Postgraduate Science

Guest · Admin login

Oscillations & Waves

Simple harmonic motion, damping, and the wave equation.

A linear restoring force $F = -kx$ gives the equation of motion $\ddot x + \omega^2 x = 0$ with $\omega = \sqrt{k/m}$. The general solution is

$$x(t) = A\cos(\omega t + \varphi).$$

Adding linear damping $-\gamma \dot x$ gives three regimes:

  • Underdamped ($\gamma < 2\omega$): oscillates with exponentially decaying amplitude.
  • Critically damped ($\gamma = 2\omega$): returns to equilibrium fastest, with no overshoot.
  • Overdamped ($\gamma > 2\omega$): slow exponential return, no oscillation.
For continuous media, the wave equation

$$\frac{\partial^2 \psi}{\partial t^2} = c^2 \frac{\partial^2 \psi}{\partial x^2}$$

admits travelling solutions $\psi(x,t) = f(x - ct) + g(x + ct)$. Superposition of solutions is exact in linear theory — the principle underlying interference, beats, and Fourier decomposition.

Interactive: mass on a spring

Quiz

1. The angular frequency of a mass $m$ on a spring with constant $k$ is:
2. In a critically damped oscillator, the system:
3. General solution of the 1D wave equation is:
4. Small-angle period of a simple pendulum of length $L$:
5. For a travelling wave $y = A\sin(kx - \omega t)$, the phase velocity is: