Wavelength, frequency and speed
Ocean crests 10 m apart that lift a gull once every 5 s are moving at 2 m/s.
Counting waves: period and frequency
Stand on a pier and watch the waves come in. Two simple measurements describe the timing. The period T is the time for one complete wave to pass a fixed point, or for a floating cork to go up, down and back up again. The frequency f is the number of waves that pass each second. They are two views of the same thing, so f = 1/T. A wave with a period of 0.5 s has a frequency of 2 per second. The unit of frequency is the hertz (Hz), meaning one cycle per second, named after Heinrich Hertz, who first made and detected radio waves in a laboratory.
The frequency of a wave is set by whatever makes it. A loudspeaker cone that vibrates 440 times a second sends out a 440 Hz sound. A hand that shakes a rope twice a second makes a 2 Hz wave. When a wave moves from one material into another, OpenStax notes that the frequency usually stays the same, because the new material is being driven at the rate of the old one. The speed can change, and then the wavelength must change to fit.
The wave equation v = fλ
In one period, a wave moves forward exactly one wavelength: the crest that was here is replaced by the next crest. Speed is distance divided by time, so the wave speed is v = λ/T. Since 1/T is the frequency, this becomes v = fλ. OpenStax calls it a fundamental relationship that holds for all types of waves: water waves, sound, light and waves on strings. Their worked example uses the gull: crests 10.0 m apart and 5.00 s between bobs give v = 10.0 m ÷ 5.00 s = 2.00 m/s. The gull itself moves up and down at a speed that keeps changing; the 2.00 m/s belongs to the pattern.
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What sets the speed?
It is tempting to think that shaking a rope faster makes the wave go faster. It does not. In the nondispersive model used here, speed is fixed by the medium itself: how stiff or tight it is, and how heavy. On a rope or a guitar string, more tension means a faster wave, which is why tightening a string raises its note. In air the speed of sound depends on temperature but hardly at all on frequency. So if you shake the rope faster, raising f, the speed stays the same and the wavelength gets shorter. If you shake it slower, the waves get longer.
This explains a pattern you can see in any orchestra. OpenStax points out that small instruments such as the piccolo make high notes and large ones such as the tuba make low notes. All the sounds travel through the same air at the same speed, so a high frequency must come with a short wavelength and a low frequency with a long one, and the size of an instrument is tied to the wavelengths it produces. At 30 °C, sound at the bottom of human hearing (20 Hz) has a wavelength of about 17 m, and at the top (20 000 Hz) about 1.7 cm.
| Wave | f | v | λ |
|---|---|---|---|
| Ocean swell (OpenStax) | 0.2 Hz | 2.00 m/s | 10.0 m |
| Concert A in air, 20 °C | 440 Hz | 343 m/s | about 0.78 m |
| FM radio, 105.1 MHz | 1.051 × 10⁸ Hz | 3.00 × 10⁸ m/s | about 2.85 m |
| Red light, 633 nm laser | about 4.7 × 10¹⁴ Hz | 3.00 × 10⁸ m/s | 633 nm |
Working with the equation
Most wave problems give you two of the three quantities and ask for the third. Rearranged, the equation reads f = v/λ and λ = v/f. Keep the units consistent: metres, seconds and hertz give metres per second. Very large and very small numbers turn up quickly. Light travels at about 3.00 × 10⁸ m/s, so a radio station broadcasting at 105.1 MHz sends out waves 3.00 × 10⁸ ÷ 1.051 × 10⁸ ≈ 2.85 m long, a result OpenStax works out in its chapter on electromagnetic waves. The same equation gives the 0.78 m wavelength of the A that orchestras tune to.
- Ocean crests are 10.0 m apart and a floating gull bobs once every 5.00 s. What is the wave speed?
- A wave has frequency 4 Hz and wavelength 0.5 m. What is its speed?
- What is the period of a 50 Hz wave?
- Sound travels at 343 m/s in air at 20 °C. What is the wavelength of a 440 Hz note? Give it to two decimal places.
- In the same air, you double the frequency of a sound. What happens to its wavelength?
- It halves
- It doubles
- It stays the same
- It becomes four times longer
- A wave passes from one material into another where it travels faster. Which usually stays the same?
- Frequency
- Wavelength
- Speed
- All three change
- Made-up example: 12 crests pass the end of a pier in 60 s. What is the frequency?
- Those same crests are 15 m apart. How fast are the waves moving?
- An FM station broadcasts at 105.1 MHz. Taking c = 3.00 × 10⁸ m/s, what is the wavelength? Give two decimal places.
- One hertz means…
- One cycle per second
- One metre per second
- One wavelength per metre
- One second per cycle
- Period T: time for one wave. Frequency f = 1/T, in hertz.
- v = fλ for every kind of wave.
- In this nondispersive model the medium sets the speed; the source sets the frequency.
- Same medium, higher frequency: shorter wavelength.
Sources for this course (11)
- OpenStax College Physics 2e (openstax.org/books/college-physics-2e): 16.9 Waves (disturbance that propagates, gull example λ 10.0 m, T 5.00 s, v = 2.00 m/s, transverse and longitudinal, amplitude independent of speed)
- 16.10 Superposition and Interference (constructive, destructive, standing waves, f1 = v/2L, beats fB = |f1 − f2|, 256 Hz fork with 2 beats -> 254 or 258 Hz)
- 17.2 Speed of Sound (Table 17.1: air 331 m/s at 0 °C, helium 965, fresh water 1480, sea water 1540, steel 5960, vulcanized rubber 54; v = 331 m/s × √(T/273 K); 343 m/s at 20 °C; audible range 20 to 20 000 Hz; P-waves 4 to 7 km/s, S-waves 2 to 5 km/s; Oldham and Lehmann)
- 17.3 Sound Intensity (I0 = 10⁻¹² W/m², Table 17.2, factor 10 = 10 dB, factor 2 = 3 dB, 85 dB for 8 h footnote, bel named for Bell)
- 17.5 Sound Interference and Resonance (noise-cancelling headphones 30 dB or more, closed tube λ = 4L and f_n = n v/4L with n odd, open tube f_n = n v/2L, 128 Hz tube at 22 °C = 0.672 m, ninth harmonic 1.15 kHz, trumpet vs clarinet overtones)
- 24.1 (Maxwell; Hertz from 1887)
- 24.3 The Electromagnetic Spectrum (c = fλ, Table 24.1, three rules of thumb, AM 540 to 1600 kHz, FM 88 to 108 MHz, microwave oven 2.45 GHz, visible 400 to 750 nm, UV 400 to about 10 nm, UV-A 320 to 400, UV-B 290 to 320, UV-C 220 to 290 nm, 99% of surface UV is UV-A, X-rays below 0.01 nm)
- 27.3 Young's Double Slit (1801, d sin θ = mλ, 0.0100 mm and 633 nm example)
- NASA science.nasa.gov/ems (Introduction, Visible Light: 380 to 700 nm; atmospheric windows)
- NIST, c = 299 792 458 m/s exact (nist.gov/si-redefinition/meter)
- ISO 16:1975 (A = 440 Hz, iso.org/standard/3601.html).
Checked: 2026-10-17