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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 14.2

Periodic-Wave Measures and Graphs

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

1. Topic Lens

Periodic-Wave Measures and Graphs 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 Periodic-Wave Measures and GraphsOriginal schematic connecting Periodic-Wave Measures and Graphs to Waves, Sound, and Physical Optics.
Periodic-Wave Measures and Graphs: 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 14.2 Lesson

5. Learning Targets

  • Measure amplitude, wavelength, period, and frequency from correctly labeled graphs.
  • Distinguish a spatial snapshot from the time record of one location.
  • Translate among \(T\), \(f\), \(\lambda\), \(v\), angular frequency \(\omega\), and wave number \(k\).
  • Read amplitude, direction, speed, and initial phase from a sinusoidal traveling-wave equation.
  • Compare the phase of two points or two observations and support conclusions with calculations.

6. Measurement Dictionary

QuantityMeaningHow to measure itSI unit
Amplitude \(A\)Largest displacement from equilibriumVertical distance from the midline to a crest or troughmetre
Wavelength \(\lambda\)Shortest spatial repeat distanceDistance between adjacent equal-phase points on a \(y\)-versus-\(x\) snapshotmetre
Period \(T\)Time for one complete cycleTime between identical states on a \(y\)-versus-\(t\) recordsecond
Frequency \(f\)Cycles completed per second\(f=N/\Delta t=1/T\)hertz
Wave speed \(v\)Rate at which a fixed phase travels\(v=\Delta x/\Delta t=\lambda/T=f\lambda\)metre per second
Wave number \(k\)Spatial phase change per metre\(k=2\pi/\lambda\)radian per metre
Angular frequency \(\omega\)Temporal phase change per second\(\omega=2\pi f=2\pi/T\)radian per second

Common measurement error: crest-to-trough height is \(2A\), not \(A\). Crest-to-adjacent-trough distance is \(\lambda/2\), not \(\lambda\).

7. One Wave, Two Different Graphs

Comparison of a spatial wave snapshot and a time-history graph The upper graph plots displacement against position and labels amplitude and wavelength. The lower graph plots displacement at one position against time and labels amplitude and period. snapshot: y versus x at one timehistory: y versus t at one positionAλTxtyy
Both graphs can look sinusoidal, but the horizontal axes answer different questions. The upper graph compares many locations at one time; the lower graph follows one location through many times.
From \(y(x)\)Read \(A\) vertically and \(\lambda\) horizontally. Period requires timing or speed information.
From \(y(t)\)Read \(A\) vertically and \(T\) horizontally. Wavelength requires spatial or speed information.

8. Build the Core Relationships

  1. One cycle occupies \(2\pi\) radians, so \(f=1/T\) and \(\omega=2\pi/T=2\pi f\).
  2. One wavelength occupies \(2\pi\) radians of spatial phase, so \(k=2\pi/\lambda\).
  3. During one period, a crest advances one wavelength; therefore \(v=\lambda/T=f\lambda\).
  4. Combining the angular quantities gives \(v=\omega/k\).
\[f=\frac{1}{T},\qquad k=\frac{2\pi}{\lambda},\qquad \omega=2\pi f,\qquad v=f\lambda=\frac{\omega}{k}\]

\(k\) measures how rapidly phase changes with position, while \(\omega\) measures how rapidly it changes with time.

9. Sinusoidal Traveling-Wave Model

\[y(x,t)=A\sin(kx-\omega t+\phi_0)\]
  • \(A\) sets the maximum displacement.
  • \(k\) determines the spatial repeat distance.
  • \(\omega\) determines the temporal repeat interval.
  • \(\phi_0\) sets the phase at \(x=0,t=0\).
  • \(kx-\omega t\) moves toward \(+x\); \(kx+\omega t\) moves toward \(-x\).

Hold phase constant to verify direction. From \(kx-\omega t+\phi_0=C\), \(x=(\omega/k)t+\text{constant}\), so a fixed phase moves toward increasing \(x\).

10. Phase Comparisons

Two observations are in phase when their phase difference is an integer multiple of \(2\pi\). A phase difference of \(\pi\) places them half a cycle apart.

\[\Delta\phi_x=k\Delta x=2\pi\frac{\Delta x}{\lambda}\]
\[\Delta\phi_t=\omega\Delta t=2\pi\frac{\Delta t}{T}\]
  • Points separated by \(\lambda\) are in phase.
  • Points separated by \(\lambda/2\) are \(180^\circ\) out of phase.
  • Observations separated by \(T/4\) differ by \(\pi/2\).

Equal displacement alone does not guarantee equal phase: one point may be moving upward while the other moves downward.

11. Extract Every Quantity from an Equation

A wave is described in SI units by

\[y(x,t)=0.120\sin\left(4\pi x-20\pi t+\frac{\pi}{6}\right)\]
Amplitude\(A=0.120\,\mathrm m\)
Wavelength\(\lambda=2\pi/k=0.500\,\mathrm m\)
Frequency\(f=\omega/(2\pi)=10.0\,\mathrm{Hz}\)
Period\(T=1/f=0.100\,\mathrm s\)
Speed\(v=\omega/k=5.00\,\mathrm{m/s}\)
DirectionThe minus sign before \(\omega t\) means \(+x\).
Initial phase\(\phi_0=\pi/6\)
At the origin\(y(0,0)=0.120\sin(\pi/6)=0.060\,\mathrm m\)

Unit check: a sine argument is dimensionless, so \(kx\), \(\omega t\), and \(\phi_0\) must all represent angles.

12. Worked Graph Example

A snapshot shows 4 complete wavelengths spanning 3.20 m. At one detector, 15 cycles pass during 6.00 s. Crest-to-trough height is 0.18 m.

\[\lambda=\frac{3.20}{4}=0.800\,\mathrm m,\qquad f=\frac{15}{6.00}=2.50\,\mathrm{Hz}\]
\[T=\frac{1}{f}=0.400\,\mathrm s,\qquad v=f\lambda=2.00\,\mathrm{m/s}\]
\[A=\frac{0.18}{2}=0.090\,\mathrm m\]

Measuring several cycles reduces fractional reading error because a nearly fixed endpoint uncertainty is divided across a longer interval.

13. Predict Graph Transformations

  • Double \(A\): vertical extrema double, but \(T\), \(f\), \(\lambda\), and ideal speed do not automatically change.
  • Double \(f\) in the same nondispersive medium: \(T\) and \(\lambda\) halve while \(v\) stays constant.
  • Increase \(\phi_0\): phase shifts without changing amplitude, period, or wavelength.
  • Reverse the sign between spatial and temporal terms: travel direction reverses, but speed magnitude remains \(\omega/k\).

Graph shape alone does not identify which physical control changed. Use scales and stated constraints.

14. Measurement Investigation

  1. Open PhET Wave on a String. Select Oscillate, No End, low damping, and one fixed tension.
  2. Pause after several cycles. Measure amplitude vertically and the span of three wavelengths horizontally; divide the span by three.
  3. At a fixed reference line, time ten cycles. Calculate \(T=\Delta t/10\) and \(f=1/T\).
  4. Calculate \(v=f\lambda\). Change only frequency, repeat, and compare calculated speeds.
  5. Change only amplitude and test whether \(f\), \(\lambda\), or \(v\) changes beyond measurement uncertainty.

Record settings, raw multi-cycle measurements, calculations, repeated trials, and uncertainty. Separate a physical trend from ruler or stopwatch resolution.

15. AP-Style Reasoning Checks

  • A graph shows displacement against time. Can wavelength be measured directly?
  • Two points are separated by \(3\lambda/4\). What is their phase difference modulo \(2\pi\)?
  • For \(y=A\cos(6x+18t)\), find speed magnitude and direction.
  • A wave's crest-to-trough height is 8 cm. State its amplitude.
  • Source frequency triples in an unchanged nondispersive medium. Predict the new wavelength.

Answers: no, spatial or speed information is needed; \(3\pi/2\); \(3\,\mathrm{m/s}\) toward \(-x\); \(4\,\mathrm{cm}\); one third of the original wavelength.

Checkpoint · Topic 14.2

Explain how periodic-wave measures and graphs 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.