The Wave Equation v = fλ — Sound and Light Examples
Three symbols, one multiplication sign, and yet v = fλ produces more wrong answers than almost any other formula in school physics. Not because the algebra is hard — a Class 8 student can rearrange it — but because students do not know which of the three quantities is fixed by the source and which is fixed by the medium. Get that straight and the whole chapter, including refraction, sound in water and the colours of light, becomes one idea instead of twenty facts.
What each symbol means
| Symbol | Name | SI unit | Decided by |
|---|---|---|---|
| v | Wave speed | m/s | The medium the wave travels through |
| f | Frequency — cycles per second | hertz (Hz) = s−1 | The source that made the wave |
| λ | Wavelength — length of one full cycle | m | Whatever v and f force it to be |
| T | Time period — time for one cycle | s | T = 1/f |
That third row is the key. Wavelength is never chosen. It is the leftover. The source sets the frequency, the medium sets the speed, and λ = v/f is whatever those two demand.
The formulas
f = 1 / T and T = 1 / f
ω = 2πf (angular frequency, rad/s)
k = 2π / λ (wave number, rad/m)
v = ω / k (the same equation, written in ω and k)
The last line surprises people, so check it once: ω/k = (2πf) ÷ (2π/λ) = fλ = v. It is the same statement in different clothing, and it is the form that appears in JEE and GATE questions written as y = A sin(ωt − kx).
Worked example 1 — sound in air
Question: A tuning fork of frequency 680 Hz is sounded in air where the speed of sound is 340 m/s. Find the wavelength and the time period.
Wavelength: λ = v / f = 340 ÷ 680 = 0.5 m
Check: f = v / λ = 340 ÷ 0.5 = 680 Hz. ✔
Time period: T = 1 / f = 1 ÷ 680 = 0.001470 s, i.e. 1.47 ms
A sanity check worth doing every time: half a metre is a believable wavelength for a fairly high musical note. If you had got 500 m or 0.0005 m, something went wrong.
Worked example 2 — the same sound entering water
Question: That same 680 Hz sound now travels into water, where the speed of sound is about 1480 m/s. What are its new frequency and wavelength?
Frequency: unchanged — 680 Hz. The tuning fork is still vibrating 680 times a second, and every one of those pushes has to arrive in the water. The medium cannot invent or destroy cycles.
Wavelength: λ = v / f = 1480 ÷ 680 = 2.176 m
Check: 680 × 2.176 = 1479.7 ≈ 1480 m/s. ✔
So the wave stretched to more than four times its length in air. This is exactly why light bends when it enters glass: the frequency (and therefore the colour) is fixed, the speed drops, so the wavelength must shrink — and a wavefront arriving at an angle is forced to change direction.
Worked example 3 — visible light, and the photon energy
Question: Orange light has a wavelength of 600 nm in vacuum. Find its frequency and the energy of one photon. Take c = 3.00 × 108 m/s and h = 6.626 × 10−34 J·s.
Convert first: 600 nm = 600 × 10−9 m = 6.00 × 10−7 m
Frequency: f = c / λ = (3.00 × 108) ÷ (6.00 × 10−7) = 5.00 × 1014 Hz
Check: λ = c / f = (3.00 × 108) ÷ (5.00 × 1014) = 6.00 × 10−7 m. ✔
Photon energy: E = hf = (6.626 × 10−34) × (5.00 × 1014) = 3.313 × 10−19 J
In electronvolts: 3.313 × 10−19 ÷ 1.602 × 10−19 = 2.068 eV
Check with the shortcut E(eV) = 1240 / λ(nm), which is just hc written in convenient units: 1240 ÷ 600 = 2.067 eV. ✔ The tiny difference is only rounding in the shortcut constant.
Worked example 4 — an FM radio station
Question: An FM station broadcasts at 100 MHz. What is the wavelength of its signal?
100 MHz = 100 × 106 Hz = 1.00 × 108 Hz
λ = c / f = (3.00 × 108) ÷ (1.00 × 108) = 3.00 m
Check: 1.00 × 108 × 3.00 = 3.00 × 108 m/s = c. ✔
Three metres is a useful number to remember, because it explains why an FM aerial is roughly a metre long — antennas are usually built as a simple fraction of the wavelength they are designed to receive.
A note on the two values of c you will meet
The speed of light in vacuum is defined as exactly 299,792,458 m/s. School and entrance-exam papers almost always tell you to use 3.00 × 108 m/s, which is 0.07% larger. Use whichever value the question gives you, and say which one you used. Do not mix them inside one solution. The same care applies to the speed of sound in air: 340 m/s and 343 m/s are both quoted, because the speed genuinely depends on temperature — take the figure printed in your question.
Common mistakes that cost marks
- Believing frequency changes in a new medium. It does not. Speed and wavelength change together; frequency is set by the source and is carried across every boundary unchanged.
- Not converting nm, µm, MHz or GHz. 600 nm is 6.00 × 10−7 m, not 600 or 6 × 10−9. One nanometre is 10−9 m; one megahertz is 106 Hz.
- Using c for sound. c is the speed of light. Sound in air is around 340 m/s — nearly a million times slower.
- Confusing period and frequency. T = 1/f. A 500 Hz note has a period of 0.002 s, not 500 s.
- Confusing amplitude with wavelength. Amplitude is the height of the wave and controls loudness or brightness; wavelength is its length along the direction of travel. Neither appears in the other's formula.
- Forgetting that ω and k carry 2π. In y = A sin(ωt − kx), reading ω as the frequency directly is wrong by a factor of 2π.
Where the wave equation appears in exams
| Exam | Typical use |
|---|---|
| CBSE/ICSE Class 9–10 | Sound: speed, frequency, wavelength, echo and range-of-hearing questions |
| CBSE/ICSE Class 11–12 | Waves, superposition, standing waves in pipes and strings, and the whole of wave optics |
| JEE Main & Advanced | y = A sin(ωt − kx) problems, beats, Doppler effect, thin-film interference |
| NEET | Direct v = fλ substitutions, plus photon energy in modern physics |
| Chemistry (Class 11, JAM, NET) | Every spectroscopy question — UV-visible, IR and NMR all convert between λ, f and E |
That last row is worth underlining for chemistry students. When an IR spectrum is labelled in wavenumbers (cm−1), that number is 1/λ with λ in centimetres — the same wave equation, rearranged one more time.
Check your working in seconds. The Waves tool takes any two of speed, frequency and wavelength and returns the third, so you can confirm a unit conversion before it costs you the whole question.
Open the Wave Calculator →Waves sit at the exact point where Class 11–12 physics and chemistry meet, and students often learn them twice without joining them up. ABC Chemistry runs Class 11–12 coaching at the Gurugram centre and online classes across India — details at abcchemistry.in.