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Progressive waves, wave graphs and phase difference

Key idea: Waves transfer energy without net matter transfer, and v = fλ. A time graph gives period; a distance graph gives wavelength, so phase differences must use the matching interval.

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  • GCE A-Level H2 Physics 2027
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Pause after each idea, then explain it in your own words before moving on.

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Enlarged diagram

A wave carries energy from one place to another without carrying the material along with it. This lesson defines the quantities that describe a wave, from amplitude to speed, and derives v = fλ. It then shows how to read the two kinds of wave graph, and how to find the phase difference between two oscillations, which you will need for every later idea in this topic: superposition, interference and standing waves.

Waves transfer energy, not matter

In a mechanical wave, such as a wave on a string, sound in air or a ripple on water, the particles of a material medium oscillate about their equilibrium positions. Each particle pulls or pushes on its neighbours as it moves, doing work on them, so energy passes along the medium. In the ideal small-amplitude model, each particle returns after a full cycle: the disturbance and its energy travel, the matter does not.

In an electromagnetic wave, such as light or microwaves, electric and magnetic fields oscillate in space and time. No material medium is required: the wave can travel through a vacuum.

A wave that carries energy away from its source like this is a progressive wave. Waves come in two types:

  • in a transverse wave, the oscillations are perpendicular to the direction the wave travels: waves on a string, and all electromagnetic waves;
  • in a longitudinal wave, the oscillations are parallel to the direction of travel: sound in air. The particles crowd together in compressions and spread apart in rarefactions.

Follow one particle and one crest

Hz
m
m/s

Pulling the rope tighter makes waves travel faster.

t = 0.00 s

A transverse wave: the marked particle stays at x = 1.2 metres and its displacement is 0.088 metres. The tracked crest is at x = 1.50 metres. Play runs one period; the window scales with wavelength to keep the tracked crest visible.

Particle equilibrium x
1.20 m
Particle displacement y
0.088 m
Tracked crest x
1.50 m
Frequency, f
1.00 Hz
Period, T
1.00 s
Wave speed, v
2.00 m/s
Displacement against distance (now)
Displacement of the red particle against time

Try this

0 of 1 done

  1. Predict where the particle and crest will be after one period, then play one cycle. (not done yet)

This transverse-string model uses y(x,t) = A sin[2πf(t − x/v)]. The marked material point keeps its x coordinate while the tracked crest travels; vertical scale is exaggerated. The focused view retunes the whole wave and restarts its cycle when you change a setting. Its window adjusts to keep the tracked crest visible for one period. The ordinary Rope mode instead lets a source change spread along the medium.

The quantities that describe a wave

QuantitySymbol and unitMeaning
displacementy, mdistance of a particle from its equilibrium position, in a stated direction (so it can be negative)
amplitudeA, mthe maximum magnitude of displacement
periodT, sthe time for one complete oscillation of a particle
frequencyf, Hzthe number of oscillations per unit time; f = 1/T
wavelengthλ, mthe distance between two neighbouring points that are in phase, such as two crests
wave speedv, m s⁻¹the distance the wave pattern travels per unit time

The wave equation follows from these definitions. In one period, each particle completes one oscillation, and in that time the pattern moves forward by exactly one wavelength. So the speed of the pattern is

v = λ/T = fλ

This is the speed of the wave, not of any particle. The particles move back and forth, with speeds that change continually through each cycle.

Two kinds of wave graph

A wave changes with both position and time, so a graph has to hold one of them fixed.

  • A displacement–distance graph is a snapshot: the displacement of every particle at one instant. One cycle along the horizontal axis is the wavelength λ.
  • A displacement–time graph follows one particle: its displacement as time passes. One cycle along the horizontal axis is the period T.

The two graphs can look identical. Always read the horizontal axis before naming a spacing.

Read the horizontal axis first

Displacement against position
Record a snapshot at one instant. One complete cycle along the position axis spans wavelength λ, measured in metres.
Displacement against time
Follow one fixed position. One complete cycle along the time axis takes period T, measured in seconds.

Both graphs show signed displacement vertically. For a longitudinal wave, a positive plotted value represents displacement along the chosen positive direction, not upward motion.

Longitudinal particle motion and propagationA row of particles has three equally spaced crowded regions. Wavelength is marked between the centres of two adjacent compressions. The wave travels right, while particle oscillation is shown by a separate horizontal two-way arrow.Longitudinal waveWave and energy travelλCrowded regions are compressions.Particle oscillationparallel to wave travel
In a longitudinal wave, particles oscillate parallel to propagation. The compressions travel, while each particle oscillates about its own equilibrium position. This particle sketch is schematic, not to scale.

The graphs below show a wave with amplitude 0.020 m, wavelength 0.80 m and period 0.25 s, travelling to the right. The first shows the profile at t = 0 and a quarter of a period later; the second follows the particle at x = 0.

A wave profile advances through space

Two sinusoidal displacement profiles with amplitude 0.020 m and wavelength 0.80 m. At x = 0 displacement changes from zero to −0.020 m.

If the figure is wider than its frame, scroll across to read all labels.

Two sinusoidal displacement profiles with amplitude 0.020 m and wavelength 0.80 m. At x = 0 displacement changes from zero to −0.020 m.Two sinusoidal displacement profiles with amplitude 0.020 m and wavelength 0.80 m. At x = 0 displacement changes from zero to −0.020 m.
The later profile is shifted right by 0.20 m, one quarter of the wavelength.
Open full-size graphOpen full-size graph
What to noticeIllustrative model y = 0.020 sin[2π(x/0.80 − t/0.25)], travelling right. Displacement and position are different quantities.
View figure data

This table is included in print while it is open.

Figure data: A wave profile advances through space
Position (m)t = 0 st = 0.0625 s (T/4)
0.000.000000-0.020000
0.020.003129-0.019754
0.040.006180-0.019021
0.060.009080-0.017820
0.080.011756-0.016180
0.100.014142-0.014142
0.120.016180-0.011756
0.140.017820-0.009080
0.160.019021-0.006180
0.180.019754-0.003129
0.200.0200000.000000
0.220.0197540.003129
0.240.0190210.006180
0.260.0178200.009080
0.280.0161800.011756
0.300.0141420.014142
0.320.0117560.016180
0.340.0090800.017820
0.360.0061800.019021
0.380.0031290.019754
0.400.0000000.020000
0.42-0.0031290.019754
0.44-0.0061800.019021
0.46-0.0090800.017820
0.48-0.0117560.016180
0.50-0.0141420.014142
0.52-0.0161800.011756
0.54-0.0178200.009080
0.56-0.0190210.006180
0.58-0.0197540.003129
0.60-0.0200000.000000
0.62-0.019754-0.003129
0.64-0.019021-0.006180
0.66-0.017820-0.009080
0.68-0.016180-0.011756
0.70-0.014142-0.014142
0.72-0.011756-0.016180
0.74-0.009080-0.017820
0.76-0.006180-0.019021
0.78-0.003129-0.019754
0.800.000000-0.020000
0.820.003129-0.019754
0.840.006180-0.019021
0.860.009080-0.017820
0.880.011756-0.016180
0.900.014142-0.014142
0.920.016180-0.011756
0.940.017820-0.009080
0.960.019021-0.006180
0.980.019754-0.003129
1.000.0200000.000000
1.020.0197540.003129
1.040.0190210.006180
1.060.0178200.009080
1.080.0161800.011756
1.100.0141420.014142
1.120.0117560.016180
1.140.0090800.017820
1.160.0061800.019021
1.180.0031290.019754
1.200.0000000.020000
1.22-0.0031290.019754
1.24-0.0061800.019021
1.26-0.0090800.017820
1.28-0.0117560.016180
1.30-0.0141420.014142
1.32-0.0161800.011756
1.34-0.0178200.009080
1.36-0.0190210.006180
1.38-0.0197540.003129
1.40-0.0200000.000000
1.42-0.019754-0.003129
1.44-0.019021-0.006180
1.46-0.017820-0.009080
1.48-0.016180-0.011756
1.50-0.014142-0.014142
1.52-0.011756-0.016180
1.54-0.009080-0.017820
1.56-0.006180-0.019021
1.58-0.003129-0.019754
1.600.000000-0.020000

Follow one position as the wave passes

A displacement–time trace at x = 0 starts at zero, falls to −0.020 m after a quarter period, and returns to the same phase after 0.25 s.

If the figure is wider than its frame, scroll across to read all labels.

A displacement–time trace at x = 0 starts at zero, falls to −0.020 m after a quarter period, and returns to the same phase after 0.25 s.A displacement–time trace at x = 0 starts at zero, falls to −0.020 m after a quarter period, and returns to the same phase after 0.25 s.
This trace follows one fixed position. Compare its initial slope with the passing spatial profile, and read the period from the horizontal axis.
Open full-size graphOpen full-size graph
What to noticeSame illustrative right-travelling wave as the spatial profiles. Position is held fixed; the horizontal axis is time.
View figure data

This table is included in print while it is open.

Figure data: Follow one position as the wave passes
Time (s)x = 0 m
0.000000.000000
0.00625-0.003129
0.01250-0.006180
0.01875-0.009080
0.02500-0.011756
0.03125-0.014142
0.03750-0.016180
0.04375-0.017820
0.05000-0.019021
0.05625-0.019754
0.06250-0.020000
0.06875-0.019754
0.07500-0.019021
0.08125-0.017820
0.08750-0.016180
0.09375-0.014142
0.10000-0.011756
0.10625-0.009080
0.11250-0.006180
0.11875-0.003129
0.125000.000000
0.131250.003129
0.137500.006180
0.143750.009080
0.150000.011756
0.156250.014142
0.162500.016180
0.168750.017820
0.175000.019021
0.181250.019754
0.187500.020000
0.193750.019754
0.200000.019021
0.206250.017820
0.212500.016180
0.218750.014142
0.225000.011756
0.231250.009080
0.237500.006180
0.243750.003129
0.250000.000000
0.25625-0.003129
0.26250-0.006180
0.26875-0.009080
0.27500-0.011756
0.28125-0.014142
0.28750-0.016180
0.29375-0.017820
0.30000-0.019021
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0.32500-0.019021
0.33125-0.017820
0.33750-0.016180
0.34375-0.014142
0.35000-0.011756
0.35625-0.009080
0.36250-0.006180
0.36875-0.003129
0.375000.000000
0.381250.003129
0.387500.006180
0.393750.009080
0.400000.011756
0.406250.014142
0.412500.016180
0.418750.017820
0.425000.019021
0.431250.019754
0.437500.020000
0.443750.019754
0.450000.019021
0.456250.017820
0.462500.016180
0.468750.014142
0.475000.011756
0.481250.009080
0.487500.006180
0.493750.003129
0.500000.000000

In a quarter of a period, the whole profile shifts a quarter of a wavelength to the right. That gives a quick way to find which way any particle is moving: for a wave travelling right, each particle is about to take the displacement that the particle just to its left has now.

Check your understanding 1

Translate between wave snapshots and time graphs

Another wave travels to the left with wavelength 1.20 m and period 0.40 s. In a snapshot, the displacement at P is zero and the particles immediately to its right have positive displacement. How far does the profile move in 0.10 s? Which way does P move initially, and what sign should its time-graph gradient have?

Show the steps Optional working
Unit: m s⁻¹
Unit: m
Initial motion of P and time-graph gradient
Show answer

The wave speed is 1.20/0.40 = 3.0 m s⁻¹, so the profile moves 3.0(0.10) = 0.30 m to the left. P takes the displacement currently just to its right, so its displacement initially becomes positive. Its time graph must therefore initially have a positive gradient. The 0.30 m describes the travelling profile, rather than a particle’s travel.

Spot the mistake 1

“Crest spacing is always the wavelength”

About 3 min

Learner claim

A learner says: “The horizontal distance between neighbouring crests on a wave graph is always the wavelength.” When is this true, and what does the spacing mean otherwise?

Diagnose the claim

On a displacement–time graph, the crest spacing is
Show solution step by step

  1. Read the axis

    Method

    Check whether the horizontal axis is position or time.

    Reason

    A cycle along space is a wavelength; a cycle along time is a period.

    Working

    Answer

    On a displacement–distance graph at one instant, the crest spacing is λ.

  2. Name the other case

    Working

    Answer

    On a displacement–time graph at one position, the crest spacing is the period T, measured in seconds.

Graphs of longitudinal waves

A displacement–distance graph of a longitudinal wave still plots displacement upwards, even though the particles move along the direction of travel. Take displacement to the right as positive. Where the graph slopes most steeply downwards, the particle on the left is displaced right and the particle on the right is displaced left: they have moved towards each other, so that is the centre of a compression. Where the graph slopes most steeply upwards, neighbouring particles have moved apart: the centre of a rarefaction. The peaks and troughs of the graph are neither.

Compression occurs at a downward zero crossing

An ideal longitudinal displacement snapshot of wavelength 0.80 m. Compression centres are at x = 0, 0.80 and 1.60 m; rarefaction centres are at x = 0.40 and 1.20 m. The displacement extrema are between these centres.

If the figure is wider than its frame, scroll across to read all labels.

An ideal longitudinal displacement snapshot of wavelength 0.80 m. Compression centres are at x = 0, 0.80 and 1.60 m; rarefaction centres are at x = 0.40 and 1.20 m. The displacement extrema are between these centres.An ideal longitudinal displacement snapshot of wavelength 0.80 m. Compression centres are at x = 0, 0.80 and 1.60 m; rarefaction centres are at x = 0.40 and 1.20 m. The displacement extrema are between these centres.
Circles mark compression centres; squares mark rarefaction centres. Both occur at zero displacement, with opposite spatial slopes.
Open full-size graphOpen full-size graph
What to noticeIllustrative small-amplitude model ξ = −0.020 sin(2πx/0.80). The vertical coordinate represents rightward particle displacement, not pressure or transverse motion. Horizontal position is the equilibrium position.
View figure data

This table is included in print while it is open.

Figure data: Compression occurs at a downward zero crossing
SeriesEquilibrium position (m)Rightward displacement (m)
Longitudinal displacement0.000.000000
Longitudinal displacement0.02-0.003129
Longitudinal displacement0.04-0.006180
Longitudinal displacement0.06-0.009080
Longitudinal displacement0.08-0.011756
Longitudinal displacement0.10-0.014142
Longitudinal displacement0.12-0.016180
Longitudinal displacement0.14-0.017820
Longitudinal displacement0.16-0.019021
Longitudinal displacement0.18-0.019754
Longitudinal displacement0.20-0.020000
Longitudinal displacement0.22-0.019754
Longitudinal displacement0.24-0.019021
Longitudinal displacement0.26-0.017820
Longitudinal displacement0.28-0.016180
Longitudinal displacement0.30-0.014142
Longitudinal displacement0.32-0.011756
Longitudinal displacement0.34-0.009080
Longitudinal displacement0.36-0.006180
Longitudinal displacement0.38-0.003129
Longitudinal displacement0.400.000000
Longitudinal displacement0.420.003129
Longitudinal displacement0.440.006180
Longitudinal displacement0.460.009080
Longitudinal displacement0.480.011756
Longitudinal displacement0.500.014142
Longitudinal displacement0.520.016180
Longitudinal displacement0.540.017820
Longitudinal displacement0.560.019021
Longitudinal displacement0.580.019754
Longitudinal displacement0.600.020000
Longitudinal displacement0.620.019754
Longitudinal displacement0.640.019021
Longitudinal displacement0.660.017820
Longitudinal displacement0.680.016180
Longitudinal displacement0.700.014142
Longitudinal displacement0.720.011756
Longitudinal displacement0.740.009080
Longitudinal displacement0.760.006180
Longitudinal displacement0.780.003129
Longitudinal displacement0.800.000000
Longitudinal displacement0.82-0.003129
Longitudinal displacement0.84-0.006180
Longitudinal displacement0.86-0.009080
Longitudinal displacement0.88-0.011756
Longitudinal displacement0.90-0.014142
Longitudinal displacement0.92-0.016180
Longitudinal displacement0.94-0.017820
Longitudinal displacement0.96-0.019021
Longitudinal displacement0.98-0.019754
Longitudinal displacement1.00-0.020000
Longitudinal displacement1.02-0.019754
Longitudinal displacement1.04-0.019021
Longitudinal displacement1.06-0.017820
Longitudinal displacement1.08-0.016180
Longitudinal displacement1.10-0.014142
Longitudinal displacement1.12-0.011756
Longitudinal displacement1.14-0.009080
Longitudinal displacement1.16-0.006180
Longitudinal displacement1.18-0.003129
Longitudinal displacement1.200.000000
Longitudinal displacement1.220.003129
Longitudinal displacement1.240.006180
Longitudinal displacement1.260.009080
Longitudinal displacement1.280.011756
Longitudinal displacement1.300.014142
Longitudinal displacement1.320.016180
Longitudinal displacement1.340.017820
Longitudinal displacement1.360.019021
Longitudinal displacement1.380.019754
Longitudinal displacement1.400.020000
Longitudinal displacement1.420.019754
Longitudinal displacement1.440.019021
Longitudinal displacement1.460.017820
Longitudinal displacement1.480.016180
Longitudinal displacement1.500.014142
Longitudinal displacement1.520.011756
Longitudinal displacement1.540.009080
Longitudinal displacement1.560.006180
Longitudinal displacement1.580.003129
Longitudinal displacement1.600.000000
Compression centres0.000.000000
Compression centres0.800.000000
Compression centres1.600.000000
Rarefaction centres0.400.000000
Rarefaction centres1.200.000000

At x = 0.80 m, neighbours are displaced towards this compression centre; at x = 0.40 m they are displaced away from the rarefaction centre. These displacements from equilibrium do not state the particles’ instantaneous velocities. The vertical axis represents horizontal particle displacement, not a transverse motion or the pressure. Compare the circle and square markers at the two kinds of zero crossing.

Phase and phase difference

The phase of an oscillating particle says how far through its cycle it is. One complete cycle is 2π radians, or 360°, so a particle a quarter of the way through its cycle has moved on by π/2.

The phase difference between two oscillations of the same frequency is how far one is ahead of the other, as a fraction of a cycle expressed as an angle:

  • two oscillations in phase (phase difference 0, 2π, …) reach their maxima at the same time;
  • two oscillations in antiphase (phase difference π) have opposite displacements: one is at a maximum when the other is at a minimum. When moving, they move in opposite directions.

A constant phase difference requires equal frequencies. If they differ, the oscillations drift in and out of step and the phase difference keeps changing.

Two waves with a phase difference of π/2

Example of two sine waves with the same frequency but a constant phase difference.

If the figure is wider than its frame, scroll across to read all labels.

Example of two sine waves with the same frequency but a constant phase difference.Example of two sine waves with the same frequency but a constant phase difference.
For equal-frequency normalised sine oscillations, Wave 2 leads Wave 1 by π/2. Their corresponding maxima are separated by a quarter-period.
Open full-size graphOpen full-size graph
What to noticeThe horizontal variable is the common time phase ωt. Curves are normalised by their equal amplitudes; a lead is a signed timing statement, not a spatial direction.
View figure data

This table is included in print while it is open.

Figure data: Two waves with a phase difference of π/2
Time phase, ωt (rad)Wave 1Wave 2
0.000000.000e+001.000e+00
0.052365.234e-029.986e-01
0.104721.045e-019.945e-01
0.157081.564e-019.877e-01
0.209442.079e-019.781e-01
0.261802.588e-019.659e-01
0.314163.090e-019.511e-01
0.366523.584e-019.336e-01
0.418884.067e-019.135e-01
0.471244.540e-018.910e-01
0.523605.000e-018.660e-01
0.575965.446e-018.387e-01
0.628325.878e-018.090e-01
0.680686.293e-017.771e-01
0.733046.691e-017.431e-01
0.785407.071e-017.071e-01
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0.942488.090e-015.878e-01
0.994848.387e-015.446e-01
1.047208.660e-015.000e-01
1.099568.910e-014.540e-01
1.151929.135e-014.067e-01
1.204289.336e-013.584e-01
1.256649.511e-013.090e-01
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1.361369.781e-012.079e-01
1.413729.877e-011.564e-01
1.466089.945e-011.045e-01
1.518449.986e-015.234e-02
1.570801.000e+001.225e-16
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1.675529.945e-01-1.045e-01
1.727889.877e-01-1.564e-01
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2.199118.090e-01-5.878e-01
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2.460916.293e-01-7.771e-01
2.513275.878e-01-8.090e-01
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2.827433.090e-01-9.511e-01
2.879792.588e-01-9.659e-01
2.932152.079e-01-9.781e-01
2.984511.564e-01-9.877e-01
3.036871.045e-01-9.945e-01
3.089235.234e-02-9.986e-01
3.141595.666e-16-1.000e+00
3.19395-5.234e-02-9.986e-01
3.24631-1.045e-01-9.945e-01
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3.40339-2.588e-01-9.659e-01
3.45575-3.090e-01-9.511e-01
3.50811-3.584e-01-9.336e-01
3.56047-4.067e-01-9.135e-01
3.61283-4.540e-01-8.910e-01
3.66519-5.000e-01-8.660e-01
3.71755-5.446e-01-8.387e-01
3.76991-5.878e-01-8.090e-01
3.82227-6.293e-01-7.771e-01
3.87463-6.691e-01-7.431e-01
3.92699-7.071e-01-7.071e-01
3.97935-7.431e-01-6.691e-01
4.03171-7.771e-01-6.293e-01
4.08407-8.090e-01-5.878e-01
4.13643-8.387e-01-5.446e-01
4.18879-8.660e-01-5.000e-01
4.24115-8.910e-01-4.540e-01
4.29351-9.135e-01-4.067e-01
4.34587-9.336e-01-3.584e-01
4.39823-9.511e-01-3.090e-01
4.45059-9.659e-01-2.588e-01
4.50295-9.781e-01-2.079e-01
4.55531-9.877e-01-1.564e-01
4.60767-9.945e-01-1.045e-01
4.66003-9.986e-01-5.234e-02
4.71239-1.000e+00-2.449e-16
4.76475-9.986e-015.234e-02
4.81711-9.945e-011.045e-01
4.86947-9.877e-011.564e-01
4.92183-9.781e-012.079e-01
4.97419-9.659e-012.588e-01
5.02655-9.511e-013.090e-01
5.07891-9.336e-013.584e-01
5.13127-9.135e-014.067e-01
5.18363-8.910e-014.540e-01
5.23599-8.660e-015.000e-01
5.28835-8.387e-015.446e-01
5.34071-8.090e-015.878e-01
5.39307-7.771e-016.293e-01
5.44543-7.431e-016.691e-01
5.49779-7.071e-017.071e-01
5.55015-6.691e-017.431e-01
5.60251-6.293e-017.771e-01
5.65487-5.878e-018.090e-01
5.70723-5.446e-018.387e-01
5.75959-5.000e-018.660e-01
5.81195-4.540e-018.910e-01
5.86431-4.067e-019.135e-01
5.91667-3.584e-019.336e-01
5.96903-3.090e-019.511e-01
6.02139-2.588e-019.659e-01
6.07375-2.079e-019.781e-01
6.12611-1.564e-019.877e-01
6.17847-1.045e-019.945e-01
6.23083-5.234e-029.986e-01
6.28319-1.133e-151.000e+00

In this graph, wave 2 reaches its maximum a quarter of a period before wave 1. Wave 2 leads wave 1 by π/2; wave 1 lags wave 2 by π/2.

Finding a phase difference from a graph

A phase difference is a fraction of a cycle, so find what fraction of a cycle separates the two oscillations, then multiply by 2π:

  • two particles a distance Δ x apart on one wave, at one instant: Δφ = (Δ x)/λ × 2π;
  • one particle at two times, or two oscillations on one displacement–time graph, separated by a time Δ t: Δφ = (Δ t)/T × 2π.

On a progressive wave, the oscillation reaches the particle nearer the source first, so that particle leads. Particles one whole wavelength apart are in phase; particles half a wavelength apart are in antiphase.

Adding or removing a whole cycle changes nothing, so subtract 2π until the phase difference lies between 0 and 2π. If a question asks for the smallest phase difference and your answer is more than π, subtract it from 2π: going round the cycle the other way is shorter. Reverse the lead/lag description when using that complementary angle: leading by 3π/2 is equivalent to lagging by π/2.

Worked example 1

Speed and phase from two graphs

A displacement–distance graph of a wave shows a wavelength of 0.80 m. A displacement–time graph for one point on the wave shows successive crests 0.25 s apart. Find the wave speed and the phase difference between two points 0.30 m apart along the wave.

Show full solution

  1. Read each cycle from its own graph

    Method

    Take the wavelength from the distance graph and the period from the time graph.

    Reason

    Crest spacing is a wavelength only along distance and a period only along time.

    Working

    Answer

    λ = 0.80 m and T = 0.25 s, so f = 1/T = 4.0 Hz.

  2. Find the speed

    Reason

    The pattern moves one wavelength in each period.

    Working

    Answer

    v = fλ = (4.0)(0.80) = 3.2 m s⁻¹

  3. Convert the separation to phase

    Working

    Answer

    The separation is 0.30/0.80 of a cycle, so

    Δφ = (0.30/0.80)(2π)

    This gives 0.75π rad, or 135°.

Check your understanding 2

Find phase difference from crest timings

A displacement–time graph shows two oscillations of period 8.0 ms. Oscillation P reaches a crest at t = 2.0 ms and oscillation Q reaches its next crest at t = 5.0 ms. Find the phase difference, and say which oscillation leads.

Show the steps Optional working
Unit: rad
Which oscillation leads?
Show answer

Δ t = 3.0 ms, so Δφ = 3.0/8.0 × 2π = 3π/4 rad. P reaches its crest first, so P leads Q by 3π/4.

Check your understanding 3

Find the smallest phase difference between points

Two points on a wave are 0.75λ apart. What is the smallest phase difference between them?

Show answer

0.75 × 2π = 3π/2 rad. Going round the cycle the other way, the same pair of oscillations differ by 2π-3π/2 = π/2 rad, so the smallest phase difference is π/2. The point nearer the source leads by 3π/2, which is the same as lagging by π/2.

Guided practice 1

Phase difference in a sound wave

About 4 min

The centres of neighbouring compressions in a sound wave are 0.45 m apart, and the frequency is 8.0 × 10² Hz. Find the speed of the wave and the phase difference between two air particles 0.15 m apart along the direction of travel.

Find the speed and the phase difference

Show the steps Optional working
Unit: m
Unit: m s⁻¹
Unit: rad
Hint 1

what is one wavelength here?

What spacing marks one complete cycle in this sound wave?

Hint 2

fraction of a cycle

0.15 m is what fraction of 0.45 m?

Show solution step by step

  1. Find the speed

    Reason

    Compressions are one wavelength apart, so λ = 0.45 m.

    Working

    Answer

    v = fλ = (800)(0.45) = 360 m s⁻¹

  2. Find the phase difference

    Working

    Answer

    The particle separation is one-third of a wavelength:

    Δφ = (0.15/0.45)(2π)

    This is 2π/3 = 2.09 rad, or 120°.

Try it yourself 1

Which point leads?

Minimal support

A wave of frequency 4.0 Hz travels at 3.2 m s⁻¹ along a rope, passing point P and then point Q, 1.0 m further on. Find the smallest phase difference between the oscillations at P and Q, and state which point leads.

Find the phase difference

Which point leads by the smallest phase difference?
Show the steps Optional working
Unit: m
Unit: rad
Hint 1

find the wavelength

Use λ = v/f. How many wavelengths is 1.0 m?

Hint 2

remove whole cycles

Whole wavelengths of separation leave the oscillations in phase, so only the part left over matters.

Show solution step by step

  1. Find the wavelength

    Reason

    The phase difference depends on the separation as a fraction of a wavelength.

    Working

    Answer

    Use λ = v/f:

    λ = 3.2/4.0 = 0.80 m

    The separation is 1.0/0.80 = 1.25 wavelengths.

  2. Remove the whole cycle and decide the lead

    Working

    Answer

    1.25 cycles is one whole cycle plus 0.25 of a cycle. Removing the whole cycle gives

    Δφ = π/2 = 1.57 rad

    The wave reaches P first, so P leads Q by π/2.

Explaining wave graphs in an exam

Exam-style question 1

An unlabelled wave graph

6 marks

A student is given a sinusoidal wave graph with no axis labels and says that the spacing of its peaks is the wavelength.

(a) Explain what must be known before this conclusion is valid, and what the spacing would be otherwise. [2 marks]

(b) Describe how a displacement–distance graph and a displacement–time graph of the same wave can be used to find the wave speed and the phase difference between two points on the wave. [4 marks]

Write your answer before viewing the mark scheme

Show solution step by step

  1. Identify the graph

    2 marks

    Method

    Say which quantity is on the horizontal axis.

    Reason

    The same sinusoid can be a snapshot in space or a record in time.

    Working

    Answer

    The peak spacing is the wavelength only if the graph shows displacement against distance at one instant. If it shows displacement against time at one position, the spacing is the period.

  2. Find speed and phase

    4 marks

    Working

    Answer

    Read λ from the displacement–distance graph and T from the displacement–time graph. Then f = 1/T and v = fλ. For two points a distance Δ x apart, the phase difference is Δφ = 2πΔ x/λ.

Common mistakes

  • Reading a wavelength from a displacement–time graph, or a period from a displacement–distance graph.
  • Confusing the wave speed with the speed of the particles. The pattern moves at a steady v = fλ; each particle speeds up and slows down as it oscillates.
  • Quoting a phase difference without saying which oscillation leads, when the question asks.
  • Using 2πΔ x/λ for two oscillations shown on a time graph. A time separation needs 2πΔ t/T.
  • Calling the peaks of a longitudinal displacement graph compressions. Compressions are where the graph slopes most steeply downwards.

Before you move on

Check your understanding 4

Relate crest motion to marker oscillation

A wave crest travels 3.6 m in 0.30 s. A floating marker completes one oscillation every 0.20 s and returns to its starting position each cycle. Find the wave speed, wavelength and smallest phase difference between two markers 0.60 m apart along the wave. Does the crest’s speed describe the marker’s net motion?

Show answer

The wave speed is 3.6/0.30 = 12 m s⁻¹. The marker’s period is 0.20 s, so λ = vT = 12(0.20) = 2.4 m. The separation is one-quarter of a wavelength, giving a phase difference of 2π(0.60/2.4) = π/2 rad. The crest carries the wave pattern along; the marker oscillates and has zero net displacement over each cycle, so 12 m s⁻¹ is not its net velocity.

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