# Oscillations and Waves

## Alexei Gilchrist

The classical theory for oscillations and waves at a second year university level.

❦

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.

- 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.