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Phase Difference

Key idea: Define phase and phase difference, and calculate phase difference from path difference or time delay (A Level Physics).

  • GCE A-Level H2 Physics 2027
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Learning objectives

  • Describe wave models, use wave quantities and interpret wave graphs in space and time.
  • Relate phase difference to separations in time and position.
  • Explain single-aperture diffraction and apply first-minimum and Rayleigh criteria.
  • Explain coherent two-source interference using phase and path difference.

1. Definitions (Must Know)

A. Phase, φ (rad)

Phase, φ, tells you where you are within a cycle of oscillation.

It is measured in radians.

B. Phase difference, Δ φ (rad)

Phase difference, Δ φ, is the difference in phase between two oscillations (two points on the same wave, or two waves).

It tells you how much one oscillation leads or lags another.

C. In phase and antiphase

RelationshipPhase difference
In phaseΔφ = 2π n
Antiphase (half a cycle apart)Δφ = (2n + 1)π

where n is an integer.

D. Linking phase difference to time delay or path difference

For the same wave (or two coherent waves of the same frequency):

  • time delay Δ t at the same position: Δφ = 2π(Δ t)/T
  • path difference Δ x at the same time: Δφ = 2π(Δ x)/λ
2π means “same phase again”

Phase is periodic. Adding or subtracting 2π does not change the physical state.

2. Key Ideas (What Earns Marks)

  • Always state what you are comparing (same position? same time?).
  • Use the fraction-of-a-cycle idea:
    • Δ t/T is the fraction of a period.
    • Δ x/λ is the fraction of a wavelength.
  • Multiply the fraction by 2π to get phase difference.
  • Reduce answers to a standard range if needed (e.g. 0 to 2π), but remember that 2π multiples are equivalent.

3. Detailed Explanations

A. Time delay → phase difference

If one oscillation reaches the same state Δ t later:

Δφ = 2π(Δ t)/T

Example: if Δ t = (1/4)T, then Δφ = π/2.

B. Path difference → phase difference

If two points on the same progressive wave are separated by distance Δ x (same time snapshot):

Δφ = 2π(Δ x)/λ

Example: if Δ x = (1/2)λ, then Δφ = π (antiphase).

C. Visualising phase difference (example)

This plot shows two sine waves with a fixed phase difference of Δφ = π/2.

Two waves with a phase difference of π/2

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

Scroll across the graph 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.
Two waves with a phase difference of π/2 rad.
Open full-size graph
View figure data
Values for Two waves with a phase difference of π/2
Phase angle (rad)Wave 1Wave 2
001
0.052359877559829880.052335956242943830.9986295347545738
0.104719755119659770.104528463267653460.9945218953682734
0.157079632679489660.156434465040230870.9876883405951378
0.209439510239319530.207911690817759310.9781476007338057
0.26179938779914940.258819045102520740.9659258262890683
0.31415926535897930.30901699437494740.9510565162951536
0.36651914291880920.358367949545300270.9335804264972017
0.418879020478639060.406736643075800150.913545457642601
0.471238898038468970.453990499739546750.8910065241883679
0.52359877559829880.499999999999999940.8660254037844387
0.57595865315812870.5446390350150270.8386705679454243
0.62831853071795860.58778525229247310.8090169943749475
0.68067840827778860.62932039104983750.7771459614569708
0.73303828583761840.66913060635885820.7431448254773942
0.78539816339744820.70710678118654750.7071067811865476
0.83775804095727810.74314482547739420.6691306063588583
0.89011791851710810.77714596145697090.6293203910498374
0.94247779607693790.80901699437494750.5877852522924732
0.99483767363676780.83867056794542390.5446390350150273
1.04719755119659760.86602540378443860.5000000000000003
1.09955742875642760.89100652418836780.45399049973954686
1.15191730631625730.91354545764260090.40673664307580043
1.20427718387608730.93358042649720170.35836794954530066
1.25663706143591720.95105651629515350.3090169943749475
1.30899693899574720.96592582628906830.2588190451025206
1.36135681655557720.97814760073380570.20791169081775931
1.4137166941154070.98768834059513780.15643446504023098
1.46607657167523690.99452189536827330.10452846326765329
1.51843644923506680.99862953475457380.05233595624294381
1.570796326794896311.2246467991473532e-16
1.62315620435472630.9986295347545738-0.052335956242943564
1.67551608191455630.9945218953682734-0.10452846326765305
1.72787595947438620.9876883405951378-0.15643446504023073
1.78023583703421620.9781476007338057-0.2079116908177595
1.8325957145940460.9659258262890683-0.25881904510252035
1.88495559215387590.9510565162951536-0.3090169943749473
1.93731546971370580.9335804264972017-0.35836794954530043
1.98967534727353560.913545457642601-0.4067366430757998
2.04203522483336550.8910065241883679-0.4539904997395467
2.09439510239319530.8660254037844387-0.4999999999999997
2.1467549799530250.8386705679454243-0.5446390350150268
2.1991148575128550.8090169943749475-0.587785252292473
2.2514747350726850.777145961456971-0.6293203910498373
2.30383461263251470.7431448254773945-0.6691306063588579
2.3561944901923450.7071067811865476-0.7071067811865475
2.40855436775217460.6691306063588583-0.743144825477394
2.46091424531200480.6293203910498374-0.7771459614569706
2.51327412287183450.5877852522924732-0.8090169943749473
2.56563400043166420.5446390350150273-0.838670567945424
2.61799387799149440.49999999999999994-0.8660254037844388
2.6703537555513240.45399049973954686-0.8910065241883678
2.72271363311115430.40673664307580004-0.913545457642601
2.7750735106709840.3583679495453002-0.9335804264972016
2.8274333882308140.3090169943749475-0.9510565162951535
2.8797932657906440.2588190451025206-0.9659258262890683
2.93215314335047370.20791169081775931-0.9781476007338057
2.98451302091030350.15643446504023098-0.9876883405951377
3.03687289847013360.10452846326765329-0.9945218953682734
3.08923277602996340.05233595624294381-0.9986295347545738
3.14159265358979275.66553889764798e-16-1
3.193952531149623-0.052335956242943564-0.9986295347545738
3.2463124087094526-0.10452846326765305-0.9945218953682734
3.2986722862692828-0.15643446504023073-0.9876883405951378
3.3510321638291125-0.20791169081775907-0.9781476007338058
3.4033920413889422-0.25881904510252035-0.9659258262890684
3.4557519189487724-0.3090169943749473-0.9510565162951536
3.508111796508602-0.3583679495453-0.9335804264972017
3.5604716740684323-0.4067366430758002-0.9135454576426008
3.612831551628262-0.4539904997395467-0.891006524188368
3.665191429188092-0.4999999999999997-0.866025403784439
3.717551306747922-0.5446390350150271-0.8386705679454243
3.7699111843077517-0.587785252292473-0.8090169943749476
3.8222710618675815-0.6293203910498373-0.7771459614569708
3.8746309394274117-0.6691306063588582-0.743144825477394
3.9269908169872414-0.7071067811865475-0.7071067811865477
3.979350694547071-0.743144825477394-0.6691306063588588
4.031710572106901-0.7771459614569706-0.6293203910498378
4.084070449666731-0.8090169943749473-0.5877852522924734
4.136430327226561-0.838670567945424-0.544639035015027
4.1887902047863905-0.8660254037844385-0.5000000000000004
4.241150082346221-0.8910065241883678-0.45399049973954697
4.29350995990605-0.9135454576426005-0.40673664307580093
4.345869837465881-0.9335804264972019-0.35836794954529994
4.39822971502571-0.9510565162951535-0.3090169943749476
4.4505895925855405-0.9659258262890683-0.2588190451025207
4.50294947014537-0.9781476007338056-0.20791169081775987
4.5553093477052-0.9876883405951377-0.15643446504023112
4.607669225265029-0.9945218953682733-0.1045284632676543
4.66002910282486-0.9986295347545739-0.05233595624294348
4.71238898038469-1-2.4492935982947064e-16
4.76474885794452-0.99862953475457380.05233595624294388
4.817108735504349-0.99452189536827340.10452846326765293
4.869468613064179-0.98768834059513780.15643446504023062
4.9218284906240095-0.97814760073380560.20791169081775937
4.97418836818384-0.96592582628906820.2588190451025211
5.026548245743669-0.95105651629515360.3090169943749472
5.078908123303498-0.93358042649720210.3583679495452995
5.1312680008633285-0.91354545764260110.4067366430757997
5.183627878423159-0.8910065241883680.4539904997395466
5.235987755982989-0.86602540378443860.5
5.288347633542818-0.83867056794542430.5446390350150266
5.340707511102648-0.80901699437494760.5877852522924729
5.393067388662478-0.77714596145697130.6293203910498368
5.445427266222309-0.7431448254773940.6691306063588585
5.497787143782138-0.70710678118654770.7071067811865474
5.550147021341968-0.66913060635885810.7431448254773942
5.602506898901797-0.62932039104983780.7771459614569706
5.654866776461628-0.58778525229247340.8090169943749472
5.707226654021457-0.54463903501502780.8386705679454236
5.759586531581288-0.499999999999999670.8660254037844388
5.811946409141117-0.453990499739546970.8910065241883678
5.8643062867009474-0.406736643075800150.9135454576426009
5.916666164260777-0.358367949545300770.9335804264972015
5.969026041820607-0.30901699437494760.9510565162951535
6.021385919380437-0.25881904510252070.9659258262890683
6.073745796940267-0.207911690817758980.9781476007338057
6.126105674500097-0.156434465040231120.9876883405951377
6.178465552059927-0.104528463267653420.9945218953682733
6.230825429619756-0.052335956242944370.9986295347545738
6.283185307179585-1.133107779529596e-151

The graph shows Wave 2 leading Wave 1 by π/2: at the same horizontal coordinate, Wave 2 is one quarter-cycle further through its oscillation. Always name which wave leads or lags when the question asks for direction, because the magnitude π/2 alone does not specify it.

4. Common Mistakes

  • Using degrees in a formula that expects radians (phase differences are in rad).
  • Mixing up Δ x (path difference) and x (position).
  • Forgetting that 2π multiples represent the same phase.
  • Giving only a positive phase magnitude when the question asks which oscillation leads or lags.

5. Exam Tips

  • Write the fraction first, then multiply by 2π:
    • Δφ = 2π(Δ x/λ) or 2π(Δ t/T).
  • If the question is about “in phase” / “antiphase”, convert your answer to a multiple of π.

6. Worked Examples

Modelled example 1

Phase difference from path difference

Core

Problem

Two points on a wave are separated by 0.30 m when λ = 1.2 m. Find their phase difference.
Study the worked solution

  1. Find the cycle fraction

    Method

    The separation is one quarter of a wavelength.

    Reason

    A full wavelength represents one complete 2π phase cycle.

    Working

    Δ x/λ = 0.30/1.2 = 0.25

  2. Convert fraction to phase

    Method

    The phase difference is π/2 rad.

    Reason

    Multiply the fraction of a spatial cycle by 2π.

    Working

    Δφ = 2π(0.25) = π/2 rad

Guided practice 2

Time delay from phase difference

About 4 min

Problem

Two oscillations have phase difference 2π/3 and period 0.60 s. Find the time delay.

Try this before viewing the solution

Unit: s

Hints

Hint 1: convert phase to cycle fraction
Use Δ t/T = Δφ/(2π).
View solution step by step

  1. Extract the cycle fraction

    Method

    2π/3 is one third of a complete phase cycle.

    Reason

    One full cycle is 2π radians.

    Working

    Δφ/2π = 1/3

  2. Apply the fraction to time

    Method

    The delay is 0.20 s.

    Reason

    A one-third-cycle delay is one third of the period.

    Working

    Δ t = (1/3)(0.60) = 0.20 s

Common misconception 3

In phase or antiphase?

Find and correct the mistake

Learner claim

Two points are separated by 1.5λ. A learner says they are in phase because the separation contains one complete wavelength. Diagnose the claim.

Try this before viewing the solution

Phase relationship

View solution step by step

  1. Include the full separation

    Method

    The phase difference is 3π.

    Reason

    The remaining half-wavelength after one full wavelength contributes another π.

    Working

    Δφ = 2π(1.5) = 3π

  2. Reduce the physical relation

    Method

    3π is equivalent to π modulo 2π.

    Reason

    Adding a full cycle does not change the relative state.

    Working

    3π-2π = π

  3. Classify

    Method

    The points are antiphase, not in phase.

    Reason

    An odd multiple of π represents half a cycle of separation.

    Working

    Δφ = (2n + 1)π

Examiner practice 4

Path difference from phase difference

3 marks

Examination question

Two waves have phase difference 3π/2. Find their path difference in terms of λ. [3 marks]

Try this before viewing the solution

View solution step by step

  1. State the relation

    1 mark

    Method

    Use Δφ = 2πΔ x/λ.

    Reason

    Phase is being related to a spatial path difference.

    Working

    Δφ = 2π(Δ x)/λ

  2. Find the path fraction

    1 mark

    Method

    Δ x/λ = 3/4.

    Reason

    Divide the stated phase by one full 2π cycle.

    Working

    (Δ x)/λ = (3π/2)/2π = 3/4

  3. State the path difference

    1 mark

    Method

    Δ x = 3λ/4.

    Reason

    The result is requested relative to wavelength.

    Working

    Δ x = (3/4)λ

Challenge 5

Phase difference from frequency and time delay

Minimal support

Independent transfer

Two coherent waves have frequency 250 Hz. One arrives 1.0 ms later than the other. Find the phase-difference magnitude.

Try this before viewing the solution

Hints

Hint 1: create the missing period
Use T = 1/f, then compare the delay with that period.
View solution step by step

  1. Find the period

    Method

    T = 4.0 × 10⁻³ s.

    Reason

    Frequency gives cycles per second, so its reciprocal gives seconds per cycle.

    Working

    T = 1/250 = 4.0 × 10⁻³ s

  2. Form the delay fraction

    Method

    The 1.0 ms delay is one quarter of a period.

    Reason

    1.0 ms = 1.0 × 10⁻³ s.

    Working

    Δ t/T = 0.25

  3. Convert to phase

    Method

    The phase-difference magnitude is π/2 rad.

    Reason

    A quarter cycle corresponds to a quarter of 2π.

    Working

    Δφ = 2π(0.25) = π/2 rad

7. Mind Stretchers

Mind stretcher 1: Same phase, different answersExtension

A student finds Δφ = 5π for a phase difference question. Another student writes Δφ = π.

Are they both correct? Explain.

Show Answer

Yes.

5π and π differ by 4π = 2(2π), which is two full cycles. Adding/subtracting 2π does not change the physical state, so the phase relationship is the same.

Mind stretcher 2: Lag vs lead (equivalent phase differences)Extension

Wave 2 lags wave 1 by 3π/2 rad.

Write an equivalent phase difference in the range 0 to 2π, and state whether you could also describe wave 2 as leading wave 1.

Show Answer

Lagging by 3π/2 is equivalent to a phase difference of: 2π-3π/2 = π/2

So wave 2 lags by 3π/2 rad is the same as wave 2 leading by π/2 rad (a quarter cycle).

Mind stretcher 3: Optional (Enrichment)Extension

Phase difference is used heavily in interference:

  • constructive interference: Δφ = 2π n
  • destructive interference: Δφ = (2n + 1)π

See: Principle of Superposition and Interference Patterns.

Continue with the next resource in this course.

Course and syllabus information
Course
GCE A-Level H2 Physics
Edition
GCE A-Level H2 Physics 2027