Why speed equals frequency times wavelength
Stand at a fixed point and count crests going past. If f crests pass every second and each crest is λ metres behind the last, then the wave train has advanced fλ metres in that second. That is the whole derivation: v = fλ is a bookkeeping identity, not a physical law about any particular medium.
The physics enters through v. What sets the speed is always the same competition: a restoring force that pulls the medium back, divided by an inertia that resists the return. For a stretched string it is v = √(T/μ), tension over linear density. For a gas it is v = √(γRT/M), which is why sound in air depends on temperature but not on pressure, and why helium — light molecules, low M — carries sound at about 1,007 m/s and raises the pitch of your voice. For a solid rod it is v = √(E/ρ), giving 5,960 m/s in steel. For light in a material it is c/n.
The consequence that trips people up is what happens when a wave crosses a boundary. Frequency is fixed by the source and never changes. The last particle in medium 1 drives the first particle in medium 2 at exactly the frequency it is oscillating at. Speed changes, so wavelength must change by the same factor. A 589 nm sodium line entering water at n = 1.333 becomes 442 nm inside the water, but it is still the same yellow line and still 5.09 × 1014 Hz. This is also why the refraction angle changes — it is the same fact viewed geometrically.
Period, angular frequency and wavenumber
Three companion quantities appear constantly in wave equations, and all are simple rearrangements of what you have already entered.
Period T = 1/f is the time for one full cycle. A 440 Hz tone has a period of 2.273 ms. Oscilloscopes measure period, spectrum analysers measure frequency, and confusing the two by an inverse is a common slip in laboratory work.
Angular frequency ω = 2πf is what appears inside the sine: a wave is written y = A sin(kx − ωt), and the 2π converts cycles into radians. Filter and resonance formulas are almost always written in ω, which is why a 1 kHz corner frequency appears as 6,283 rad/s in a transfer function.
Angular wavenumber k = 2π/λ is the spatial counterpart, radians of phase per metre. It gives the compact identity v = ω/k. Note that spectroscopists use a different quantity also called wavenumber, ν̃ = 1/λ in cm−1, without the 2π — an infrared band at 1,700 cm−1 is a wavelength of 5.88 µm. Check which convention a source uses before comparing numbers; they differ by a factor of 2π and a factor of 100.
One further distinction matters for dispersive media. The phase velocity ω/k is what this calculator returns; the group velocity dω/dk is the speed at which energy and information actually travel. They coincide only when the speed is independent of frequency. In air and in vacuum they are the same for practical purposes; in optical fibre, deep water waves and waveguides they are not, and pulse spreading is the direct consequence.
Worked example: sizing a bass trap and a Wi-Fi antenna
Part 1 — a room mode at 60 Hz. You want to know how big an absorber must be to work at 60 Hz in a room at 20 °C.
- Wavelength. λ = v / f = 343.2 ÷ 60 = 5.72 m.
- Quarter wavelength. Porous absorbers work on particle velocity, which peaks a quarter wavelength from a rigid wall: 5.72 ÷ 4 = 1.43 m.
- Interpret. A 100 mm panel is 7% of a quarter wavelength at 60 Hz, which is why thin foam does nothing for bass. At 1 kHz the wavelength is 0.343 m and a quarter wavelength is 86 mm, so the same panel is genuinely effective there.
Part 2 — a 2.45 GHz antenna. Wi-Fi's 2.4 GHz band centres near 2.45 GHz.
- Wavelength in vacuum. λ = 299,792,458 ÷ 2.45 × 109 = 0.12236 m, about 122 mm.
- Half-wave dipole. A resonant dipole is close to λ/2 = 61.2 mm tip to tip, and a quarter-wave whip over a ground plane is 30.6 mm.
- Velocity factor. Inside coaxial cable with solid polyethylene dielectric the wave travels at about 0.66c, so the same frequency has a wavelength of 0.12236 × 0.66 = 80.8 mm in the cable. Cutting a matching stub to the free-space length would make it 51% too long.
Both parts are the same arithmetic, six orders of magnitude apart in frequency and one in speed. That generality is the point of v = fλ.
Sanity checks and what the numbers imply physically
Start with an order-of-magnitude check. In air, wavelength in metres is roughly 343 divided by frequency in hertz: 20 Hz is 17 m, 1 kHz is 34 cm, 20 kHz is 1.7 cm. For light in vacuum, wavelength in metres is 300 divided by frequency in megahertz: FM radio near 100 MHz is 3 m, and a 500 THz optical wave is 600 nm. If your answer is far from those anchors, a unit prefix has gone astray.
Wavelength then tells you how the wave interacts with objects. A wave diffracts strongly around obstacles much smaller than λ and casts sharp shadows behind obstacles much larger than λ. That single rule explains why you hear the bass from a party through a wall while the vocals are muffled, why a 2.4 GHz signal at 122 mm passes through a doorway but struggles with a foil-backed wall, and why optical microscopy cannot resolve detail much finer than about half a wavelength of visible light.
Wavelength also sets resonator dimensions. An open-ended pipe resonates when its length is a multiple of λ/2; a pipe closed at one end resonates at odd multiples of λ/4, which is why a stopped organ pipe sounds an octave below an open one of the same length. Antennas follow the same rule with electromagnetic waves. The half-wavelength column in the results table gives you that dimension directly.
Finally, be careful with temperature in gases. The speed of sound in dry air is close to 331.3 + 0.606×T m/s with T in degrees Celsius, so a wind instrument tuned in a cold hall will play flat when the hall warms: a 10 K rise raises the speed by 1.8% and every resonant frequency with it, roughly 30 cents.
Wave speed in common media, with wavelengths at two frequencies
| Medium | Speed (m/s) | λ at 100 Hz | λ at 1 kHz |
|---|---|---|---|
| Air, 0 °C, dry | 331.3 | 3.313 m | 0.3313 m |
| Air, 20 °C, dry | 343.2 | 3.432 m | 0.3432 m |
| Helium, 20 °C | 1,007 | 10.07 m | 1.007 m |
| Fresh water, 20 °C | 1,481 | 14.81 m | 1.481 m |
| Seawater, 13 °C, 35 ‰ | 1,500 | 15.00 m | 1.500 m |
| Ice | 3,200 | 32.00 m | 3.200 m |
| Glass (crown) | 5,640 | 56.40 m | 5.640 m |
| Steel | 5,960 | 59.60 m | 5.960 m |
| Aluminium | 6,420 | 64.20 m | 6.420 m |
| Light in vacuum | 299,792,458 | 2,998 km | 299.8 km |
Solid-medium figures are longitudinal bulk speeds; a thin rod carries extensional waves more slowly, and shear waves slower again. Seawater speed varies with temperature, salinity and depth by tens of metres per second.
Mistakes that produce a wrong wavelength
- Using the vacuum speed of light inside a material. In glass at n = 1.5168 the wavelength is a third shorter; in coaxial cable with a velocity factor of 0.66 it is a third shorter again. Cutting cable to a free-space quarter wave is the classic RF beginner's error.
- Mixing up period and frequency. They are reciprocals, so a factor-of-f² error follows if you substitute one for the other in
v = λ/T. - Assuming frequency changes at a boundary. It never does for a linear medium. Speed and wavelength change together, leaving f untouched.
- Ignoring temperature in a gas. Sound in air gains about 0.6 m/s per kelvin. Between a 0 °C morning and a 30 °C afternoon the speed changes by 5.4%.
- Confusing the two definitions of wavenumber. Physicists use
k = 2π/λin rad/m; spectroscopists use1/λin cm−1. The two differ by 2π and by a factor of 100. - Applying phase velocity to a pulse in a dispersive medium. Energy travels at the group velocity. In optical fibre that difference is what limits data rate over long spans.
- Dropping the SI prefix. Entering 2.45 rather than 2.45 GHz gives 299,792,458 ÷ 2.45 = 1.224 × 108 m, a wavelength of about 122,000 kilometres instead of 122 millimetres.
Where this relation leads next
Once you have wavelength and frequency, most wave problems become geometry. Interference and diffraction depend on path differences measured in wavelengths: two paths differing by a whole number of wavelengths reinforce, and by an odd number of half wavelengths cancel. That is the basis of thin-film coatings, diffraction gratings and the comb filtering you hear when a reflection arrives shortly after a direct sound.
When source or observer moves, the frequency you measure is no longer the frequency emitted, and the Doppler shift gives the new value; the wavelength ahead of a moving source is compressed by exactly the distance it advances in one period. When a wave crosses into another medium at an angle, the change of wavelength computed here is what bends the ray, which is Snell's law restated. And once you know the amplitude as well as the frequency, the energy carried per second per square metre is the intensity that sets the decibel level.
Standing waves connect this page back to oscillators. A string or air column fixed at its ends supports only wavelengths that fit a whole number of half wavelengths, which quantises the allowed frequencies into a harmonic series — the same discreteness that makes musical instruments play notes rather than noise, and the classical ancestor of the quantised energy levels of a bound electron. The single-oscillator version of that story is the pendulum period, and the restoring-force version is Hooke's law, since a wave on a string is nothing more than a line of masses connected by springs.
Key terms
- Wavelength
- The distance over which the wave repeats — crest to crest, or trough to trough. It is a property of the wave in a particular medium, not of the source alone.
- Frequency
- Cycles per second, in hertz. Set by the source and preserved across boundaries between linear media.
- Phase velocity
- The speed at which a point of constant phase, such as a crest, advances: ω/k. This is what v = fλ returns.
- Group velocity
- The speed at which a wave packet and its energy travel: dω/dk. Equal to the phase velocity only in a non-dispersive medium.
- Velocity factor
- For a transmission line, the ratio of propagation speed to the vacuum speed of light — typically 0.66 for solid polyethylene coax and 0.8–0.88 for foam dielectric.