Learn9702Paper 2 · AS Level Structured QuestionsWavesProgressive waves and intensity

Waves · Subtopic 1 of 3

Progressive Waves and Intensity

A progressive wave transfers energy without transferring matter. Points one wavelength apart oscillate in phase; points half a wavelength apart are in antiphase. Phase difference (in degrees or radians) $=\dfrac{x}{\lambda}\times360^\circ$ . Intensity, the power per unit area, is proportional to amplitude squared, and falls with the inverse square of distance from a point source.

Cambridge 9702 Paper 1Cambridge 9702 Paper 2Edexcel YPH11 WPH12AQA 7408 Paper 1≈ 35 minRevision 6 · updated 2026-10-27

Section 1 of 14

Learning objectives

Define the quantities that describe a wave and use $v=f\lambda$ .
Calculate phase difference between points on a wave.
Use an oscilloscope to measure period and amplitude.
Use $I=\dfrac PA$ and $I\propto A^2$ .

Pause and think

Turn up the volume on a speaker so the amplitude of the sound wave doubles. Is the sound twice as intense?

Section 2 of 14

Core idea

Start hereTwo corks on a pond

Drop a stone in a pond and watch two corks bob. If they're a whole wavelength apart, they rise and fall together; half a wavelength apart, one rises as the other falls. How "out of step" they are depends only on their separation compared with the wavelength.

The big idea

$v=f\lambda$ . Phase difference for separation $x$ : $\phi=\frac{2\pi x}\lambda$ . Intensity $I=\frac PA\propto(\text{amplitude})^2$ .

1. Describing waves

Displacement–distance at one instant (wave moving right)

time = 0.0

P and Q are a quarter-wavelength apart, so they are

90^\circ

(

\tfrac\pi2

rad) out of phase. Points one whole wavelength apart move in phase.

v=f\lambda\qquad T=\frac1f\qquad\phi=\frac{2\pi x}\lambda\qquad I=\frac PA

$\phi$: phase difference (rad).
$x$: separation along the wave.
$I$: intensity (W m⁻²).

Phase difference

$x=\lambda\Rightarrow\phi=2\pi$
One wavelength apart: one full cycle.
$\phi=\frac x\lambda\times2\pi$
Fraction of a cycle.

Worked examplef = 50 Hz, λ = 0.40 m

Speed; phase difference between points 0.10 m apart.

Goal: $v$ , $\phi$ .
Strategy: Formulas.

$v=50\times0.40=20\text{ m s}^{-1}$
$f\lambda$ .
$\phi=\frac{2\pi(0.10)}{0.40}=\frac\pi2$
Quarter wavelength.

Answer

20

m s

^{-1}

;

\frac\pi2

rad (

90^\circ

✓ Check

\frac14

wavelength ↔

\frac14

cycle ✓.

Worked exampleCRO trace

One cycle spans 4 divisions at 2 ms/div.

Goal: Frequency.
Strategy: Period first.

$T=4\times2=8\text{ ms}$
Time base.
$f=\frac1{0.008}=125\text{ Hz}$
Reciprocal.

Answer

125

Hz.

✓ Checkms → s conversion ✓.

Inverse square law for a point source

$A=4\pi r^2$
Energy spreads over a sphere.
$I=\frac P{4\pi r^2}$
Double $r$ : quarter intensity.

Common misconception $I\propto A$ Intensity ∝ amplitude squared: double amplitude, ×4 intensity.

2. Transverse and longitudinal

Transverse vs longitudinal

Light and water waves are transverse; sound is longitudinal. Only transverse waves can be polarised.

Pause and think

Points half a wavelength apart: phase difference?

Summary

$v=f\lambda$ ; $\phi=2\pi x/\lambda$ .
$I\propto A^2$ ; point source $1/r^2$ .

Phase difference: How far out of step two oscillations are.
Intensity: Power per unit area.

HistoryHuygens' wavelets. In

1678

Christiaan Huygens proposed that every point on a wavefront acts as a source of tiny spherical wavelets, and the new wavefront is the surface touching them all. His principle explains reflection, refraction and diffraction, and is still taught today.

Fun factSound intensity spans a trillion times. Your ears can detect sound with an intensity of about

10^{-12}

W m⁻² and tolerate up to about

1

W m⁻² before pain: a range of a trillion. That's why loudness is measured on the logarithmic decibel scale.

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