Wave Equation

Physics· Unit 7: Waves, Topic 4· 15 min read

1. Derivation of the Wave Equation★★☆☆☆⏱ 5 min

The wave equation follows directly from the fundamental definitions of speed, frequency, wavelength and period. Speed is always distance divided by time, so we can adapt this definition for waves.

📘 Definition

Wave Period

T

Time taken for one complete wave cycle to pass a fixed point

Example:

A wave of 5 Hz frequency has a period of 0.2 s

🔬 Derivation

Goal:

Derive the wave equation $v = f λ$

Starting from:

Basic definitions of speed, period and frequency

1
In one full period $T$ , a wave travels exactly one wavelength $λ$ :
2
Wave speed equals distance travelled divided by time taken:
3
$v = \frac{\text{distance}}{\text{time}} = \frac{\lambda}{T}$
4
We know frequency $f$ is the inverse of period: $f = \frac{1}{T}$
5
Substitute $\frac{1}{T} = f$ into the speed equation:

Result:

This gives the standard wave equation: $v = f λ$

Exam tip:

When asked for a derivation, always start from basic definitions to earn full marks in CIE exams.

2. Applying the Wave Equation★★★☆☆⏱ 6 min

For any wave, if you know two of the three quantities ( $v$ , $f$ , $λ$ ), you can calculate the third by rearranging the wave equation. Always use SI units (metres for wavelength, hertz for frequency) to get the correct speed in m/s.

📐 Worked Example

A microwave has frequency 12 GHz. Given the speed of microwaves is $3.0 \times 1 0^{8}$ m/s, calculate the wavelength.

1
Convert frequency from GHz to Hz (SI unit):
2
$12 \text{ GHz} = 12 \times 10^9 \text{ Hz} = 1.2 \times 10^{10} \text{ Hz}$
3
Rearrange $v = f λ$ to solve for $λ$ :
4
$\lambda = \frac{v}{f}$
5
Substitute values and calculate:
6
$\lambda = \frac{3.0 \times 10^8}{1.2 \times 10^{10}} = 0.025 \text{ m} = 2.5 \text{ cm}$

✓ Quick check

Test your unit conversion and rearrangement skills

A sound wave of wavelength 1.7 m travels at 340 m/s. What is its frequency?
- 20 Hz
- 200 Hz
- 578 Hz
- 0.005 Hz
Reveal answer
200 Hz —
Correct! $f = v / λ = 340/1.7 = 200$ Hz.

Exam tip:

Unit conversion for prefixes (M, G, c, k) is a common trap: always check your units before substituting.

3. Wave Speed vs Particle Speed★★★☆☆⏱ 4 min

One of the most commonly tested misconceptions in CIE exams is confusing the speed of wave propagation through the medium with the speed of the individual particles of the medium.

📐 Worked Example

A wave generator on a fixed tension string produces 5 Hz waves with wavelength 0.4 m. If frequency is increased to 10 Hz, what is the new wave speed?

1
The tension (medium property) is unchanged, so wave speed stays constant. Calculate original speed:
2
$v = f\lambda = 5 \times 0.4 = 2 \text{ m/s}$
3
The new wave speed is still 2 m/s; the new wavelength becomes 0.2 m.

Exam tip:

If a question says the wave is travelling in the same medium, wave speed is constant regardless of frequency change.

4. Common Pitfalls

Wrong move:

Forgetting to convert units from MHz/cm to SI units (Hz/m)

Why:

This leads to answers off by orders of magnitude, which are marked wrong

Correct move:

Always confirm wavelength is in metres and frequency in hertz before substituting into the equation

Wrong move:

Claiming increasing frequency increases wave speed in a fixed medium

Why:

Wave speed depends only on medium properties, not frequency of the wave

Correct move:

Remember $v$ is constant for a fixed medium, so increasing $f$ decreases $λ$ proportionally

Wrong move:

Deriving the wave equation by just writing $v = f λ$ with no steps

Why:

CIE awards marks for the derivation process starting from first principles

Correct move:

Start from $v = λ / T$ , substitute $f = 1/ T$ to get the final result

Wrong move:

Confusing wave propagation speed with particle oscillation speed

Why:

Particles oscillate around equilibrium so their speed changes over a cycle, while wave speed is constant

Correct move:

Wave speed describes how fast energy travels through the medium, not how fast particles move

5. Quick Reference Cheatsheet

Quantity	Symbol	SI Unit	Relation
Wave speed	$v$	m/s	$v = f λ$
Frequency	$f$	Hz	$f = v / λ = 1/ T$
Wavelength	$λ$	m	$λ = v / f$
Period	$T$	s	$T = 1/ f = λ / v$

6. Frequently Asked

Does the wave equation apply to all types of waves?

Yes, the relation $v = f λ$ holds for all mechanical and electromagnetic waves, regardless of the medium they travel through.

Why doesn't changing frequency change wave speed in a constant medium?

Wave speed depends only on the properties of the medium (tension, density for mechanical waves; permittivity for EM waves). Changing frequency changes wavelength inversely, so speed remains constant.

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2022 · 11
Calculate wave speed from f and λ
2023 · 22
Find wavelength for radio waves

Going deeper

syllabusCIE 9702 Official Syllabus

What's Next

The wave equation is the foundational relation for all advanced wave topics in CIE A-Level Physics. It is used constantly to solve problems involving standing waves, interference, diffraction gratings, and electromagnetic radiation. A solid understanding of this equation and its underlying assumptions is required for almost all wave-related questions, including many high-mark extended response problems. Mastery of this sub-topic makes all subsequent wave topics much easier to grasp.