Oscillations and Wave Characteristics

An oscillation is SHM if the restoring force (and acceleration) is proportional to displacement and directed towards the equilibrium position:

$a = -\omega^2 x$

$x = A\cos(\omega t)$; $v = -A\omega\sin(\omega t)$; $a = -A\omega^2\cos(\omega t)$

Where $\omega = 2\pi f = 2\pi/T$.

$v_{\max} = A\omega$ (at $x = 0$); $a_{\max} = A\omega^2$ (at $x = \pm A$).

Energy in SHM

$E_T = \frac{1}{2}m\omega^2 A^2$ (constant for undamped SHM) At $x = 0$: all KE. At $x = \pm A$: all PE.

Transverse: Oscillation perpendicular to direction of propagation (EM waves, water waves, string waves). Can be polarised.

Longitudinal: Oscillation parallel to propagation (sound, P-waves). Compressions and rarefactions. Cannot be polarised.

$y = A\sin(\omega t - kx)$

Where $k = 2\pi/\lambda$ (wave number), $\omega = 2\pi f$.

$I = \frac{P}{A} \propto A^2$

For a point source: $I \propto 1/r^2$ (inverse square law).

• SHM: $a = -\omega^2 x$; $v_{\max} = A\omega$; $a_{\max} = A\omega^2$
• $v = f\lambda$; $T = 1/f$; $\omega = 2\pi f$
• Transverse (polarisable) vs longitudinal (compressions/rarefactions)
• $I \propto A^2 \propto 1/r^2$ for point source