# Oscillations and Waves

## Alexei Gilchrist

These are notes that have or are being used at Macquarie University in one of the core 2nd year physics units. The fist half examines the oscillatory behaviour that is ubiquitous in Nature, and in particular the harmonic oscillator as one of the cornerstone models of physics. The second half introduces quantum wave mechanics, in the playground of Schrodinger’s equation in one-dimension, and mostly piecewise-constant potentials.

*Status:* Many of the notes on classical oscillations are mostly typeset and listed below. The wavefunction notes are being created and currently only available as *mindmaps* (see end of page).

### 1 First-half on *Classical oscillations and waves*

- The motion for a harmonic oscillator is derived using Newton’s second law. Different parametrizations of the solution, the velocity, acceleration and energy are also determined.
- The equations of motion and solutions are derived for the simple pendulum and a general pendulum. Dynamical maps are introduced as a way of handling nonlinear oscillators.
- The Lotka-Voltera equations are a very simple model of predator-prey dynamics. They lead to oscillatory behaviour in both populations.
- The solutions to the harmonic oscillator with a velocity dependant drag force are derived. The relaxation time for the oscillator is defined.
- Energy of the damped harmonic oscillator is described. The Q of an oscillator.
- The map of phase space of a dynamical system is a really convenient way of summarising the behaviour of a system. In particular it doesn’t require a solution to the equations of motion.
- The solution to a damped oscillator with a periodic driving force is derived.
- When an oscillator is driven at just the right frequency it hits resonance and absorbs energy from the driving. The details of resonance for a driven and damped harmonic oscillator are explored.
- Coupling together two or more oscillators introduces a whole new level of complexity. It turns out that there is a particular way of looking at the system that makes it simple to solve the motion.
- The loaded string is a classic problem of coupled oscillators where \(N\) small masses are threaded onto a light string. It makes a nice transition for considering the continuum case and waves.
- The continuum limit of the loaded string is derived, arriving at the one dimensional wave equation.

### 2 Second-half on *Quantum Wavefunctions*

These notes are available as mindmap SVG images, with mostly hand-crafted math. They are in the process of being created so some links make not yet work, and all maps may change as I improve the material, fix typos etc. Some of the maps can get large so please be patient while they are loading. They are SVG files with some javascript to help navigate around the map.

**Mindmaps:** /map/quantum-wave-mechanics/.

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