Wave Speed, Frequency and Wavelength — v = fλ Explained

The three quantities

Three numbers describe any repeating wave:

• Frequency ($f$): how many complete cycles pass a point each second, in hertz (Hz).
• Wavelength ($\lambda$): the length of one complete cycle, in meters — crest to crest, or compression to compression.
• Speed ($v$): how fast the wave pattern travels through the medium, in m/s.

They are locked together by the single most-used equation in wave physics.

The wave equation: v = fλ

In one period ($T = 1/f$), the wave advances exactly one wavelength. So distance over time gives:

Frequency vs period

Period ($T$) is the time for one cycle; frequency is cycles per second. They are inverses:

$$f = \frac{1}{T}$$

A guitar string vibrating 440 times per second has period $T = 1/440 \approx 0.0023\ \mathrm{s}$. High frequency always means short period — they are two views of the same rhythm.

Speed depends on the medium

What sets the speed? The medium, not the source. Sound moves at ~343 m/s in air but ~1500 m/s in water and ~5000 m/s in steel — same shout, wildly different speeds.

Because the source fixes the frequency, entering a new medium changes the wavelength instead: $\lambda = v/f$. Shout from air into water and your voice's wavelength more than quadruples while the pitch (frequency) stays identical.

Worked mini-example

Ocean waves with wavelength $\lambda = 30\ \mathrm{m}$ roll past a buoy 6 times per minute. How fast are they moving?

Frequency: $f = 6/60 = 0.10\ \mathrm{Hz}$. Then:

$$v = f\lambda = 0.10 \times 30 = 3.0\ \mathrm{m/s}$$

A gentle 3 m/s — about walking speed. Note the method: count cycles, convert to per-second, multiply by wavelength.

Variables and units

Common mistakes

Real-world applications

• Piano tuners adjust string tension until the frequency matches the target note — frequency is what the ear hears as pitch.

• Ultrasound imaging chooses frequencies whose wavelengths in tissue give the needed resolution: higher f, smaller λ, finer detail.

• Radio stations are assigned frequencies; your receiver tunes to one, and the wavelength follows from v = fλ automatically.