The Wave Equation

This lesson covers the wave equation — a key formula that links wave speed, frequency and wavelength. It is required by AQA GCSE Combined Science Trilogy (8464), Physics Paper 2, section 6.1. You must be able to recall this equation, rearrange it, and use it in calculations.

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The Wave Equation

$v = f \lambda$v=fλ

where:

• $v$v = wave speed in metres per second (m/s)
• $f$f = frequency in hertz (Hz)
• $\lambda$λ = wavelength in metres (m)

This equation tells us that the speed of a wave equals its frequency multiplied by its wavelength.

Exam Tip (AQA 8464): You must memorise this equation — it is not given on the AQA equation sheet. Write it at the top of your exam paper as soon as you open it.

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Rearranging the Equation

You need to be confident rearranging the wave equation to find any of the three quantities:

To find	Rearranged equation
Wave speed (v)	$v = f \lambda$v=fλ
Frequency (f)	$f = \dfrac{v}{\lambda}$f=λv​
Wavelength (λ)	$\lambda = \dfrac{v}{f}$λ=fv​

graph TD
    WE["Wave Equation: v = f × λ"] --> F["Find v: v = f × λ"]
    WE --> G["Find f: f = v ÷ λ"]
    WE --> H["Find λ: λ = v ÷ f"]

    style WE fill:#2c3e50,color:#fff
    style F fill:#27ae60,color:#fff
    style G fill:#2980b9,color:#fff
    style H fill:#8e44ad,color:#fff

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Worked Examples

Worked Example 1 — Finding wave speed

A sound wave has a frequency of 256 Hz and a wavelength of 1.3 m. Calculate the wave speed.

$v = f \lambda = 256 \times 1.3 = 332.8 \text{ m/s}$v=fλ=256×1.3=332.8 m/s

Worked Example 2 — Finding wavelength

A radio wave travels at $3 \times 10^{8}$3×108 m/s and has a frequency of $1 \times 10^{6}$1×106 Hz. Calculate its wavelength.

$\lambda = \frac{v}{f} = \frac{3 \times 10^{8}}{1 \times 10^{6}} = 300 \text{ m}$λ=fv​=1×1063×108​=300 m

Worked Example 3 — Finding frequency

A water wave has a wavelength of 2.5 m and travels at 5 m/s. Calculate the frequency.

$f = \frac{v}{\lambda} = \frac{5}{2.5} = 2 \text{ Hz}$f=λv​=2.55​=2 Hz

Worked Example 4 — Combining with T = 1/f

A wave has a period of 0.005 s and a wavelength of 1.7 m. Calculate the wave speed.

Step 1: Find the frequency.

$f = \frac{1}{T} = \frac{1}{0.005} = 200 \text{ Hz}$f=T1​=0.0051​=200 Hz

Step 2: Use the wave equation.

$v = f \lambda = 200 \times 1.7 = 340 \text{ m/s}$v=fλ=200×1.7=340 m/s

Exam Tip: When a question gives you the period instead of frequency, you need to use $f = 1/T$f=1/T first before substituting into $v = f\lambda$v=fλ. Show both steps clearly for full marks.

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Units Check

Quantity	Correct unit	Common wrong unit
Wave speed (v)	m/s	km/h (must convert)
Frequency (f)	Hz	kHz or MHz (must convert)
Wavelength (λ)	m	cm or nm (must convert)

Unit Conversions

• 1 kHz = 1 000 Hz = $10^{3}$103 Hz
• 1 MHz = 1 000 000 Hz = $10^{6}$106 Hz
• 1 cm = 0.01 m = $10^{-2}$10−2 m
• 1 nm = $10^{-9}$10−9 m

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Applying the Wave Equation to Different Waves

Wave	Typical speed	Notes
Sound in air	~340 m/s	Varies with temperature
Sound in water	~1 500 m/s	Faster in denser media
Sound in steel	~6 000 m/s	Fastest in solids
Light / EM waves in vacuum	$3 \times 10^{8}$3×108 m/s	The same for all EM waves

Exam Tip: All electromagnetic waves travel at the same speed in a vacuum: $3 \times 10^{8}$3×108 m/s. Sound travels much slower and its speed depends on the medium.

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Common Mistakes

Mistake	Correction
Using the wrong units (e.g. cm instead of m)	Always convert to SI units before substituting
Confusing wave speed with frequency	Wave speed = how fast the wave travels; frequency = how many waves per second
Forgetting to convert period to frequency	If given $T$T, calculate $f = 1/T$f=1/T first
Writing the equation as $v = f / \lambda$v=f/λ	The correct equation is $v = f \times \lambda$v=f×λ (multiplication, not division)

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Practice Questions

1. A wave has a frequency of 50 Hz and a wavelength of 4 m. Calculate its speed.

   • $v = 50 \times 4 = 200$v=50×4=200 m/s

2. An electromagnetic wave has a wavelength of $6 \times 10^{-7}$6×10−7 m and travels at $3 \times 10^{8}$3×108 m/s. Calculate its frequency.

   • $f = 3 \times 10^{8} \div 6 \times 10^{-7} = 5 \times 10^{14}$f=3×108÷6×10−7=5×1014 Hz

3. A sound wave has a frequency of 440 Hz and travels at 340 m/s. Calculate its wavelength.

   • $\lambda = 340 \div 440 = 0.773$λ=340÷440=0.773 m (3 s.f.)

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Summary

• The wave equation is $v = f \lambda$v=fλ — you must memorise this for AQA (8464).
• Rearrange to find: $f = v / \lambda$f=v/λ or $\lambda = v / f$λ=v/f.
• If given the period, use $f = 1/T$f=1/T first.
• Always convert to SI units (m, s, Hz) before calculating.
• Show your working clearly: write the equation, substitute values, calculate, give units.
• All EM waves travel at $3 \times 10^{8}$3×108 m/s in a vacuum.

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Extended Worked Examples

The wave equation v = fλ is one of only a handful of equations AQA expects you to recall from memory. Exam questions often combine it with unit conversions or with T = 1/f. These extended worked examples show every step — this is also how you should set out your answers for full marks.

Worked Example 5 — Unit conversion first

A radio station broadcasts at 98 MHz. The wave travels at the speed of light. Calculate the wavelength.