Use the wave equation and intensity relationships
Derive and apply v = fλ, then relate power per area to amplitude and intensity.
AS Level; Cambridge International Physics 9702, 2025–2027 syllabus, Topic 7. These original wave examples and diagrams are conceptual study aids; full timed assessment and human subject-teacher review remain outstanding.
Before you begin
- Use frequency, period, wavelength, amplitude, speed and trigonometric ratios with consistent units.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
How far does a wave's phase point travel in one period?
What you will learn
- Derive the wave equation from distance, frequency and wavelength
- Calculate wave speed, frequency or wavelength using v = fλ
- Use intensity = power/area and intensity proportional to amplitude squared
Derive the speed-frequency-wavelength relation
In one period T, a fixed phase point advances by one wavelength λ. Its speed is therefore v = λ/T. Since frequency f = 1/T, substituting gives v = fλ.
Use compatible units: v in m s⁻¹, f in Hz (s⁻¹) and λ in m. If the medium changes, wave speed can change; when frequency is fixed by the source, wavelength changes with it.
Intensity describes power spread over area
Intensity is power transferred per unit area: I = P/A. For a progressive wave, intensity is proportional to the square of its amplitude, I ∝ a², for the same wave conditions.
If amplitude doubles under those same conditions, intensity becomes four times as large. Do not assume power is unchanged when area or amplitude changes unless the question states the comparison model.
Pause and explain
If amplitude doubles under the same wave conditions, how does intensity change?
Worked example
A wave has frequency 4.0 Hz and wavelength 0.75 m. Find its speed. A 2.0 W source spreads power uniformly over 0.50 m²; find intensity.
Show the worked solution
- Wave speed v = fλ = 4.0 × 0.75 = 3.0 m s⁻¹.
- Intensity I = P/A = 2.0/0.50 = 4.0 W m⁻².
- The wave equation and intensity equation describe different quantities.
Answer Wave speed 3.0 m s⁻¹; intensity 4.0 W m⁻².
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Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
A sound wave has frequency 500 Hz and wavelength 0.68 m. Find its speed.
Give me a hint
Multiply frequency by wavelength using SI units.
Compare my reasoning
- v = fλ.
- v = 500 × 0.68.
- State m s⁻¹.
340 m s⁻¹.
Look for these in your work
- Correct relation used
- Product calculated
- Speed unit stated
A wave travels at 12 m s⁻¹ with frequency 8.0 Hz. Find its wavelength.
Give me a hint
Rearrange v = fλ to λ = v/f.
Compare my reasoning
- λ = 12/8.0 m.
- λ = 1.5 m.
- Check that the unit is length.
1.5 m.
Look for these in your work
- Equation rearranged
- Value calculated
- Length unit stated
A transmitter's wave amplitude changes from 2.0 mm to 6.0 mm while all other conditions stay fixed. Its initial intensity is 5.0 W m⁻². Find the new intensity.
Give me a hint
Intensity scales with the square of the amplitude ratio.
Compare my reasoning
- Amplitude ratio = 6.0/2.0 = 3.
- Intensity factor = 3² = 9.
- New intensity = 9 × 5.0 W m⁻².
45 W m⁻².
Look for these in your work
- Amplitude ratio calculated
- Square applied
- Intensity scaled correctly
Common mistakes
- Using period in place of frequency without taking its reciprocal
- Leaving wavelength in centimetres when speed is required in SI units
- Doubling amplitude and assuming intensity only doubles
What can you explain now?
How far does a wave's phase point travel in one period?
Compare with the explanation
One wavelength
Adjacent identical phase points are separated by λ.
After trying it yourself, choose your next review. This is your self-assessment.
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Make it stick
Why does one period correspond to one wavelength when deriving v = fλ?
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.