Waves & Oscilations Summary

$F = -kx$ : Restoring force proportional to displacement.
$m\ddot{x} = -kx$ : Equation of motion.
$\ddot{x} + \frac{k}{m}x = 0$ : Simplified equation.
$\omega^2 = \frac{k}{m}$ : Defines angular frequency.
$\ddot{x} + \omega^2x = 0$ : Final form of SHM equation.
General solution: $x(t) = Acos(\omega t + \phi)$ , where $A$ is amplitude and $\phi$ is phase.
Angular frequency: $\omega = 2\pi f = \frac{2\pi}{T}$ , where $f$ is frequency and $T$ is period.
Phase: $\phi(t) = \omega t + \phi_0$ .

Friction force: $F_{fric} \sim -\alpha \dot{x}$ .
Equation of motion: $m\ddot{x} = -kx - \alpha \dot{x}$ .
$\ddot{x} + 2\beta \dot{x} + \omega0^2x = 0$ : Standard form with damping coefficient $\beta = \frac{\alpha}{2m}$ and natural frequency $\omega0^2 = \frac{k}{m}$ .
Solutions depend on the relationship between $\beta$ and $\omega_0$ .
- Underdamped (\beta < \omega_0): Oscillations with exponentially decaying amplitude.
- Critically damped ( $\beta = \omega_0$ ): Quickest return to equilibrium without oscillation.
- Overdamped (\beta > \omega_0): Slow return to equilibrium without oscillation.

External driving force: $F(t) = F_0cos(\omega t)$ .
Equation of motion: $m\ddot{x} + \alpha \dot{x} + kx = F_0cos(\omega t)$ .
$\ddot{x} + 2\beta \dot{x} + \omega_0^2x = f(t)$ , where $f(t) = \frac{F(t)}{m}$ .
General solution: $x(t) = xh(t) + xp(t)$ , where $xh(t)$ is the homogeneous solution and $xp(t)$ is the particular solution.

Wave function: $y(x, t) = \hat{y} sin(kx + \omega t + \phi_0)$ , where $k$ is wave number.
Wave number: $k = \frac{2\pi}{\lambda}$ , related to wavelength $\lambda$ .
Wave equation: $\frac{\partial^2y(x, t)}{\partial t^2} - c^2\frac{\partial^2y(x, t)}{\partial x^2} = 0$ , where $c$ is wave speed.
Wave speed: $c = \frac{\omega}{k} = \lambda \nu$ , where $\nu$ is frequency.
Solutions to the wave equation: $f(x, t) = f(kx + \omega t)$ or $g(x, t) = g(kx - \omega t)$ .

If $y1$ and $y2$ are solutions, then $y = y1 + y2$ is also a solution (linearity).
For two waves: $u1(x, t) = A1 sin(\omega t - kx + \phi1)$ , $u2(x, t) = A2 sin(\omega t - kx + \phi2)$ .
Resultant wave: $u(x, t) = A sin(\omega t - kx + \varphi)$ , where $A = \sqrt{A1^2 + A2^2 + 2A1A2 cos(\phi2 - \phi1)}$ and $tan \varphi = \frac{A1 sin \phi1 + A2 sin \phi2}{A1 cos \phi1 + A2 cos \phi2}$ .

Change in observed frequency due to relative motion between source and observer.
$\Delta f = \pm \frac{\Delta v}{c} f_0$ , where $\Delta v$ is the relative velocity.
$f = f_0 (1 \pm \frac{v}{c})$ .

Macroscopic variables: temperature (T), volume (V), pressure (p), number of particles (N).
Internal energy (U) is a state function.
First Law of Thermodynamics: $\Delta U = \Delta Q + \Delta W$ , where $\Delta Q$ is heat added and $\Delta W$ is work done on the system.
Heat capacity: $\Delta Q = c m \Delta \vartheta$ .

Relates macroscopic properties to microscopic behavior.
Average translational kinetic energy: <E{kin}> = \frac{3}{2} kB T.
Internal energy: $U = N <E_{kin}> = \frac{3}{2}nRT$ .
Maxwell-Boltzmann distribution: $n(v) = n0 v^2 exp(-\frac{v^2}{v0^2})$ , where $v0^2 = \frac{2kBT}{m}$ .
Root-mean-square speed: $v_{rms} = \sqrt{<v^2>}$ .

Transverse waves on a string: $c = \sqrt{\frac{F}{\mu}}$ , where $F$ is tension and $\mu$ is mass per unit length.
Longitudinal waves in a solid: $c = \sqrt{\frac{E}{\rho}}$ , where $E$ is Young's modulus and $\rho$ is density.
Speed of sound in a gas: $\sqrt{ \frac{\gamma P}{\rho}}$ , where $\gamma$ is adiabatic index and $P$ is pressure.