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Principle Of Superposition

Key idea: Use the principle of superposition to add displacements and explain constructive and destructive interference (A Level Physics).

  • GCE A-Level H2 Physics 2027
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Learning objectives

  • Apply the principle of superposition to resultant displacement.

1. Definitions (Must Know)

A. Principle of superposition

When waves of the same kind overlap, the resultant displacement at a point is the algebraic (vector) sum of the individual displacements at that point.

B. Interference

Interference is the superposition of coherent waves that produces a stable pattern of maxima (large amplitude) and minima (small amplitude).

C. Constructive and destructive interference (two waves)

For two coherent waves of the same frequency:

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

where n is an integer.

Link: phase difference

See: Phase Difference.

2. Key Ideas (What Earns Marks)

  • Superposition works point-by-point: add displacements at the same position and time.
  • Displacement has sign (up/down). Add it algebraically.
  • Complete cancellation at a point requires:
    • phase difference of π (antiphase), and
    • equal amplitudes (or approximately equal).
  • Interference patterns require coherent sources (constant phase difference).
Exam pitfall: adding intensities instead of displacements

At a specific point, superposition adds instantaneous displacements with sign. Intensity relations are secondary results and should not replace displacement addition.

3. Detailed Explanations

A. What “add displacements” means

If two waves overlap at a point:

y = y₁ + y₂

So if one wave produces + 2 mm at that instant and the other produces -1 mm, the resultant is + 1 mm.

B. Why coherence matters

If the phase difference between two sources changes randomly, the maxima/minima move around quickly, and no stable pattern is observed.

Coherent sources keep a constant phase difference, so the pattern is steady.

C. Visual example (superposition of two sine waves)

The resultant wave is the point-by-point sum of the two component waves.

Superposition of two sine waves (example)

The resultant displacement at each point is the algebraic sum of the two displacements.

Scroll across the graph to read all labels.

The resultant displacement at each point is the algebraic sum of the two displacements.The resultant displacement at each point is the algebraic sum of the two displacements.
Resultant displacement is the sum of the two displacements at each point.
Open full-size graph
View figure data
Values for Superposition of two sine waves (example)
Phase angle (rad)Wave 1Wave 2Resultant
000.60.6
0.052359877559829880.052335956242943830.59917772085274420.6515136770956881
0.104719755119659770.104528463267653460.5967131372209640.7012416004886174
0.157079632679489660.156434465040230870.59261300435708260.7490474693973135
0.209439510239319530.207911690817759310.58688856044028340.7948002512580427
0.26179938779914940.258819045102520740.5795554957734410.8383745408759617
0.31415926535897930.30901699437494740.57063390977709220.8796509041520395
0.36651914291880920.358367949545300270.56014825589832110.9185162054436213
0.418879020478639060.406736643075800150.54812727458556050.9548639176613607
0.471238898038468970.453990499739546750.53460391451302070.9885944142525674
0.52359877559829880.499999999999999940.51961524227066321.0196152422706632
0.57595865315812870.5446390350150270.50320234076725461.0478413757822815
0.62831853071795860.58778525229247310.48541019662496841.0731954489174416
0.68067840827778860.62932039104983750.466287576874182431.0956079679240198
0.73303828583761840.66913060635885820.445886895286436551.1150175016452948
0.78539816339744820.70710678118654750.42426406871192851.131370849898476
0.83775804095727810.74314482547739420.4014783638153151.1446231892927092
0.89011791851710810.77714596145697090.377592234629902431.1547381960868734
0.94247779607693790.80901699437494750.352671151375483941.1616881457504313
0.99483767363676780.83867056794542390.326783421009016361.1654539889544404
1.04719755119659760.86602540378443860.30000000000000021.1660254037844389
1.09955742875642760.89100652418836780.27239429984372811.1634008240320959
1.15191730631625730.91354545764260090.244041985845480261.157587443488081
1.20427718387608730.93358042649720170.21502076972718041.148601196224382
1.25663706143591720.95105651629515350.18541019662496851.136466712920122
1.30899693899574720.96592582628906830.155291427061512331.1212172533505806
1.36135681655557720.97814760073380570.124747014490655591.1028946152244612
1.4137166941154070.98768834059513780.093860679024138581.0815490196192763
1.46607657167523690.99452189536827330.062717077960591981.0572389733288652
1.51843644923506680.99862953475457380.0314015737457662841.0300311085003402
1.570796326794896317.347880794884119e-171
1.62315620435472630.9986295347545738-0.031401573745766140.9672279610088077
1.67551608191455630.9945218953682734-0.062717077960591820.9318048174076816
1.72787595947438620.9876883405951378-0.093860679024138430.8938276615709994
1.78023583703421620.9781476007338057-0.12474701449065570.85340058624315
1.8325957145940460.9659258262890683-0.15529142706151220.8106343992275561
1.88495559215387590.9510565162951536-0.185410196624968350.7656463196701853
1.93731546971370580.9335804264972017-0.215020769727180260.7185596567700214
1.98967534727353560.913545457642601-0.244041985845479870.6695034717971211
2.04203522483336550.8910065241883679-0.2723942998437280.6186122243446399
2.09439510239319530.8660254037844387-0.29999999999999980.5660254037844389
2.1467549799530250.8386705679454243-0.326783421009016030.5118871469364082
2.1991148575128550.8090169943749475-0.35267115137548380.4563458429994636
2.2514747350726850.777145961456971-0.37759223462990240.39955372682706863
2.30383461263251470.7431448254773945-0.40147836381531470.34166646166207976
2.3561944901923450.7071067811865476-0.424264068711928450.2828427124746191
2.40855436775217460.6691306063588583-0.44588689528643640.22324371107242197
2.46091424531200480.6293203910498374-0.46628757687418230.16303281417565507
2.51327412287183450.5877852522924732-0.485410196624968370.10237505566750488
2.56563400043166420.5446390350150273-0.50320234076725450.041436694247772854
2.61799387799149440.49999999999999994-0.5196152422706632-0.019615242270663302
2.6703537555513240.45399049973954686-0.5346039145130207-0.08061341477347383
2.72271363311115430.40673664307580004-0.5481272745855605-0.1413906315097605
2.7750735106709840.3583679495453002-0.560148255898321-0.20178030635302074
2.8274333882308140.3090169943749475-0.570633909777092-0.26161691540214455
2.8797932657906440.2588190451025206-0.579555495773441-0.32073645067092044
2.93215314335047370.20791169081775931-0.5868885604402834-0.3789768696225241
2.98451302091030350.15643446504023098-0.5926130043570825-0.4361785393168516
3.03687289847013360.10452846326765329-0.596713137220964-0.49218467395331067
3.08923277602996340.05233595624294381-0.5991777208527442-0.5468417646098004
3.14159265358979275.66553889764798e-16-0.6-0.5999999999999994
3.193952531149623-0.052335956242943564-0.5991777208527442-0.6515136770956877
3.2463124087094526-0.10452846326765305-0.596713137220964-0.7012416004886171
3.2986722862692828-0.15643446504023073-0.5926130043570826-0.7490474693973134
3.3510321638291125-0.20791169081775907-0.5868885604402835-0.7948002512580425
3.4033920413889422-0.25881904510252035-0.579555495773441-0.8383745408759613
3.4557519189487724-0.3090169943749473-0.5706339097770922-0.8796509041520395
3.508111796508602-0.3583679495453-0.5601482558983211-0.9185162054436211
3.5604716740684323-0.4067366430758002-0.5481272745855604-0.9548639176613607
3.612831551628262-0.4539904997395467-0.5346039145130208-0.9885944142525676
3.665191429188092-0.4999999999999997-0.5196152422706634-1.019615242270663
3.717551306747922-0.5446390350150271-0.5032023407672546-1.0478413757822818
3.7699111843077517-0.587785252292473-0.48541019662496854-1.0731954489174416
3.8222710618675815-0.6293203910498373-0.46628757687418243-1.0956079679240198
3.8746309394274117-0.6691306063588582-0.4458868952864364-1.1150175016452946
3.9269908169872414-0.7071067811865475-0.4242640687119286-1.131370849898476
3.979350694547071-0.743144825477394-0.40147836381531526-1.1446231892927092
4.031710572106901-0.7771459614569706-0.3775922346299027-1.1547381960868732
4.084070449666731-0.8090169943749473-0.352671151375484-1.1616881457504313
4.136430327226561-0.838670567945424-0.3267834210090162-1.1654539889544402
4.1887902047863905-0.8660254037844385-0.30000000000000027-1.1660254037844386
4.241150082346221-0.8910065241883678-0.27239429984372815-1.1634008240320959
4.29350995990605-0.9135454576426005-0.24404198584548054-1.157587443488081
4.345869837465881-0.9335804264972019-0.21502076972717996-1.1486011962243818
4.39822971502571-0.9510565162951535-0.18541019662496858-1.136466712920122
4.4505895925855405-0.9659258262890683-0.15529142706151242-1.1212172533505806
4.50294947014537-0.9781476007338056-0.12474701449065592-1.1028946152244614
4.5553093477052-0.9876883405951377-0.09386067902413867-1.0815490196192763
4.607669225265029-0.9945218953682733-0.06271707796059257-1.0572389733288658
4.66002910282486-0.9986295347545739-0.03140157374576609-1.03003110850034
4.71238898038469-1-1.4695761589768238e-16-1.0000000000000002
4.76474885794452-0.99862953475457380.031401573745766326-0.9672279610088075
4.817108735504349-0.99452189536827340.06271707796059175-0.9318048174076816
4.869468613064179-0.98768834059513780.09386067902413837-0.8938276615709994
4.9218284906240095-0.97814760073380560.12474701449065562-0.85340058624315
4.97418836818384-0.96592582628906820.15529142706151264-0.8106343992275555
5.026548245743669-0.95105651629515360.1854101966249683-0.7656463196701854
5.078908123303498-0.93358042649720210.21502076972717968-0.7185596567700224
5.1312680008633285-0.91354545764260110.24404198584547981-0.6695034717971213
5.183627878423159-0.8910065241883680.2723942998437279-0.6186122243446401
5.235987755982989-0.86602540378443860.3-0.5660254037844386
5.288347633542818-0.83867056794542430.326783421009016-0.5118871469364084
5.340707511102648-0.80901699437494760.3526711513754837-0.45634584299946385
5.393067388662478-0.77714596145697130.3775922346299021-0.39955372682706924
5.445427266222309-0.7431448254773940.40147836381531504-0.341666461662079
5.497787143782138-0.70710678118654770.4242640687119284-0.2828427124746193
5.550147021341968-0.66913060635885810.44588689528643655-0.22324371107242158
5.602506898901797-0.62932039104983780.4662875768741823-0.16303281417565552
5.654866776461628-0.58778525229247340.4854101966249683-0.10237505566750504
5.707226654021457-0.54463903501502780.5032023407672541-0.04143669424777363
5.759586531581288-0.499999999999999670.51961524227066320.01961524227066358
5.811946409141117-0.453990499739546970.53460391451302070.08061341477347372
5.8643062867009474-0.406736643075800150.54812727458556050.1413906315097604
5.916666164260777-0.358367949545300770.56014825589832080.20178030635302008
5.969026041820607-0.30901699437494760.5706339097770920.26161691540214443
6.021385919380437-0.25881904510252070.5795554957734410.3207364506709203
6.073745796940267-0.207911690817758980.58688856044028340.3789768696225244
6.126105674500097-0.156434465040231120.59261300435708250.4361785393168514
6.178465552059927-0.104528463267653420.5967131372209640.49218467395331056
6.230825429619756-0.052335956242944370.59917772085274420.5468417646097998
6.283185307179585-1.133107779529596e-150.60.5999999999999989

4. Common Mistakes

  • Adding amplitudes without checking the phase difference.
  • Mixing up amplitude and intensity (intensity is proportional to amplitude squared).
  • Saying “destructive interference means zero always” (zero only if amplitudes match).

5. Exam Tips

  • Use the exact phrase: “resultant displacement is the algebraic sum of the individual displacements”.
  • For interference questions, state the condition for coherence: “constant phase difference”.
  • If asked for “minimum intensity”, check whether the amplitudes are equal.

6. Worked Examples

Modelled example 1

In phase (maximum resultant displacement)

Core

Problem

Two waves of amplitudes 3.0 mm and 2.0 mm meet in phase. Find the maximum resultant displacement.
Study the worked solution

  1. Interpret in phase

    Method

    The two instantaneous displacements reach their positive maxima together.

    Reason

    In-phase waves have zero phase difference modulo 2π.

    Working

    y₁ = +3.0 mm, y₂ = +2.0 mm

  2. Apply superposition

    Method

    The maximum resultant displacement is 5.0 mm.

    Reason

    Signed displacements at the same point and instant add algebraically.

    Working

    y = 3.0 + 2.0 = 5.0 mm

Guided practice 2

Antiphase (minimum resultant displacement)

About 3 min

Problem

Two waves of amplitudes 4.0 mm and 1.5 mm meet in antiphase. Find the minimum resultant amplitude.

Try this before viewing the solution

Unit: mm

Hints

Hint 1: add signed displacements
At the instant one wave is + 4.0 mm, the other is -1.5 mm.
View solution step by step

  1. Assign opposing signs

    Method

    The peak displacements oppose each other.

    Reason

    Antiphase means a phase difference of an odd multiple of π.

    Working

    y = +4.0 + (-1.5) mm

  2. Find the remaining amplitude

    Method

    The minimum resultant amplitude is 2.5 mm.

    Reason

    Unequal amplitudes cancel only partially.

    Working

    |4.0-1.5| = 2.5 mm

Common misconception 3

Complete cancellation condition

Find and correct the mistake

Learner claim

Two waves meet with phase difference π. A learner says complete cancellation is guaranteed regardless of their amplitudes. Diagnose the claim when one amplitude is 6.0 mm.

Try this before viewing the solution

Unit: mm

View solution step by step

  1. Use the phase condition

    Method

    A phase difference of π makes the instantaneous displacements oppose.

    Reason

    Antiphase supplies opposite signs at corresponding points in the cycle.

    Working

    y₂ = -A₂ when y₁ = +A₁

  2. Add the amplitude condition

    Method

    The second amplitude must also be 6.0 mm.

    Reason

    Only equal and opposite displacements sum to zero at every corresponding instant.

    Working

    6.0-A₂ = 0 ⇒ A₂ = 6.0 mm

Examiner practice 4

Adding instantaneous displacements

2 marks

Examination question

At one point and instant, y₁ = +2.0 mm and y₂ = -5.0 mm. Find and interpret the resultant displacement. [2 marks]

Try this before viewing the solution

Unit: mm

View solution step by step

  1. Add algebraically

    1 mark

    Method

    The resultant is -3.0 mm.

    Reason

    Superposition adds displacements at the same point and time, including their signs.

    Working

    y = (+2.0) + (-5.0) = -3.0 mm

  2. Interpret the sign

    1 mark

    Method

    The particle is displaced 3.0 mm in the defined negative direction.

    Reason

    The minus sign carries directional information; it is not discarded as an amplitude.

    Working

    |y| = 3.0 mm, negative direction

Challenge 5

Resultant amplitude for a general phase difference

Minimal support

Independent transfer

Two coherent waves have amplitudes 4.0 mm and 3.0 mm with phase difference 60°. Find the resultant amplitude using the general phasor result supplied in this lesson.

Try this before viewing the solution

Unit: mm

Hints

Hint 1: use the non-extreme phase relation
Substitute into A² = A₁² + A₂² + 2A₁A₂ cos Δφ.
View solution step by step

  1. Select the general relation

    Method

    Use the cosine-rule form for amplitudes with a non-zero, non-π phase difference.

    Reason

    The simple sum and difference are limiting cases only.

    Working

    A = square root of (A₁² + A₂² + 2A₁A₂ cos Δφ)

  2. Substitute and calculate

    Method

    The resultant amplitude is 6.08 mm.

    Reason

    cos 60° = 0.5, giving A² = 37 mm².

    Working

    A = square root of (16 + 9 + 12) = square root of 37 = 6.08 mm

7. Mind Stretchers

Mind stretcher 1: Why “amplitude same” mattersExtension

Two coherent waves meet at a point with phase difference π, but their amplitudes are A and 0.2A.

Will the resultant displacement be zero? Explain.

Show Answer

No.

Antiphase means the displacements oppose, but because the amplitudes are not equal, there is incomplete cancellation. The resultant amplitude is A-0.2A = 0.8A.

Mind stretcher 2: Superposition vs stable interferenceExtension

Two sound waves from different sources overlap in air. Their frequencies are almost equal but not exactly equal.

Does the principle of superposition still apply? Will you observe a stable interference pattern at a fixed point? Explain.

Show Answer

Superposition always applies: the instantaneous resultant displacement is the sum of the individual displacements.

However, a stable interference pattern requires a constant phase difference. If the frequencies differ slightly, the phase difference changes with time, so the “maxima/minima” at a point drift (you may hear beats rather than a steady pattern).

Mind stretcher 3: Optional (Enrichment)Extension

A. Resultant amplitude formula (phasor result)

If two waves of amplitudes A₁ and A₂ have phase difference Δφ, the resultant amplitude is: A = square root of (A₁² + A₂² + 2A₁A₂ cos(Δφ))

This is useful when Δφ is not 0 or π, but it is not always needed for exam questions.

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