Deep Physics: Waves and Oscillations for Talented High Schoolers
Real waves and oscillations, taught from the simple harmonic oscillator up through standing waves, interference, and resonance. For talented high schoolers.
Online for homeschool families anywhere · or in-person in Princeton, NJ
A World Through the Lens of Waves and Oscillations
Anything that can vibrate, will. Anything that can carry a wave, does. The everyday world is full of motion that swings, ripples, and rings.
Push a child on a swing once a second. If your timing matches the swing’s natural rhythm, it climbs higher every push. Mismatch the timing and the swing dies. The energy you supply is the same. Only the timing changed.
Pluck a guitar string. It does not produce one frequency. It produces an entire harmonic series, a fundamental and its overtones, all from one motion. The shape of the string decides which frequencies survive.
Drop two stones into still water. The two ripples pass straight through each other. They emerge unchanged on the other side, as if neither had ever met the other.
Stand by a doorway. You hear someone speaking in the next room before you can see them. Sound bends around the door’s edge. Light, with a much shorter wavelength, barely bends at all.
All four phenomena share a single mathematical structure. A restoring force gives every oscillator the same equation of motion. A wave equation governs every linear wave: mechanical, acoustic, electromagnetic, quantum. Superposition makes them add. With these tools, you have the language in which every wave in physics is written.
You will:
- Derive the period of a simple harmonic oscillator from one principle, then show why every system near stable equilibrium is one.
- Predict the speed of a wave from the physical properties of the medium it travels through.
- Decompose any periodic motion into its Fourier (normal-mode) components.
- Build a wave packet from a sum of plane waves close in frequency, and find the speed at which the packet itself moves.
What You Will Actually Understand
1. Simple Harmonic Motion
The cleanest oscillator in physics, and the model for everything else. The mass on a spring. The simple pendulum at small angles. Restoring forces, period, frequency, amplitude, and phase. Energy traded between kinetic and potential, conserved across the cycle. The universality of SHM: every system near a stable equilibrium behaves as one.
2. Damped, Driven, and Nonlinear Oscillations
What friction does, what a periodic push can undo, and what happens past the linear regime. Damped oscillations, resonance, and the quality factor. The step past linearity into anharmonic, parametric, and self-oscillations: where the clean sine wave breaks down and what replaces it. The bridge from textbook oscillators to the ones that turn up in the real world.
3. Mechanical Waves and the Wave Equation
How a disturbance travels without the medium going anywhere. Transverse and longitudinal waves. The wave equation as the shape that every traveling disturbance obeys. Wave speed predicted from the properties of the medium. Wave energy and impedance at boundaries: what passes through, what reflects.
4. Superposition, Interference, and Fourier Analysis
How waves combine, and how any periodic motion decomposes. Linearity and the superposition principle. Constructive and destructive interference, beats, the double-slit experiment. Fourier decomposition: any periodic motion as a sum of normal modes.
5. Standing Waves and Normal Modes
Why a string of fixed length sings in only certain notes. Standing waves built from counter-propagating traveling waves. Boundary conditions, the discrete spectrum of allowed modes, fundamental and overtones. The first appearance of an eigenvalue problem: a linear operator (the wave equation with boundary conditions) admits only a discrete set of solutions.
6. Refraction, Diffraction, Doppler, and Dispersion
What waves do at obstacles, when the source moves, and when the medium is choosy. Wavefronts and Huygens’ principle. Refraction and Snell’s law from a change in wave speed alone. Diffraction at apertures. The Doppler effect. Dispersion: when different frequencies travel at different speeds, and how a steady wave turns into a moving packet. Polarization of transverse waves.
The specific topics, and the depth given to each, may shift depending on class priorities and the dynamics of the cohort.
Schedule, Pricing & Enrollment
Formats: Fall, Spring, and Summer semesters.
Schedule, format, tuition, refund policy, and certificates apply to every Lyceum course. They live on the Physics Lyceum: High School overview.
To enroll, schedule a call. We confirm fit, prerequisites, and the right semester.
Part of the SoTS Physics Lyceum
Waves and Oscillations is one of six classical core courses in the SoTS Physics Lyceum: a multi-year curriculum in Princeton, NJ. Students earning the Mastery in Classical and Modern Physics complete the six classical core courses plus any two of the four modern electives.
Classical core: Mechanics of motion. Mechanics of bodies and fluids. Waves and oscillations. Thermodynamics. Electricity and magnetism. Geometric optics.
Modern electives: Special Relativity. Quantum mechanics. Nuclear and particle physics. Astronomy and cosmology.
The Lyceum is built on the Deep Physics methodology.