.”

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