Physics 9702 · Paper 2

The Wave Equation v = fλ

One equation ties a wave's speed, frequency and wavelength together: $v = f\lambda$ .
Know any two of the three and you can find the third.

Where v = fλ comes from

In one period $T$ , the wave moves forward by exactly one wavelength $\lambda$ .
Speed is distance over time, so $v = \dfrac{\lambda}{T}$ . Since $f = 1/T$ , this is $v = f\lambda$ .

v = f\lambda

Symbols

$v$ = wave speed (m s⁻¹)
$f$ = frequency (Hz)
$\lambda$ = wavelength (m)

Wave	Speed v	Frequency f	Wavelength λ
Ripple on water	≈ 0.12 m s⁻¹	≈ 6 Hz	≈ 0.02 m
Sound in air	≈ 330 m s⁻¹	20 Hz to 20 kHz	15 m to 0.017 m
Wave on a slinky	≈ 1 m s⁻¹	≈ 2 Hz	≈ 0.5 m

Using the wave equation

Worked example

Frequency of a wave on a cord

A transverse wave has wavelength 12 cm and travels at 48 cm s⁻¹. Find its frequency.

Keep the units matched (both in cm here, so the answer is still in Hz): $f = \dfrac{v}{\lambda} = \dfrac{48}{12}$ .
$f = 4.0\text{ Hz}$ .

Answer

f = 4.0\text{ Hz}

Show that: deriving v = fλ from the definitions

An exam “show that” asks for the reasoning, not just the answer. In one period the wave advances one wavelength, so

v = \frac{\text{distance}}{\text{time}} = \frac{\lambda}{T}.

Frequency is the reciprocal of the period, $f = 1/T$ , so substituting gives $v = f\lambda$ .

(9702/21/M/J/25 Q5(a))

Common mistake

Mixing units. If

\lambda

is in cm and

v

in m s⁻¹, convert one first. Missing a factor of 100 here is the most common lost mark.

Your turn— tap to reveal the worked answer (9702/12/F/M/24 Q24)

Blue whales make sounds of frequency 10 Hz to 40 Hz. The speed of sound in seawater is about 1500 m s⁻¹. Find the range of wavelengths. (9702/12/F/M/24 Q24)

Answer: $\lambda = v/f$ , so from $1500/10 = 150\text{ m}$ down to $1500/40 = 37.5\text{ m}$ .

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