Oscillations and Waves

Oscillations and Waves

Oscillations describe repetitive motion back and forth around an equilibrium position. Waves act as a mechanism to transport energy through a medium without transporting matter.

Simple Harmonic Motion (SHM)

SHM occurs when the restoring force is directly proportional to the displacement and acts in the opposite direction. $F = -kx$ This leads to the differential equation: $\frac{d^2x}{dt^2} + \frac{k}{m}x = 0$

The solution is sinusoidal: $x(t) = A \cos(\omega t + \phi)$ where:

• $A$: Amplitude (maximum displacement)
• $\omega$: Angular frequency ($\omega = \sqrt{k/m}$)
• $\phi$: Phase constant

Period ($T$) and Frequency ($f$): $T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{m}{k}}$ $f = \frac{1}{T} = \frac{\omega}{2\pi}$

Pendulums

Simple Pendulum

A mass $m$ suspended by a string of length $L$. For small angles: $T = 2\pi \sqrt{\frac{L}{g}}$ Notice the period is independent of mass.

Physical Pendulum

A rigid body oscillating about a pivot. $T = 2\pi \sqrt{\frac{I}{mgd}}$ where $I$ is the moment of inertia and $d$ is the distance from the pivot to the center of mass.

Engineering Context: Understanding natural periods of oscillation is critical in structural dynamics, especially for designing buildings to withstand earthquakes. Resonance occurs when the ground motion frequency matches the building's natural frequency, potentially causing catastrophic failure.

Mechanical Waves and Sound

A mechanical wave is a disturbance that travels through a material medium.

Wave Equation: $y(x,t) = A \sin(kx - \omega t)$ where $k = 2\pi/\lambda$ is the wave number and $\omega = 2\pi f$.

Wave Speed ($v$): $v = \lambda f = \frac{\omega}{k}$

Sound Waves: Longitudinal pressure waves. The speed of sound depends on the medium's properties (bulk modulus $B$ and density $\rho$). $v = \sqrt{\frac{B}{\rho}}$