The classical theory for oscillations and waves at a second year university level.
Level: 2, Subjects: Classical

The Lotka-Voltera equations. They are a simple simulation of the interaction of predators with prey and lead to oscillatory population dynamics.

Oscillatory behaviour is ubiquitous in Nature, and in particular the harmonic oscillator is one of the cornerstone models of physics. This material is used at Macquarie University in the unit phys201 Physics IIA where it forms the first half.

Status: In development. Missing many solved exercises and a few more articles.

Curriculum:

2 Oscillators  

1 Harmonic Oscillator

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.

2 Oscillators  

2 The Pendulum

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.

2 Oscillators  

3 Lotka-Voltera Equations

The Lotka-Voltera equations are a very simple model of predator-prey dynamics. They lead to oscillatory behaviour in both populations.

2 Oscillators  

4 Damped Harmonic Oscillator

The solutions to the harmonic oscillator with a velocity dependant drag force are derived. The relaxation time for the oscillator is defined.

2 Oscillators  

5 Energy in a Damped Harmonic Oscillator

Energy of the damped harmonic oscillator is described. The Q of an oscillator.

2 Oscillators  

6 Dynamical system maps

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.

2 Oscillators  

7 Driven and Damped Oscillator

The solution to a damped oscillator with a periodic driving force is derived.

2 Oscillators  

8 Resonance

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.

2 Coupled Oscillators  

9 Coupled Oscillators

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.

2 Coupled Oscillators  

10 The Loaded String

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.

2 Coupled Oscillators Waves  

11 Continuum Limit of the Loaded String

The continuum limit of the loaded string is derived, arriving at the one dimensional wave equation.