The Wave Equation

This lesson covers the wave equation and how to use it to calculate wave speed, frequency and wavelength, as required by the Edexcel GCSE Combined Science specification (1SC0). This equation applies to all types of wave — transverse and longitudinal — and is one of the most frequently examined equations in the physics papers.

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The Wave Equation

The wave equation links three quantities:

$v = f \lambda$v=fλ

Symbol	Quantity	SI Unit
v	Wave speed	metres per second (m/s)
f	Frequency	hertz (Hz)
λ	Wavelength	metres (m)

What Each Quantity Means

• Wave speed (v) — the distance travelled by the wave per second.
• Frequency (f) — the number of complete waves passing a point per second.
• Wavelength (λ) — the distance between two consecutive points in phase (e.g. crest to crest).

Exam Tip: The wave equation is on the physics equation sheet, but practising rearrangements will save you valuable time in the exam.

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Rearranging the Wave Equation

You need to be able to rearrange the equation for any of the three variables.

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

The Formula Triangle

A useful memory aid:

graph TD
    A["v"] --- B["f × λ"]

Cover the quantity you want to find:

• Cover v → f × λ (multiply)
• Cover f → v ÷ λ (divide)
• Cover λ → v ÷ f (divide)

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Worked Examples

Worked Example 1

A wave has a frequency of 250 Hz and a wavelength of 1.32 m. Calculate the wave speed.

Step 1: Write the equation.

$v = f \lambda$v=fλ

Step 2: Substitute.

$v = 250 \times 1.32$v=250×1.32

Step 3: Calculate.

$v = 330 \text{ m/s}$v=330 m/s

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Worked Example 2

A sound wave travels at 340 m/s and has a frequency of 680 Hz. Calculate the wavelength.

Step 1: Rearrange.

$\lambda = \frac{v}{f}$λ=fv​

Step 2: Substitute.

$\lambda = \frac{340}{680}$λ=680340​

Step 3: Calculate.

$\lambda = 0.5 \text{ m}$λ=0.5 m

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Worked Example 3

A water wave has a wavelength of 2.4 m and a wave speed of 6.0 m/s. Calculate the frequency.

Step 1: Rearrange.

$f = \frac{v}{\lambda}$f=λv​

Step 2: Substitute.

$f = \frac{6.0}{2.4}$f=2.46.0​

Step 3: Calculate.

$f = 2.5 \text{ Hz}$f=2.5 Hz

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Worked Example 4

A radio wave has a frequency of 9.0 × 10⁸ Hz. All electromagnetic waves travel at 3.0 × 10⁸ m/s in a vacuum. Calculate the wavelength.

Step 1: Rearrange.

$\lambda = \frac{v}{f}$λ=fv​

Step 2: Substitute.

$\lambda = \frac{3.0 \times 10^{8}}{9.0 \times 10^{8}}$λ=9.0×1083.0×108​

Step 3: Calculate.

$\lambda = 0.33 \text{ m (2 s.f.)}$λ=0.33 m (2 s.f.)

Exam Tip: When dealing with electromagnetic waves, always use v = 3.0 × 10⁸ m/s (the speed of light in a vacuum). This value is usually given in the question or on the data sheet.

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Connecting Frequency, Wavelength and Speed

An important relationship to understand:

• If the frequency increases and the speed stays the same, the wavelength decreases (and vice versa).
• This is because v = fλ is a constant for a given medium.

Change	Effect (speed constant)
f increases	λ decreases
f decreases	λ increases
λ increases	f decreases
λ decreases	f increases

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Wave Speed in Different Media

The speed of a wave depends on the medium it is travelling through.

Medium	Approximate speed of sound
Air (20 °C)	340 m/s
Water	1500 m/s
Steel	5000 m/s

• Sound travels fastest in solids and slowest in gases because the particles are closer together and can pass on vibrations more quickly.
• When a wave enters a different medium, its speed and wavelength change, but its frequency stays the same.

Exam Tip: "Frequency stays the same when a wave changes medium" is a commonly examined statement. Remember: the source determines the frequency, and the medium determines the speed.

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Using the Wave Equation with Period

Sometimes a question gives you the period (T) instead of the frequency. Since:

$f = \frac{1}{T}$f=T1​

You can substitute into the wave equation:

$v = \frac{\lambda}{T}$v=Tλ​

Worked Example 5

A wave has a period of 0.004 s and a wavelength of 1.36 m. Calculate the wave speed.

Step 1: Calculate frequency.

$f = \frac{1}{T} = \frac{1}{0.004} = 250 \text{ Hz}$f=T1​=0.0041​=250 Hz

Step 2: Use the wave equation.

$v = f \lambda = 250 \times 1.36 = 340 \text{ m/s}$v=fλ=250×1.36=340 m/s

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Common Mistakes

Mistake	How to avoid it
Forgetting to convert units (e.g. kHz to Hz, cm to m)	Always convert to SI units before substituting
Confusing period and frequency	Remember: f = 1/T; they are reciprocals
Using the wrong speed for EM waves	All EM waves travel at 3.0 × 10⁸ m/s in a vacuum
Writing the wrong unit for the answer	v → m/s, f → Hz, λ → m

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Summary

• The wave equation is $v = f \lambda$v=fλ.
• It can be rearranged to find frequency $\left(f = \frac{v}{\lambda}\right)$(f=λv​) or wavelength $\left(\lambda = \frac{v}{f}\right)$(λ=fv​).
• When a wave changes medium, its speed and wavelength change, but its frequency remains constant.
• Sound travels faster in solids than in liquids, and faster in liquids than in gases.
• All electromagnetic waves travel at 3.0 × 10⁸ m/s in a vacuum.
• Always convert to SI units before substituting into the equation.

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Unit Conversion Reference

Many exam calculations fail because of unit mismatches rather than arithmetic errors. Make sure every quantity is converted into SI base units before substituting into v = fλ.