# 3: Linear Oscillators

- Page ID
- 9578

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- 3.1: Introduction to Linear Oscillators
- Oscillations are a ubiquitous feature in nature.

- 3.2: Linear Restoring Forces
- An oscillatory system implies that there be a stable equilibrium about which the oscillations occur.

- 3.3: Linearity and Superposition
- An important aspect of linear systems is that the solutions obey the Principle of Superposition, that is, for the superposition of different oscillatory modes, the amplitudes add linearly.

- 3.4: Geometrical Representations of Dynamical Motion
- The powerful pattern-recognition capabilities of the human brain, coupled with geometrical representations of the motion of dynamical systems, provide a sensitive probe of periodic motion. The geometry of the motion often can provide more insight into the dynamics than inspection of mathematical functions.

- 3.5: Linearly-damped Free Linear Oscillator
- This is a ubiquitous feature in nature.

- 3.6: Sinusoidally-driven, linearly-damped, linear oscillator
- This occurs frequently in nature.

- 3.7: Wave equation
- Wave motion is a ubiquitous feature in nature. Mechanical wave motion is manifest by transverse waves on fluid surfaces, longitudinal and transverse seismic waves travelling through the Earth, and vibrations of mechanical structures such as suspended cables.

- 3.8: Travelling and standing wave solutions of the wave equation
- The wave equation can have both travelling and standing-wave solutions.

- 3.9: Waveform Analysis
- A Fourier decomposition into a sum of harmonic terms can be used to analyze signals.

- 3.10: Signal Processing
- Use of Fourier decomposition to analyze a system response.

- 3.11: Wave Propagation
- Phase, group, and signal velocities in wave packets.

*Thumbnail: A girl on a garden swing oscillating back and forth in a damped motion. (CC SA 2.0; Luiz Carlos).*