oscillations & waves
SHM incl. damping, driving & resonance, travelling waves, refraction & polarization, interference & diffraction, standing waves, Doppler.
simple harmonic motion · damped oscillations · forced oscillations & resonance · travelling waves · light & refraction · interference · diffraction · standing waves · Doppler
Simple harmonic motion
Condition. \(a=-\omega^2x\), i.e. restoring force \(F=-m\omega^2x\).
Solutions. \(x=x_0\cos(\omega t+\phi)\), \(v=-\omega x_0\sin(\omega t+\phi)\), \(a=-\omega^2x\); \(v^2=\omega^2(x_0^2-x^2)\), \(v_{\max}=\omega x_0\), \(a_{\max}=\omega^2x_0\). Period \(T=2\pi/\omega\), \(f=1/T\). From initial conditions: \(x_0=\sqrt{x(0)^2+(v(0)/\omega)^2}\), \(\tan\phi=-v(0)/(\omega x(0))\) — amplitude and phase are the two free constants; \(\omega\) belongs to the system.
Energy. Total \(E=\tfrac12m\omega^2x_0^2\); \(K=\tfrac12m\omega^2(x_0^2-x^2)\), \(U=\tfrac12m\omega^2x^2\). Over a cycle \(\avg{\sin^2}=\avg{\cos^2}=\tfrac12\), so \(\avg{K}=\avg{U}=E/2\).
Oscillators. Mass–spring: \(\omega=\sqrt{k/m}\), \(T=2\pi\sqrt{m/k}\). Simple pendulum (small angle): \(\omega=\sqrt{g/L}\), \(T=2\pi\sqrt{L/g}\); finite amplitude \(\phi_0\): \(T=2\pi\sqrt{L/g}\,\bigl(1+\phi_0^2/16+\cdots\bigr)\). Physical pendulum: \(T=2\pi\sqrt{I/(mgd)}\).
Complex-exponential method. Solve the real equation via \(z=z_0e^{i\omega t}\) and take \(x=\operatorname{Re}z\): differentiation becomes multiplication by \(i\omega\), turning linear oscillator equations into algebra; amplitude and phase ride in the one complex constant \(z_0\).
Damped oscillations
Equation. \(m\ddot x+b\dot x+kx=0\), standard form \(\ddot x+\gamma\dot x+\omega_0^2x=0\) with \(\gamma=b/m\), \(\omega_0^2=k/m\). Trying \(z=z_0e^{\alpha t}\) gives \(\alpha=-\gamma/2\pm\sqrt{(\gamma/2)^2-\omega_0^2}\).
Light damping (\(\gamma\lt2\omega_0\)). \[x=x_0e^{-\gamma t/2}\cos(\omega_1t+\phi),\qquad \omega_1=\omega_0\sqrt{1-(\gamma/2\omega_0)^2}:\] oscillation at nearly \(\omega_0\) inside an exponential envelope. Energy decays as \(E=E_0e^{-\gamma t}\), time constant \(\tau=1/\gamma\).
Critical & heavy damping. \(\gamma=2\omega_0\): fastest non-oscillatory return (metering instruments); \(\gamma\gt2\omega_0\): sum of two decaying exponentials, sluggish.
Quality factor. \(Q=\dfrac{\text{energy stored}}{\text{energy dissipated per radian}}=\dfrac{\omega_0}{\gamma}\) for light damping — the number of radians for the energy to fall by \(e\).
Forced oscillations & resonance
Undamped, driven. \(\ddot x+\omega_0^2x=(F_0/m)\cos\omega t\) has steady response \(x=A\cos\omega t\), \(A=\dfrac{F_0/m}{\omega_0^2-\omega^2}\): in phase below \(\omega_0\), antiphase above, divergent at resonance.
Forced damped oscillator. \(\ddot x+\gamma\dot x+\omega_0^2x=(F_0/m)\cos\omega t\); steady state \(x=A\cos(\omega t-\phi)\) with \[A=\frac{F_0/m}{\sqrt{(\omega_0^2-\omega^2)^2+(\gamma\omega)^2}},\qquad \tan\phi=\frac{\gamma\omega}{\omega_0^2-\omega^2}.\] The displacement lags the drive by \(\phi\): \(0\to\pi/2\) (at \(\omega_0\)) \(\to\pi\), the swing sharpening as \(\gamma\to0\). The full solution adds the decaying free oscillation — the transient that dies in time \(\sim\tau\).
Resonance. \(A(\omega_0)=\dfrac{F_0}{m\omega_0\gamma}=Q\times\dfrac{F_0}{k}\): resonant response is \(Q\) times the static deflection. The stored-energy resonance curve is a Lorentzian of full width at half maximum \(\Delta\omega=\gamma\), so \[Q=\frac{\omega_0}{\Delta\omega}\] — the two definitions of \(Q\) (energy ratio, frequency selectivity) agree.
Time vs frequency response. \(\tau\,\Delta\omega=1\): a long ring-down time and a narrow resonance are the same property measured in the two domains — high-\(Q\) systems ring long and tune sharply.
Travelling waves
Waveform. \(y=A\sin(kx\mp\omega t+\phi)\), \(k=2\pi/\lambda\), \(\omega=2\pi f\); speed \(v=f\lambda=\omega/k\).
Phase difference. \(\Delta\phi=2\pi\Delta x/\lambda=2\pi\Delta t/T\).
Types. Transverse: displacement perpendicular to propagation. Longitudinal: displacement parallel to propagation.
Intensity. \(I=P/A\propto A^2\); spherical spreading \(I=P/(4\pi r^2)\); decibel level \(\beta=10\log_{10}(I/I_0)\), \(I_0=10^{-12}\,\mathrm{W\,m^{-2}}\).
Light, refraction & polarization
Refractive index. \(n=c/v\); \(\lambda_{\mathrm{medium}}=\lambda_0/n\), frequency unchanged across a boundary.
Snell's law. \(n_1\sin\theta_1=n_2\sin\theta_2\).
Critical angle. For \(n_1\gt n_2\): \(\sin\theta_c=n_2/n_1\).
Brewster's angle. \(\tan\theta_B=n_2/n_1\).
Polarization (Malus). \(I=I_0\cos^2\theta\).
EM waves. \(c=f\lambda\), \(E=hf=hc/\lambda\); spectrum in increasing \(f\): radio, micro, IR, visible, UV, X, gamma.
Superposition & interference
Superposition. Resultant displacement \(y=y_1+y_2+\cdots\).
Coherence. Constant phase difference, same \(f\).
Path difference. Path difference \(\Delta r\) gives phase \(\Delta\phi=2\pi\Delta r/\lambda\). Constructive: \(\Delta r=m\lambda\); destructive: \(\Delta r=(m+\tfrac12)\lambda\).
Two-source intensity. Equal amplitudes: \(I=4I_0\cos^2(\Delta\phi/2)\).
Slits, gratings & diffraction
Young double slit. Fringe spacing \(s=\lambda D/d\) for \(D\gg d\); maxima at \(d\sin\theta=m\lambda\).
Diffraction grating. \(d\sin\theta=m\lambda\), \(d=1/(\text{lines per metre})\); maximum order \(m_{\max}\le d/\lambda\).
Single slit. Minima at \(a\sin\theta=m\lambda\) (\(m=1,2,\dots\)); central maximum width on screen \(\approx 2\lambda D/a\).
Rayleigh criterion. Circular aperture: \(\theta\approx1.22\lambda/D\); resolving power improves with larger aperture and shorter \(\lambda\).
Thin films. Normal incidence: extra phase \(\pi\) on reflection from higher \(n\); optical path \(2nt\); track the phase reversals.
Standing waves & resonance
Standing wave. \(y=2A\sin(kx)\cos(\omega t)\) from equal opposite travelling waves. Nodes: zero amplitude; antinodes: maximum; node–antinode separation \(\lambda/4\), adjacent nodes \(\lambda/2\).
Harmonics. String fixed at both ends, or pipe open at both ends: \(L=n\lambda_n/2\), \(f_n=nv/(2L)\), \(n=1,2,3,\dots\) Pipe closed at one end: \(L=n\lambda_n/4\), \(f_n=nv/(4L)\), \(n=1,3,5,\dots\)
Beats. \(f_{\mathrm{beat}}=|f_1-f_2|\).
Resonance. Driving at \(f\approx f_0\): large amplitude; phase changes through resonance.
Doppler effect
Sound. Medium wave speed \(v\): \(f'=f\dfrac{v\pm v_o}{v\mp v_s}\) — choose \(+\) in the numerator when the observer moves toward the source, \(-\) in the denominator when the source moves toward the observer.
Moving source wavelength. Ahead \(\lambda=(v-v_s)/f\); behind \(\lambda=(v+v_s)/f\).
Light, low speed. \(z=\dfrac{\Delta\lambda}{\lambda_0}=\dfrac{\lambda_{\mathrm{obs}}-\lambda_0}{\lambda_0}\approx v_r/c\) for recession.
Relativistic longitudinal. Receding: \(\dfrac{f_{\mathrm{obs}}}{f_{\mathrm{em}}}=\sqrt{\dfrac{1-\beta}{1+\beta}}\), \(\lambda_{\mathrm{obs}}/\lambda_{\mathrm{em}}=\sqrt{(1+\beta)/(1-\beta)}\).
Cosmology. \(v=H_0d\) for small \(z\), \(z\approx v/c\).