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AP Course

AP Physics 2

Study AP Physics 2 Units 9–15, with the Fall 2026 radioactive-decay clarification applied and official exam updates linked separately.

Study Units 9–15 through original models, derivations, experiments, quizzes, and practice sets.

Lessons
Particle Motion Behind Temperature and PressureConnecting Gas Variables with the Ideal-Gas ModelHeat Flow and the Approach to Thermal BalanceEnergy Accounting for Thermodynamic ProcessesMaterial Response to Heating and Heat ConductionEntropy, Probability, and the Direction of Thermal ChangeCharge Interactions and Coulomb-Force ModelsTracking Charge During Contact and InductionMapping Electric Fields from Source ChargesEnergy of Configurations of ChargesElectric Potential and Equipotential ReasoningCapacitance and Energy Stored in Electric FieldsConserving Energy in Charged-Particle MotionCharge Flow and Conventional CurrentModeling Sources, Wires, and Loads in Simple NetworksHow Geometry and Material Set ResistanceElectrical Energy Transfer and PowerReducing Series-Parallel DC NetworksLoop Equations from Energy ConservationJunction Equations from Charge ConservationTransient Charging and Discharging in RC NetworksSources, Direction, and Strength of Magnetic FieldsMagnetic Forces and Curved Paths of ChargesForces Between Fields and Current-Carrying ConductorsChanging Magnetic Flux and Induced EMFReflection and Absorption at Material BoundariesLocating Images with Mirror Ray ModelsRefraction, Index, and Total Internal ReflectionLocating Images with Thin-Lens ModelsPulses, Wave Types, and Propagation SpeedPeriodic-Wave Measures and GraphsBoundary Changes and PolarizationElectromagnetic-Wave BehaviorFrequency Shifts from Relative MotionSuperposition, Interference, and Standing WavesDiffraction from Openings and EdgesTwo-Slit and Grating Pattern GeometryPhase Change and Thin-Film ColorQuantum Models and Dual Wave-Particle EvidenceEnergy Levels in a Hydrogen-Like AtomConnecting Spectral Lines to Energy TransitionsThermal Spectra and Blackbody CurvesPhoton Thresholds in the Photoelectric EffectPhoton-Electron Scattering and MomentumMass-Energy Accounting in Fission and FusionRandom Nuclear Decay, Activity, and Half-Life
Quizzes
Practice Problems Formula notes, diagrams, and practice sets will be added here.

AP Physics 2 · Unit 14 · Topic 14.6

Superposition, Interference, and Standing Waves

Superposition, boundary conditions, diffraction, and phase explain sound and physical-optics patterns.

1. Topic Lens

Superposition, Interference, and Standing Waves is studied through waves, sound, and physical optics. Connect the system boundary, interacting parts, and measurable evidence before applying a formula.

\[v=f\lambda\]

2. Why the Formula Works

The relationship is built from definitions and conservation reasoning:

  1. A repeating wave advances one wavelength during one period T.
  2. Speed is distance over time, so v=λ/T.
  3. Frequency is 1/T, giving v=fλ.
\[\lambda=\frac{v}{f}\]

3. Detailed Visual Model

Pixel diagram for Superposition, Interference, and Standing WavesOriginal schematic connecting Superposition, Interference, and Standing Waves to Waves, Sound, and Physical Optics.
Superposition, Interference, and Standing Waves: an original pixel-style model. Use it as a schematic, not a literal scale drawing.

4. Worked Example and Lab Link

A 340 Hz tone travels at 340 m/s. Find its wavelength.

Answer: λ=v/f=1.00 m.

Investigation idea: Measure standing-wave nodes in an air column or string and compare allowed wavelengths with boundary conditions.

Common trap: Wave speed is determined by the medium; changing frequency usually changes wavelength, not the medium's speed.

Expanded Topic 14.6 Lesson

5. Learning Targets

  • Add overlapping wave disturbances point by point using the linear superposition principle.
  • Relate phase difference and path difference to constructive, partial, or destructive interference.
  • Distinguish the addition of amplitudes from the addition of wave intensities.
  • Derive a standing wave from two equal sinusoidal waves traveling in opposite directions.
  • Use boundary conditions to determine nodes, antinodes, wavelengths, and resonant frequencies.

6. Superposition Is Point-by-Point Addition

When waves overlap in a linear system, the resultant disturbance at each location and instant is the algebraic sum of the individual disturbances:

\[y_{\mathrm{net}}(x,t)=\sum_i y_i(x,t)\]

For a string, add signed displacements. For sound, add pressure changes. For electromagnetic waves, add electric-field vectors and magnetic-field vectors. The waves do not permanently consume or bounce off one another; after a temporary overlap, each continues according to the system's wave equation.

7. Linear-Model Limits

Superposition follows when the governing equation is linear. Small-amplitude string waves, ordinary sound, and electromagnetic fields are commonly modeled this way.

  • Two positive displacements add to a larger positive displacement.
  • A positive and negative displacement can partially or completely cancel.
  • Cancellation at one moment does not mean that the component waves or their energy have vanished permanently.
  • At very large amplitudes, nonlinear material response can distort waves and invalidate simple addition.

8. Original Pulse-Superposition Model

Point-by-point superposition of two pulse pairs Two positive pulses overlap to make a pulse of greater amplitude, while an equal positive and negative pulse overlap to produce zero displacement. After overlap the original pulses continue. two crests approachingconstructive overlap: 2Acrest and trough approachingdestructive overlap: 0component pulses continue after the overlap
At overlap, add the vertical coordinates at the same \(x\). The upper and lower cases are separate examples placed on shared guide lines; the drawing does not show pulses colliding like solid objects.

9. Phase Controls the Resultant Amplitude

For two sinusoidal waves with the same frequency and wave number but amplitudes \(A_1,A_2\) and phase difference \(\Delta\phi\), phasor addition gives

\[A_R=\sqrt{A_1^2+A_2^2+2A_1A_2\cos\Delta\phi}\]

When \(A_1=A_2=A\), this becomes

\[A_R=2A\left|\cos\frac{\Delta\phi}{2}\right|\]
\(\Delta\phi=0,2\pi,\ldots\)Complete constructive interference: \(A_R=A_1+A_2\).
\(\Delta\phi=\pi,3\pi,\ldots\)Most destructive: \(A_R=|A_1-A_2|\).
Other phase differencesPartial interference: the resultant lies between those limits.

10. Path Difference Becomes Phase Difference

For coherent, in-phase sources with the same wavelength, a path difference \(\Delta r=r_2-r_1\) produces

\[\Delta\phi=\frac{2\pi}{\lambda}\Delta r\]
\[\text{constructive: }\Delta r=m\lambda,\qquad \text{destructive: }\Delta r=\left(m+\frac12\right)\lambda\]

If the sources begin with an initial phase offset \(\phi_0\), use \(\Delta\phi=2\pi\Delta r/\lambda+\phi_0\). Path difference alone is not enough unless the source phase relation is known and stable.

11. Amplitude Is Not Intensity

For two coherent waves whose individual time-averaged intensities are \(I_1\) and \(I_2\),

\[I=I_1+I_2+2\sqrt{I_1I_2}\cos\Delta\phi\]

Two equal in-phase waves each of intensity \(I_0\) produce amplitude \(2A\) and maximum intensity \(4I_0\), not \(2I_0\). For mutually incoherent sources, the cross term averages to zero over time, so measured intensities normally add: \(I\approx I_1+I_2\).

12. Worked Interference Examples

Unequal amplitudes: two waves have \(A_1=3.0\,\mathrm{mm}\), \(A_2=4.0\,\mathrm{mm}\), and \(\Delta\phi=\pi/2\).

\[A_R=\sqrt{3.0^2+4.0^2+2(3.0)(4.0)\cos(\pi/2)}=5.0\,\mathrm{mm}\]

Path difference: two in-phase speakers emit \(\lambda=0.80\,\mathrm m\). At a point where \(\Delta r=1.20\,\mathrm m\),

\[\Delta\phi=2\pi\frac{1.20}{0.80}=3\pi\]

The waves arrive out of phase and interfere destructively. Complete cancellation additionally requires equal amplitudes at that point.

13. Nearby Frequencies Produce Beats

Superpose two equal-amplitude waves at one location with slightly different angular frequencies. A trigonometric identity gives a rapidly oscillating tone inside a slowly varying envelope:

\[y=2A\cos\left(\frac{\Delta\omega}{2}t\right)\cos(\omega_{\mathrm{avg}}t),\qquad f_{\mathrm{beat}}=|f_1-f_2|\]

For \(f_1=256\,\mathrm{Hz}\) and \(f_2=260\,\mathrm{Hz}\), loudness reaches a maximum four times per second. Beats are time-varying interference, not a third source emitting at \(4\,\mathrm{Hz}\).

14. Deriving a Standing Wave

Add equal waves traveling in opposite directions:

\[y_1=A\sin(kx-\omega t),\qquad y_2=A\sin(kx+\omega t)\]

Using \(\sin(a-b)+\sin(a+b)=2\sin a\cos b\),

\[y(x,t)=2A\sin(kx)\cos(\omega t)\]

The factor \(2A\sin(kx)\) sets a position-dependent amplitude while \(\cos(\omega t)\) makes every non-node position oscillate. The pattern does not translate, although its two component waves do.

15. Nodes, Antinodes, and Spacing

For the standing-wave form above,

\[x_{\mathrm{node}}=n\frac{\lambda}{2},\qquad x_{\mathrm{antinode}}=(2n+1)\frac{\lambda}{4}\]
  • Adjacent nodes are separated by \(\lambda/2\).
  • A neighboring node and antinode are separated by \(\lambda/4\).
  • All points between the same pair of nodes oscillate in phase.
  • Points in adjacent loops oscillate \(\pi\) radians out of phase.

16. Original Normal-Mode Map

First three normal modes of a string fixed at both ends The fundamental has one antinode, the second mode has two antinodes, and the third mode has three antinodes. Every mode has nodes at both fixed ends. n = 1 · λ₁ = 2Ln = 2 · λ₂ = Ln = 3 · λ₃ = 2L/3fixed ends
The dots mark displacement nodes. Mode \(n\) fits \(n\) half-wavelengths into length \(L\), giving \(n\) antinodes and \(n+1\) nodes when both endpoints are counted.

17. A String Fixed at Both Ends

Both endpoints must be displacement nodes. Only wavelengths that fit an integer number of half-wavelengths are allowed:

\[L=n\frac{\lambda_n}{2},\qquad \lambda_n=\frac{2L}{n},\qquad f_n=\frac{nv}{2L}=nf_1\]

These allowed patterns are normal modes. The lowest frequency \(f_1\) is the fundamental. For an ideal string the higher resonant frequencies are integer harmonics.

18. Air-Column Boundary Conditions

Open–open tubeDisplacement antinode at each end. \(f_n=nv/(2L)\), \(n=1,2,3,\ldots\)
Closed–open tubeDisplacement node at the closed end and antinode at the open end. \(f_n=(2n-1)v/(4L)\).

Pressure nodes occur where displacement antinodes occur, and pressure antinodes occur where displacement nodes occur. End corrections make a real tube's effective acoustic length slightly different from its measured length.

19. Worked Resonance Example

A string of length \(1.20\,\mathrm m\) has linear density \(\mu=0.0050\,\mathrm{kg/m}\) and tension \(45\,\mathrm N\).

\[v=\sqrt{\frac{F_T}{\mu}}=\sqrt{\frac{45}{0.0050}}=94.9\,\mathrm{m/s}\]

For the third harmonic,

\[\lambda_3=\frac{2(1.20)}{3}=0.800\,\mathrm m,\qquad f_3=\frac{3(94.9)}{2(1.20)}=119\,\mathrm{Hz}\]

The pattern has three antinodes and four nodes including the endpoints. Increasing tension by a factor of four doubles every resonant frequency because \(v\propto\sqrt{F_T}\).

20. Energy and Resonance

A pure standing wave has zero time-averaged net energy flow along the system because its equal counterpropagating components carry equal power in opposite directions. Energy still changes locally between kinetic and potential forms.

At resonance, a periodic driver supplies energy at a normal-mode frequency and can build a large response. Damping limits the amplitude and broadens the frequency range of the response. A standing-wave shape alone does not imply unlimited energy or zero damping.

21. Evidence-Building Investigations

A. Two-source interference
  1. Open PhET Wave Interference and choose two coherent sources.
  2. Keep frequency fixed and map at least three high-amplitude and three low-amplitude locations.
  3. Measure or infer \(r_1,r_2\), calculate \(\Delta r\), and test the predicted condition.
  4. Change source separation or wavelength and explain how the pattern spacing responds.
B. Standing waves on a string
  1. Open PhET Wave on a String with low damping and a fixed end.
  2. Vary frequency slowly and record settings that produce stable patterns.
  3. Count loops, measure node spacing, and infer \(\lambda\).
  4. Test whether \(f_n/f_1\approx n\), then identify finite resolution and damping as limitations.

22. Common Reasoning Traps

  • Adding intensities before fields: coherent waves interfere through signed amplitudes or fields first.
  • Assuming destructive means zero: complete cancellation requires equal amplitudes and a \(\pi\) phase difference.
  • Thinking waves stop at overlap: linear component waves pass through and re-emerge unchanged.
  • Calling every quiet point a node: a node remains zero at all times; temporary cancellation at one instant is not enough.
  • Using the wrong tube boundary: open and closed ends impose different displacement conditions.
  • Counting only internal nodes: fixed endpoints are also nodes.

23. AP-Style Reasoning Checks

  1. Two equal pulses, one \(+4\,\mathrm{cm}\) and one \(-3\,\mathrm{cm}\), overlap completely. What is the instantaneous displacement?
  2. Two in-phase sources have path difference \(3\lambda/2\). Is the interference constructive or destructive?
  3. Two equal coherent waves interfere constructively. Compare resultant amplitude and intensity with one wave.
  4. Adjacent nodes are \(0.30\,\mathrm m\) apart. Find the wavelength.
  5. A fixed-fixed string shows four antinodes. Identify the harmonic and number of nodes including endpoints.
  6. Why can a standing-wave node have zero displacement while the string near it remains sloped?

Answers: \(+1\,\mathrm{cm}\); destructive; amplitude \(2A\) and intensity \(4I_0\); \(0.60\,\mathrm m\); fourth harmonic with five nodes; the node condition fixes displacement, not spatial derivative, and nearby points still oscillate.

Checkpoint · Topic 14.6

Explain how superposition, interference, and standing waves supports or limits this conclusion: λ=v/f=1.00 m.

Official curriculum reference: College Board AP Physics 2 course page. The explanation and worked example are independently written for this study site.