In one period $T$, a wave moves forward exactly one wavelength. Speed is distance over time (average speed), so the wave's speed is one wavelength per period. Since $\frac{1}{T} = f$ (from the parts of a wave), that gives the most useful equation for waves.
Key idea
$$v = f\,\lambda$$ Wave speed (m/s) = frequency (Hz) × wavelength (m). The medium sets the speed; if you change the frequency, the wavelength changes to match.
Why it works
$v = \dfrac{\lambda}{T} = \lambda \times \dfrac{1}{T} = f\,\lambda$. Think of a line of waves marching past you: if 4 waves pass each second and each is 2 m long, then 8 m of wave passes each second, so $v = 8$ m/s. Shake a rope faster and the waves bunch up (shorter $\lambda$), but they still travel at the speed the rope allows.
Worked example: Ocean waves
Ocean waves 12 m long pass a pier at 0.5 Hz. How fast do they travel?
- $v = f\,\lambda = 0.5\,\mathrm{Hz} \times 12\,\mathrm{m} = 6\,\mathrm{m/s}$.
Answer: 6 m/s
Worked example: A musical note
Sound travels at 343 m/s in air. Find the wavelength of a 686 Hz note.
- Rearrange: $\lambda = \dfrac{v}{f}$.
- $\lambda = \dfrac{343}{686} = 0.5$ m.
Answer: 0.5 m
Common mistake
Thinking a higher frequency makes a wave travel faster. In the same medium, every sound travels at about 343 m/s, high notes and low notes alike; otherwise music from far away would arrive scrambled. A higher frequency just means a shorter wavelength. Check: $f \times \lambda$ should give the same speed for every wave in that medium.
An FM radio station at 100 MHz (100,000,000 Hz) sends radio waves at the speed of light, 300,000,000 m/s, so each wave is $\dfrac{300{,}000{,}000}{100{,}000{,}000} = 3$ m long.