UPIII · Second-year undergraduate · Calculus-based waves, optics, thermodynamics, and modern physics
Waves, light, and heat—then the places they stop working.
An eleven-unit calculus-based third course carrying one idea from end to end: energy that travels. Simple harmonic motion becomes a mechanical wave, the wave becomes light in both its ray and wave descriptions, the same energy moving molecule to molecule becomes heat and the second law, and the final four units are the experiments where that classical account fails and relativity and quantum mechanics replace it.
Independent pathway notice. GioPhysics University & College Physics is an independent learning pathway. GioPhysics is not a university, does not award academic credit or degrees, and is not accredited by or affiliated with any university or professional body. This is not a universal programme specification. Course coverage and degree requirements vary by institution; confirm requirements with your university.
Before unit 01
Bring energy and fields forward. Add differential equations.
Physics III normally follows calculus-based mechanics and electromagnetism. It leans on energy methods, sinusoidal functions, and second-order differential equations, and it introduces probability where the physics itself becomes statistical.Readiness, not gatekeeping
Know which mathematics must be refreshed.
Almost every model in this course is a sinusoid, an exponential, or a combination of the two. Fluency with both—and with the differential equations that produce them—matters more here than any individual formula.
- University Physics I and II, or equivalent calculus-based mechanics and electromagnetism
- Energy methods, momentum, and rotational dynamics from mechanics
- Sinusoidal functions, complex exponentials in outline, logarithms, and small-angle approximations
- Second-order linear differential equations with constant coefficients, at least operationally
How calculus enters: a second-order differential equation defines simple harmonic motion and the wave equation; integrals accumulate work, heat, entropy, and intensity over a cycle or an aperture; and partial derivatives appear operationally where a wave varies in both space and time.
Mathematical tools used explicitly
- Second-order differential equations for oscillators and the wave equation
- Partial derivatives, used operationally where a wave varies in space and time
- Superposition, phasor addition, and phase-difference bookkeeping
- Definite integrals for work, heat, entropy change, and intensity over a cycle or aperture
- Exponentials and logarithms for damping, decay, decibels, and Boltzmann factors
- Probability and normalisation, introduced where the physics itself becomes statistical
- Dimensional, limiting-case, and correspondence checks against the classical result
Eleven units · about fourteen weeks of teaching
One idea carries the course: energy that travels.
An oscillator stores energy and gives it back. Couple oscillators and the energy moves as a wave; make the wave electromagnetic and it is light; let it move molecule to molecule and it is heat. The last four units are where that classical account runs out.Units 1–5 form the shared waves-and-optics spine, from simple harmonic motion through interference, diffraction, and polarization.
Units 6–7 are thermal physics; many departments teach these in a separate course, and some assess them within Physics II instead.
Units 8–9 introduce relativity and quantum physics at a depth that varies widely; some courses treat them as a survey, others as full units.
Units 10–11 are an explicitly bounded survey of atomic, nuclear, and particle physics, not a complete course in any of the three.
Course scope: This GioPhysics course map is an adaptable learning sequence, not academic credit, accreditation, or a universal university syllabus. Third-semester content is the least standardised of the introductory sequence: departments place oscillations, waves, optics, and thermal physics differently, and the modern-physics units here are a bounded survey rather than a complete course in relativity, quantum mechanics, atomic, nuclear, or particle physics. Follow your institution's published scope, notation, laboratory programme, and assessment rules.
Complete interactive course map
All eleven units, searchable by concept and evidence.
Search 99 numbered subsections, inspect outcomes and prerequisites, and expand the laboratory and problem-practice plan for each unit.Mapped lesson links: linked topics open relevant existing GioPhysics material. Topics without a link are mapped but not yet written; nothing here links to a page that does not teach it.
Interactive course map
Choose one phase. Open one unit. Learn by doing.
Search includes unit titles, topics, skills, outcomes, prerequisites, and laboratory questions across all 11 units.
Private study checklist
0 of 11 units reviewed1.5 weeks · UPIII-01 · Medium-heavy
Oscillations
Rebuild simple harmonic motion from the linear restoring force and its differential equation, then extend to energy, mass–spring systems, pendulums, small oscillations in a potential well, damping, and driven resonance, stating the approximation each period model rests on.
9 topics5 outcomes2 lab directions170 min practiceOpen Oscillations: topics, evidence, and practice
Topics and mapped lessons
- Periodic motion and the simple-harmonic condition
Amplitude, period, frequency, and angular frequency; the defining relation a = −ω²x produced by a restoring force linear in displacement; isochronism; and the fact that no real system stays linear beyond a limited displacement.
- The SHM equation of motion and its general solution
Turning m d²x/dt² = −kx into d²x/dt² = −ω²x, verifying x = A cos(ωt + φ) by substitution, fixing A and φ from initial position and velocity, eliminating time to reach v = ±ω√(A² − x²), and noting that superposing solutions works only while the equation stays linear.
- Velocity, acceleration, and phase relations in SHM
Differentiating the sinusoidal solution for quarter-cycle phase shifts between x, v, and a, v(max) = ωA and a(max) = ω²A, and phase differences between oscillators sharing a frequency, all assuming undamped sinusoidal motion.
- Energy in simple harmonic motion
The quadratic potential well, a total energy ½mω²A², kinetic and potential terms exchanging at twice the motion's frequency, and speed obtained from energy rather than time; the constant total assumes a frictionless, exactly quadratic well.
- Mass–spring oscillators
ω = √(k/m) and T = 2π√(m/k) from Hooke's law, why a vertical spring oscillates about its shifted equilibrium with the same ω, series and parallel effective stiffness, and the massless-spring and Hookean-range assumptions behind the period.
- Simple and physical pendulums
Gravity as a restoring torque, d²θ/dt² = −(Mgd/I)sin θ, the small-angle step that makes it harmonic, T = 2π√(I/(Mgd)) and T = 2π√(L/g), the centre of oscillation, and the θ₀²/16 correction showing isochronism is only first order.
- Small oscillations in a potential well
Taylor-expanding U(x) about a stable equilibrium, reading ω = √(U″(x₀)/m) from the curvature, why a stable system is harmonic for small enough displacement provided U″(x₀) ≠ 0, and the cubic and quartic terms that make a real period amplitude-dependent.
- Damped oscillations
A velocity-proportional drag term added to the equation, the solution A₀e(−γt)cos(ω′t) with ω′ = √(ω₀² − γ²), the underdamped, critical, and overdamped regimes, energy decay and Q; linear drag is a model, and dry friction decays differently.
- Driven oscillations and resonance
Transient plus steady state under sinusoidal driving, response at the drive frequency, an amplitude peak near ω₀ pulled below it by damping, a phase lag near zero well below ω₀, exactly π/2 at ω₀ whatever the damping, and approaching π well above, with peak height and width set by Q and the linear single-frequency assumption the response curve rests on.
Learning outcomes
- Show that a linear restoring force produces d²x/dt² = −ω²x and fix the amplitude and phase constant from initial conditions.
- Relate displacement, velocity, acceleration, energy, and phase at any point in a simple harmonic cycle.
- Derive period models for mass–spring, simple-pendulum, and physical-pendulum oscillators and state the approximation each rests on.
- Obtain the angular frequency of small oscillations from the curvature of a potential-energy function at a stable equilibrium, and say when that curvature test fails.
- Classify damped responses and predict the steady-state amplitude and phase lag of a driven oscillator near resonance.
Prerequisite thread
- Newton's second law, free-body diagrams, and rotational analogues
- Work, potential energy, and mechanical-energy conservation
- Derivatives, integrals, and the vocabulary of second-order linear differential equations
- Taylor expansion and the small-angle approximation
Laboratory directions
hands-onWhere the spring oscillator model breaks Over what mass and amplitude range does T = 2π√(m/k) predict a real spring oscillator's period, and what does the residual pattern reveal?
Evidence: Period data across masses and amplitudes, a linearized T²-versus-m fit, an intercept converted to the spring's effective mass and compared with the predicted one third of the spring's own mass, residuals, and a stated range of validitycomputationalResonance curve of a driven damped oscillator How do the damping constant and the drive frequency reshape steady-state amplitude, phase lag, and the time the transient takes to die away?
Evidence: Numerically integrated trajectories, an amplitude-versus-frequency sweep at several damping values, a phase-versus-frequency sweep tested against the π/2 crossing at ω₀, fitted peak positions and widths, and comparison with the linear-response predictionProblem practice
Deriving the equation of motion and naming its approximation before quoting any period formula
Capstone: Take a real oscillator, derive its equation of motion from forces or torques, predict its period and its decay, then identify the one assumption whose failure the data would expose first.
- Concept checks that require an explanation, not only a numerical answer
- Multi-step quantitative modelling with unit and limiting-case checks
- A calculus or data-interpretation challenge
Interactive concept map
Follow the model from claim to evidence.
01 · Physical claimOscillations 02 · Representationx, v, and a versus time on shared axes Differential-equation solving
03 · TestPrediction before measurement Over what mass and amplitude range does T = 2π√(m/k) predict a real spring oscillator's period, and what does the residual pattern reveal?
04 · Evidence & boundaryDecide, then qualify Period data across masses and amplitudes, a linearized T²-versus-m fit, an intercept converted to the spring's effective mass and compared with the predicted one third of the spring's own mass, residuals, and a stated range of validity
Interactive diagram for Oscillations: follow the physical claim through its representation, proposed test, evidence, and model boundary. modelDifferential-equation solving turns the stated idea into a representation that can make a prediction.
Example questions
Try the reasoning before revealing the structure.
Concept Using Oscillations, explain how a physicist can: Show that a linear restoring force produces d²x/dt² = −ω²x and fix the amplitude and phase constant from initial conditions.
- Periodic motion and the simple-harmonic condition
1.5 weeks · UPIII-02 · Heavy
Mechanical Waves
Build mechanical waves from the linear wave equation: travelling wave functions, energy and intensity transport, superposition and boundary reflection, standing-wave normal modes, and sound as a paired displacement and pressure wave, with the small-amplitude linear assumption kept visible.
9 topics5 outcomes2 lab directions200 min practiceOpen Mechanical Waves: topics, evidence, and practice
Topics and mapped lessons
- Describing travelling waves
Transverse and longitudinal disturbances, amplitude, wavelength, period and frequency, v = f λ with the medium fixing v and the source fixing f, phase difference from separation along the direction of travel, and snapshot versus history graphs, for a non-dispersive medium that carries a pulse without changing its shape.
- The wave function, the wave equation, and wave speed
Wave number and angular frequency as the spatial and temporal counterparts of each other, v = w/k, and the sign of the wave-function argument fixing the direction of travel; the wave equation derived from tension and inertia, its f(x - vt) and g(x + vt) solutions, the string result v = √(FT/μ), and the elastic-over-inertial pattern in other media, valid only for small slopes in a uniform, stiffness-free medium.
- Wave energy, power, and intensity
Energy transport without net matter transport, average power on a string as one-half mu v w-squared A-squared, amplitude-squared and frequency-squared scaling, and intensity as power per unit wavefront area, spreading as inverse-square from a point source, as one-over-r from a line source, and not at all for a plane wave, assuming a sinusoidal wave in a linear, lossless medium.
- Superposition, reflection, and interference
Linearity of the wave equation, phase set by path difference, the resultant amplitude of two equal waves, and the junction coefficients r = (v2 - v1)/(v2 + v1) and t = 1 + r, giving inversion at a fixed end, assuming coherent sources and a medium that responds linearly to displacement.
- Standing waves and normal modes
Counter-propagating superposition, adjacent nodes half a wavelength apart with antinodes midway between them, boundary conditions quantising the spectrum to n v/2L or to odd multiples of v/4L, and zero net energy transport, for ideal rigid or free ends with negligible damping.
- Sound as displacement and pressure
Longitudinal displacement, the pressure change equal to minus the bulk modulus times the displacement gradient and so a quarter cycle out of phase with the displacement, and speed as √(B/ρ), with the adiabatic ideal-gas result √(γ R T/M) for molar mass M, a continuum small-amplitude model that fails at shock amplitudes.
- Sound intensity and the decibel scale
Intensity from pressure amplitude, inverse-square spreading from a point source, and levels defined logarithmically against a reference intensity fixed by definition at 1.00e-12 W per square metre, which must be converted back to intensities before sources are combined, assuming a free, non-reverberant field.
- Harmonics, air columns, and timbre
Boundary conditions selecting an integer harmonic series, open and stopped pipes, the end correction at an open end, and excitation setting how much of each mode is present and so the timbre, an idealisation that ignores stiffness-driven inharmonicity and radiation losses.
- Beats and the Doppler effect
Beat frequency from the difference of two near-equal tones, and separate moving-source and moving-observer Doppler forms, which differ because sound travels in a medium rather than depending on relative velocity alone, both restricted to source and observer speeds below the speed of sound.
Learning outcomes
- Write and interpret a travelling-wave function, relating wave number, angular frequency, phase, and direction of travel.
- Show that small-amplitude disturbances obey the linear wave equation and that its coefficient fixes the medium's wave speed.
- Calculate wave power and intensity, and relate amplitude, frequency, source geometry, and sound-intensity level.
- Apply superposition and boundary conditions to interference, reflection, standing-wave modes, and resonant spectra.
- Use beat and Doppler relations with their medium and speed assumptions stated, and explain why the moving-source and moving-observer forms differ.
Prerequisite thread
- Simple harmonic motion, phase, angular frequency, and resonance
- Newton's second law, tension, energy, power, and mass density
- Partial derivatives read graphically; no multivariable calculus assumed
- Logarithms, sinusoidal functions, and small-angle approximations
Laboratory directions
hands-onStanding-wave spectrum of a string Do the resonant frequencies of a fixed-fixed string follow the integer-mode model, and what wave speed do they imply?
Evidence: Resonant frequencies versus mode number, node positions, a fitted wave speed, comparison with the square-root tension prediction, residuals, and uncertaintyvideo analysisDoppler pass-by from a recording Can a single recorded pass-by recover both the emitted frequency and the source speed?
Evidence: A time-frequency spectrogram, approach and recede frequencies, a two-unknown solution, an independent speed estimate, sampling-resolution limits, and uncertaintyProblem practice
Deriving results from the wave function and boundary conditions before selecting a standing-wave or sound formula
Capstone: Design a mechanical-wave or acoustic measurement that determines one wave speed by two independent methods, then state which assumption - linearity, ideal boundaries, or a lossless medium - most limits the agreement.
- Concept checks that require an explanation, not only a numerical answer
- Multi-step quantitative modelling with unit and limiting-case checks
- A calculus or data-interpretation challenge
Interactive concept map
Follow the model from claim to evidence.
01 · Physical claimMechanical Waves 02 · RepresentationSnapshot and history graphs Wave-function interpretation
03 · TestPrediction before measurement Do the resonant frequencies of a fixed-fixed string follow the integer-mode model, and what wave speed do they imply?
04 · Evidence & boundaryDecide, then qualify Resonant frequencies versus mode number, node positions, a fitted wave speed, comparison with the square-root tension prediction, residuals, and uncertainty
Interactive diagram for Mechanical Waves: follow the physical claim through its representation, proposed test, evidence, and model boundary. modelWave-function interpretation turns the stated idea into a representation that can make a prediction.
Example questions
Try the reasoning before revealing the structure.
Concept Using Mechanical Waves, explain how a physicist can: Write and interpret a travelling-wave function, relating wave number, angular frequency, phase, and direction of travel.
- Describing travelling waves
1 week · UPIII-03 · Medium-heavy
Wave Phenomena
Apply linear superposition to coherent sources: phase from path difference, two-source amplitude and intensity patterns, beats as amplitude modulation in time, and Doppler shifts in a medium, with the linearity, coherence, and medium-frame assumptions that bound each result.
9 topics5 outcomes2 lab directions170 min practiceOpen Wave Phenomena: topics, evidence, and practice
Topics and mapped lessons
- The superposition principle and linear media
Linearity of the wave equation, displacement-by-displacement addition, overlapping waves emerging unchanged, and the small-amplitude restoring-force condition that fails for finite-amplitude sound and overdriven media.
- Phase difference, path difference, and coherence
Path difference converted to phase difference, source phase offsets, temporal and spatial coherence, coherence time and length, and why two independent sources at the same nominal frequency give no stable pattern.
- Resultant amplitude, intensity, and fringe visibility
Equal-amplitude superposition and the squared-cosine intensity pattern, phasor addition for unequal amplitudes, fringe visibility, the incoherent limit where intensities simply add, and energy redistributed rather than destroyed.
- Interference geometry: nodal lines and accessible orders
Nodal and antinodal hyperbolae around two point sources, the far-field angular maxima condition, the limit that source separation places on the number of orders, and the near-field breakdown of the parallel-ray approximation.
- Beats as amplitude modulation in time
Two nearby frequencies summed into a carrier at the mean frequency inside a slow envelope, the beat rate equal to the full frequency difference because the envelope peaks twice per envelope cycle, the linear sum still containing only the two original frequencies so that an intensity-sensitive detector is what makes the beat audible and heterodyne mixing needs a nonlinear element rather than superposition, tuning by nulling the beat, and the loss of a defined beat once the separation is no longer small.
- Doppler effect for a source and observer in a medium
Wavefront crowding derived for a moving source and separately for a moving observer, why the two expressions differ although they agree to first order in speed over wave speed, sign conventions for approach and recession, the combined form, and the medium rest frame the whole model requires.
- Doppler geometry, reflected shifts, and model limits
Only the line-of-sight velocity component shifts the frequency, wind and moving media changing the effective wave speed, the double shift when a moving reflector both receives and re-emits in sonar and blood-flow ultrasound, and the failure of the medium-frame formula for light, so that radar's 2v/c shift is the low-speed limit of the relativistic result rather than an instance of this derivation.
- Supersonic sources and the Mach cone
The subsonic-to-supersonic transition, wavefront envelopes forming a cone, the Mach number fixing its half-angle, sonic-boom timing and ground geometry, and a construction that fixes the cone angle but says nothing about shock strength because it assumes linear acoustics.
- Wave-phenomena data studio
Recovering wavelength, phase, beat rate, and source speed from measured or simulated signals; spectrogram window and sampling limits; competing explanations for one observed frequency structure; and honest uncertainty on each inferred parameter.
Learning outcomes
- Justify superposition from the linearity of the wave equation and state where the linear model fails.
- Convert path difference and source phase offsets into a phase difference, then predict amplitude and intensity.
- Derive the beat frequency by summing two nearby-frequency waves and interpret the modulation envelope.
- Apply moving-source and moving-observer Doppler relations with consistent signs and low-speed limiting checks.
- Distinguish coherent interference from incoherent intensity addition using measurable evidence.
Prerequisite thread
- Sinusoidal wave functions, wave number, angular frequency, and phase
- Wave speed, the wavelength-frequency relation, and wave intensity in a medium
- Trigonometric sum-to-product identities and small-angle approximations
- Simple harmonic motion and the linear one-dimensional wave equation
Laboratory directions
hands-onTwo-speaker interference map Do the loud and quiet positions in front of two driven speakers follow the path-difference model at more than one frequency?
Evidence: Position and sound-level data, measured path differences, predicted maxima and minima, residuals, uncertainty, and a stated room-reflection limitationcomputationalBeats and Doppler from one recording Can a single recording containing two sources, one of them moving, separate a beat rate from a Doppler shift?
Evidence: Time-frequency tracks, an extracted beat rate, approach and recession frequencies, an inferred source speed, uncertainty, and a window-length and sampling limitationProblem practice
Converting geometry or relative motion into a phase or frequency shift before selecting an interference, beat, or Doppler equation
Capstone: Given one recorded acoustic signal, decide whether its frequency structure comes from two-source interference, beats, or relative motion, quantify the responsible parameter, and name the measurement that rules out the other two explanations.
- Concept checks that require an explanation, not only a numerical answer
- Multi-step quantitative modelling with unit and limiting-case checks
- A calculus or data-interpretation challenge
Interactive concept map
Follow the model from claim to evidence.
01 · Physical claimWave Phenomena 02 · RepresentationPhasor and phase-difference diagrams Phase-difference accounting
03 · TestPrediction before measurement Do the loud and quiet positions in front of two driven speakers follow the path-difference model at more than one frequency?
04 · Evidence & boundaryDecide, then qualify Position and sound-level data, measured path differences, predicted maxima and minima, residuals, uncertainty, and a stated room-reflection limitation
Interactive diagram for Wave Phenomena: follow the physical claim through its representation, proposed test, evidence, and model boundary. modelPhase-difference accounting turns the stated idea into a representation that can make a prediction.
Example questions
Try the reasoning before revealing the structure.
Concept Using Wave Phenomena, explain how a physicist can: Justify superposition from the linearity of the wave equation and state where the linear model fails.
- The superposition principle and linear media
Labs and problem solving
Make every wave and every engine answer to measurement.
The 22 laboratory directions and unit-specific problem plans use one repeatable loop from prediction to defensible evidence. Simulation or video analysis supplements—not replaces—hands-on work required by a learner's institution.Predict
Sketch the oscillation, wavefront, ray path, or energy flow and state what should happen before any algebra.
Derive
Connect definitions symbolically, keeping every phase, sign, and integration constant tied to physical meaning.
Measure
Design an interference, calorimetry, spectral, or decay measurement with calibration and uncertainty visible.
Test
Compare model with data, then check limiting cases: does the quantum result reduce to the classical one where it must?
Communicate
State the model used, the regime it holds in, the evidence, and what the result does not establish.
Course-level outcomes
What successful study should make possible.
These are learning capabilities, not promises of a grade, academic credit, transfer approval, professional status, or course completion.- Model oscillating systems with differential equations and read amplitude, phase, and energy from their solutions.
- Apply superposition to mechanical and optical waves, predicting interference, beats, standing waves, and diffraction.
- Choose the ray or wave model of light from the ratio of aperture to wavelength, and justify the choice.
- Connect a molecular model of matter to temperature, pressure, and internal energy.
- Apply the first and second laws to cycles and engines, and use entropy to argue which processes can happen.
- Use Lorentz transformations and relativistic energy-momentum, checking that low-speed limits recover Newtonian results.
- Use quantised energy, matter waves, and the uncertainty principle to explain evidence classical models cannot.
- Interpret atomic spectra and nuclear decay quantitatively, stating what each model does and does not establish.
Choose the right starting point
Already met oscillations? Start at unit 02 anyway.
Unit 01 is short and is the foundation the rest of the course stands on: every wave, every optical fringe, and every quantum stationary state in this map is read back to simple harmonic motion. Skimming it costs less than rebuilding it in unit 09.