The Wave Equation

What this lesson covers in the specification

Incandio is aligned to this specification. It is not published by, endorsed by or affiliated with Pearson, and it reproduces none of Pearson's wording — the statement numbers are given so you can check every lesson against your own copy.

• 3.5 — Use v = f × λ
• 3.6 — Use f = 1 ÷ T
• 3.7 — Use those relationships in different contexts, including sound and electromagnetic waves

Specification section 3(b) Properties of waves — also taught in What a Wave Is, The Doppler Effect, The Electromagnetic Spectrum, Uses and Dangers of Electromagnetic Radiation.

1 · Understand it

No exam language yet. The only question this section answers is: do I actually understand what is happening?

The wave equation is one of the few in the course you could derive from the meanings of the words, and doing so once is worth more than memorising it ten times. Frequency is how many complete waves pass a point each second. Wavelength is how long each of those waves is. So the length of wave passing you each second — which is the speed — must be the number of them multiplied by the length of each: v = f × λ.

Think of it like a train going past a station

You are standing on a platform as a goods train passes. Two carriages go by every second, and each carriage is fifteen metres long. How fast is the train? Nobody needs an equation for this: 2 × 15 = 30 metres of train pass you each second, so the train is doing 30 m/s. The carriages per second are the frequency, the length of a carriage is the wavelength, and the speed of the train is the wave speed. If you ever forget v = f × λ, put yourself on the platform and count carriages.

The second equation is even shorter. The PERIOD is the time for one complete wave to pass, and the FREQUENCY is the number of waves per second — so if one wave takes a quarter of a second, four waves pass per second. Frequency and period are reciprocals: f = 1 ÷ T, and equally T = 1 ÷ f. A wave of period 0.02 s has a frequency of 50 Hz.

The wave equation used three ways

(a) A wave has frequency 25 Hz and wavelength 2.0 m. Find its speed. (b) A sound wave travels at 340 m/s with a frequency of 850 Hz. Find its wavelength. (c) A radio wave of wavelength 300 m travels at 3.0 × 10⁸ m/s. Find its frequency.

1. (a) v = f × λ = 25 × 2.0 = 50 m/s.
2. (b) Rearrange: λ = v ÷ f = 340 ÷ 850 = 0.40 m.
3. (c) Rearrange: f = v ÷ λ = 3.0 × 10⁸ ÷ 300 = 1.0 × 10⁶ Hz, which is 1.0 MHz.
4. In each case decide which quantity is missing and rearrange BEFORE substituting.

Answer: 50 m/s, 0.40 m and 1.0 × 10⁶ Hz.

Statement 3.7 asks for these relationships in different contexts, and the two that matter are sound and electromagnetic waves. For SOUND in air the speed is roughly 340 m/s and does not depend on the frequency — a high note and a low note reach you together, which is fortunate, since otherwise an orchestra would arrive out of order. For ALL ELECTROMAGNETIC WAVES in a vacuum the speed is 3.0 × 10⁸ m/s, again the same for every one of them from radio to gamma.

The consequence worth understanding, not memorising

1. In a given medium the SPEED of a wave is fixed — 340 m/s for sound in air, 3.0 × 10⁸ m/s for light in a vacuum.
2. Since v = f × λ and v is fixed, f and λ must be INVERSELY PROPORTIONAL to each other.
3. So a higher frequency means a shorter wavelength, always, and doubling the frequency halves the wavelength.
4. This is why gamma rays sit at one end of the electromagnetic spectrum and radio waves at the other: they are the same kind of wave, differing only in how the fixed speed is divided between frequency and wavelength.
5. And it is why a bass note has a wavelength of metres while a high whistle has one of centimetres, even though both travel at 340 m/s.

One habit will save you repeatedly: check the size of your answer against something you know. A sound wavelength should come out in centimetres or metres; if you get kilometres, something has gone wrong. A visible-light wavelength is under a millionth of a metre. A radio frequency is in the millions of hertz. Numbers in this topic span an enormous range, and a rearrangement done the wrong way round usually produces an answer that is absurd rather than merely wrong — which is a gift, if you look.

2 · Grade 9 Notes

A different job from the section above. You have already understood it; this is the precise set of things to LEARN — definitions to reproduce word for word, processes in order, equations with units, and the answers that score full marks.

v = f × λ

v is wave speed in metres per second (m/s), f is frequency in hertz (Hz), λ is wavelength in metres (m)

Units: m/s, Hz, m. Rearranged: f = v ÷ λ and λ = v ÷ f.

f = 1 ÷ T

f is frequency in hertz (Hz), T is the period in seconds (s)

Units: Hz, s. Rearranged: T = 1 ÷ f. Frequency and period are reciprocals of each other.

Learn this definition · Wave speed

The distance travelled by a wave each second, equal to the frequency multiplied by the wavelength. It is fixed by the medium the wave is travelling through, not by the frequency.

The speeds worth knowing by heart

• SOUND in air: about 340 m/s — and the same for every frequency, so all notes arrive together
• ALL ELECTROMAGNETIC WAVES in a vacuum: 3.0 × 10⁸ m/s — radio, light, X-rays, gamma, all identical
• Sound travels FASTER in liquids than in air, and faster still in solids
• Light travels SLOWER in glass or water than in a vacuum, which is what causes refraction

Answering any wave-equation question

1. Write down which of v, f and λ you have been given, with their units.
2. If you have been given a period instead of a frequency, convert it first with f = 1 ÷ T.
3. Rearrange v = f × λ for the missing quantity BEFORE substituting any numbers.
4. Substitute and calculate, keeping the units with the numbers.
5. Check the answer is a sensible size for the kind of wave described.

Frequency and wavelength in a fixed medium

A. The SPEED is fixed by the medium — 340 m/s for sound in air, whatever the note.

B. So frequency and wavelength are INVERSELY PROPORTIONAL: doubling one halves the other, because their product must stay equal to that fixed speed.

Model answer [3 marks]

A sound wave in air has a frequency of 170 Hz. The speed of sound in air is 340 m/s. Calculate its wavelength. [3]

Rearranging v = f × λ gives λ = v ÷ f. Substituting, λ = 340 ÷ 170 = 2.0 m.

Model answer [4 marks]

A wave has a period of 0.025 s and a wavelength of 12 m. Calculate its speed. [4]

First find the frequency from the period: f = 1 ÷ T = 1 ÷ 0.025 = 40 Hz. Then use the wave equation: v = f × λ = 40 × 12 = 480 m/s.

Practical — measuring wave speed in a ripple tank and on a string

1. Ripple tank: set a vibrating bar to make plane waves and shine a lamp above so the wave pattern shows on paper beneath.
2. Measure the length spanned by ten wavelengths on the paper with a metre rule, and divide by ten to find λ.
3. Count the waves passing a point in 10 s, and divide by 10 to find f; or read f from the signal generator.
4. Calculate the wave speed from v = f × λ.
5. String: stretch a string between a vibration generator and a pulley with a hanging mass, and adjust the frequency until a steady pattern of loops appears.
6. Measure across as many loops as possible; each loop is half a wavelength, so λ = 2 × the length of one loop.
7. Read the frequency from the signal generator and calculate v = f × λ.

Variables

Independent (changed) — The frequency set on the signal generator
Dependent (measured) — The wavelength measured

Sources of error

Not this: A higher-frequency sound travels faster, which is why it sounds sharper.

This: In a given medium every frequency travels at the same speed — all the notes of an orchestra reach you together. A higher frequency means a shorter wavelength, not a greater speed.

Mark-losing trap. If you are given a PERIOD, convert to frequency with f = 1 ÷ T before using v = f × λ.

Mark-losing trap. In one medium the SPEED is fixed. Higher frequency means shorter wavelength, never faster.

Mark-losing trap. Check the size of the answer: light wavelengths are tiny, radio frequencies are enormous.

3 · Prove it — the five questions

The five questions climb Grade 6 → Grade 7 → Grade 8 → Grade 9 → Grade 9 challenge, and are marked inside Incandio on your own device, by rule, with an authored diagnosis of the mistake you actually made. The mark schemes stay in the app so that the practice is worth doing; the questions themselves are here.

1. Grade 6 · Calculate [2 marks] — A wave has a frequency of 8.0 Hz and a wavelength of 3.0 m. Calculate its speed.
2. Grade 7 · Calculate [2 marks] — A wave has a period of 0.20 s. Calculate its frequency.
3. Grade 8 · Calculate [3 marks] — A radio station broadcasts at a frequency of 100 MHz. Radio waves travel at 3.0 × 10⁸ m/s. Calculate the wavelength.
4. Grade 9 · Calculate [4 marks] — A wave in a ripple tank has a period of 0.050 s and a wavelength of 1.6 cm. Calculate its speed in metres per second.
5. 9+ · Analyse [6 marks] — A student says: 'A high-pitched sound must travel faster than a low one — that is why it sounds sharper and reaches you first.' Select every statement that belongs in a full-mark analysis.

The people behind this science

Two ways into the same idea — the one whose equations fixed the speed before anyone measured it, and the one who tried to measure the speed of light and failed usefully. Inside Incandio each of them answers knowing exactly which lesson you have just finished.

James Clerk Maxwell — the one whose equations fixed the speed before anyone measured it

The most remarkable number on this page is 3.0 × 10⁸ m/s, and Maxwell did not measure it — he calculated it. His equations for electric and magnetic fields contained two constants that had been determined in entirely electrical experiments, and combining them gave the speed of a predicted wave. It came out at almost exactly the measured speed of light, and he concluded that light must be such a wave. It is one of the few moments in physics where a whole phenomenon was identified by arithmetic.

• “How did your equations give you a speed you had not measured?”
• “What made you conclude that light must be an electromagnetic wave?”
• “Why do all electromagnetic waves travel at the same speed?”
• “What would it have meant if the number had come out differently?”
• “Which is more convincing, a prediction or a measurement?”

Galileo Galilei — the one who tried to measure the speed of light and failed usefully

Galileo attempted to measure the speed of light with two lanterns on hilltops and two assistants uncovering them in turn. The method could not possibly work — light crosses a valley far too quickly for human reactions — but the experiment was not worthless: he concluded that if light was not instantaneous it must be extraordinarily fast, which is a genuine result honestly obtained. It is also a lesson this page needs, since the numbers here span a range no single method could ever measure.

• “What exactly did you and your assistant do with the lanterns?”
• “What did you conclude when the experiment gave you nothing?”
• “How fast did you think light travelled?”
• “Is a failed measurement worth anything at all?”
• “How would you have measured the speed of sound instead?”

Carry on through the course

• Previous lesson: What a Wave Is
• Next lesson: The Doppler Effect
• What a Wave Is — Something crosses the room and nothing gets to the other side — the idea that makes the rest of the topic possible.
• The Doppler Effect — Why an ambulance changes note as it passes — and why the siren itself never does.
• The Electromagnetic Spectrum — Seven names for one thing — and the two properties that change as you go along it.
• Uses and Dangers of Electromagnetic Radiation — Seven bands, seven jobs — and why the danger rises steadily as you move along the spectrum.
• Reflection and Ray Diagrams — One rule, drawn correctly — and why the image in a mirror is behind it.
• All of Physics · Physics key terms · required practicals · equations · Incandio Science · The 4PH1 specification