PHYS 123 Waves
Contents
Periodic Motion
| Quantity | Unit | Definition |
|---|---|---|
| Period | $\mathrm{s}$ | $T = \dfrac{1}{f} = \dfrac{2\pi}{\omega}$ |
| Frequency | $\mathrm{Hz}$ | $f = \dfrac{1}{T} = \dfrac{\omega}{2\pi}$ |
| Angular frequency | $\mathrm{s^{-1}}$ | $\omega = 2\pi f = \dfrac{2\pi}{T}$ |
Simple harmonic motion (SHM)
| Description | Equations |
|---|---|
| Angular frequency in SHM | $\omega = \sqrt{\dfrac{k}{m}}$ |
| Spring constant | $k = m\omega^2$ |
| Displacement in SHM | $x(t) = A\sin(\omega t + \phi)$ |
| Velocity in SHM | $v(t) = \omega A\cos(\omega t + \phi)$ |
| Acceleration in SHM | $a(t) = -\omega^2 A\sin(\omega t + \phi) = -\omega^2 x(t)$ |
| Restoring force in SHM | $F = -k\Delta x$ |
| Simple harmonic oscillator equation | $\frac{d^2 x}{dt^2} = -\omega^2 x = -\frac{k}{m}x$ |
| Conservation of energy in SHM | $E = \frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \frac{1}{2}kA^2$ |
| Amplitude | $A = \sqrt{x_0^2 + \dfrac{v_0^2}{\omega^2}}$ |
| Phase angle | $\phi = \arctan\left(\dfrac{\omega x_0}{v_0}\right)$ |
Applications of SHM
| Description | Equations |
|---|---|
| Restoring torque in angular SHM | $\tau = -\kappa\Delta\theta$ |
| Rotational displacement in angular SHM | $\theta(t) = \theta_{\mathrm{max}}\sin(\omega t + \phi)$ |
| Angular frequency in angular SHM | $\omega = \sqrt{\dfrac{\kappa}{I}}$ |
| Angular frequency in simple pendulum | $\omega = \sqrt{\dfrac{g}{L}}$ |
| Angular frequency in physical pendulum | $\omega = \sqrt{\dfrac{mgL}{I}}$ |
Damped oscillation
| Description | Equations |
|---|---|
| Drag force | $F = -bv$ |
| Time constant | $\tau = \dfrac{m}{b}$ |
| Angular frequency of damped oscillator | $\omega_d = \sqrt{\omega^2 - \left(\frac{b}{2m}\right)^2}$ |
| Displacement of damped oscillator | $x(t) = Ae^{-bt/2m}\sin(\omega_d t + \phi)$ |
1D Waves
| Description | Equations |
|---|---|
| Wave speed, wavelength, and frequency | $c = \lambda f = \dfrac{\lambda}{T}$ |
| Wave number | $k = \dfrac{2\pi}{\lambda}$ |
| Angular frequency | $\omega = kc = 2\pi f$ |
| Linear mass density | $\mu = \dfrac{m}{l}$ |
| Wave speed of strings | $c = \sqrt{\dfrac{F_T}{\mu}}$ |
| Particle speed | $v = \frac{\partial f}{\partial t}$ |
| Wave and particle speed | $c \not= v$ |
| Energy put into a wave | $E = \frac{1}{2}\mu\lambda\omega^2A^2$ |
| Average power supplied to produce waves | $\begin{aligned}\overline{P} &= \mu cv^2 \cr &= \frac{1}{2}\mu c A^2\omega^2 \cr &= \frac{1}{2}\sqrt{\mu F_T}\omega^2A^2\end{aligned}$ |
| Wave kinetic and potential energy | $K = U$ |
Wave function and boundary conditions
| Description | Equations |
|---|---|
| Traveling wave functions | $f(x,t) = f(x - ct)$ to right $f(x,t) = f(x + ct)$ to left |
| Harmonic (sinusoidal) traveling wave functions | $f(x,t) = A\sin(kx-\omega t+\phi)$ to right $f(x,t) = A\sin(kx+\omega t+\phi)$ to left |
| 1D wave equation | $\dfrac{\partial^2 f}{\partial x^2} = \dfrac{1}{c^2}\dfrac{\partial^2 f}{\partial t^2}$ |
| Principle of superposition | $f(x,t) = f_1(x,t)+f_2(x,t)$ |
| Boundary conditions | Free end: heavy $\rightarrow$ light spring Fixed end: light $\rightarrow$ heavy spring |
| Shape of reflected wave | Free end: horizontal reflection only Fixed end: horizontal reflection, vertical inversion |
| Shape of transmitted wave | Similar to incident wave |
Standing waves
| Description | Equations |
|---|---|
| Standing wave function | $f(x,t) = 2A\sin(kx)\cos(\omega t)$ |
| Standing wave of strings with two fixed ends |
$n$th harmonic, $n$ antinodes, $n+1$ nodes |
| Wavelength of $n$th harmonic | $\lambda_n = \dfrac{2L}{n}$ |
| Frequency of $n$th harmonic | $f_n = n\dfrac{c}{2L} = nf_1$ |
| Location of $m$th node of $n$th harmonic | $x_m = \dfrac{m\lambda_n}{2} \newline m \in [0, n+1]$ |
2D and 3D Waves
| Quantity | Unit | Definition |
|---|---|---|
| 2D intensity | $\mathrm{W/m}$ | $I = \dfrac{P}{L} = \dfrac{P}{2\pi r}$ |
| 3D intensity | $\mathrm{W/m^2}$ | $I = \dfrac{P}{A} = \dfrac{P}{4\pi r^2}$ |
| Intensity level | $\mathrm{dB}$ | $\beta = (10 \ \mathrm{dB})\log\left(\dfrac{I}{I_{\text{th}}}\right) \newline I_{\text{th}} = 10^{-12} \ \mathrm{W/m^2}$ |
| Description | Equations |
|---|---|
| Path difference of constructive interference (in phase, at antinodal line) | $\delta = n\lambda$ |
| Path difference of destructive interference (out of phase, at nodal line) | $\delta = (n - \frac{1}{2})\lambda$ |
| Beat frequency | $f_b = \lvert f_1 - f_2 \rvert$ |
| Average frequency | $\overline{f} = \frac{1}{2}(f_1 + f_2)$ |
| Wave function of beats | $\begin{aligned}y(x, t) &= y_1 + y_2 \cr &= 2A \cos(2\pi \frac{1}{2}f_b t)\sin(2\pi\overline{f}t)\end{aligned}$ |
| Doppler effect $(v_s < c)$ s - source; o - observer; rel to medium |
$\dfrac{f_{\mathrm{o}}}{f_{\mathrm{s}}} = \dfrac{c \pm v_{\mathrm{o}}}{c \pm v_{\mathrm{s}}}$ |
| Angle of shock wave $(v_s > c)$ | $\sin\theta = \dfrac{c}{v_s}$ |
| Mach number | $\footnotesize\text{Mach number} = \dfrac{v_s}{c}$ |
Ray Optics (Geometric Optics)
| Description | Equations |
|---|---|
| Law of reflection | $\theta_1 = \theta_2$ |
| Refraction index | $n_1 c_1 = n_2 c_2$ |
| Wavelength of light in a new medium | $n_1\lambda_1 = n_2\lambda_2$ |
| Snel’s law | $n_1\sin\theta_1 = n_2\sin\theta_2$ |
| Critical angle $(n_2>n_1)$ |
$\arcsin\left(\dfrac{n_1}{n_2}\right)$ |
| Lens equation o - object; i - image |
$\dfrac{1}{f} = \dfrac{1}{d_\text{o}} + \dfrac{1}{d_\text{i}}$ |
| Magnification | $M = \dfrac{h_\text{i}}{h_\text{o}} = -\dfrac{d_\text{i}}{d_\text{o}}$ |
| Radius of curvature (distance of center) of mirror and focal length | $R = 2\lvert f \rvert$ |
| Angular magnification | $M_\theta = \bigg\lvert\dfrac{\theta_\text{i}}{\theta_\text{o}}\bigg\rvert$ |
| Small-angle (paraxial) approximation of angular magnification | $M_\theta = \dfrac{0.25 \ \mathrm{m}}{f}$ |
| Lens strength | $d = \dfrac{1 \ \mathrm{m}}{f}$ |
Lens/mirror equation sign convention
| Sign | Lens |
|---|---|
| $f>0 \newline f<0$ | converging lens diverging lens |
| $d_\text{o}>0 \newline d_\text{o}<0$ | object in front of lens object behind lens |
| $d_\text{i}>0 \newline d_\text{i}<0$ | image behind lens (in front of mirror) image in front of lens (behind mirror) |
| $h_\text{i}>0 \newline h_\text{i}<0$ | image upright image inverted |
| $\lvert M \rvert>1 \newline \lvert M \rvert<1$ | image larger than object image smaller than object |
Converging lens images
| Object distance $d_\text{o}$ | Image distance $\lvert d_\text{i}\rvert$ | Image location | Upright Inverted |
Magnification | Real/Virtual |
|---|---|---|---|---|---|
| $(0, f)$ | $\lvert d_\text{i} \lvert>f$ | Same | Upright | Magnified | Virtual |
| $f$ | $\infty$ | - | - | - | Parallel light |
| $(f, 2f)$ | $(2f, \infty)$ | Opposite | Inverted | Magnified | Real |
| $2f$ | $2f$ | Opposite | Inverted | Same | Real |
| $(2f, \infty)$ | $(f, 2f)$ | Opposite | Inverted | Demagnified | Real |
| $\infty$ | $f$ | Opposite | - | - | Point |
Diverging lens images
| Object distance $d_\text{o}$ | Image distance $\lvert d_\text{i}\rvert$ | Image Location | Upright/ Inverted |
Magnification | Real/Virtual |
|---|---|---|---|---|---|
| $(0, \infty)$ | $\lvert d_\text{i}\rvert<d_\text{o}$ | Same | Upright | Demagnified | Virtual |
Wave and Particle Optics
Single slit interference
| Description | Equations |
|---|---|
| Variables | $n = 1, 2, 3, … \newline a =$ width of the slit |
| Destructive interference | $a\sin\theta = \pm n\lambda$ |
| Location of destructive interference (only for small angles) |
$y_n = \pm n\dfrac{\lambda L}{a}$ |
Double slit interference
| Description | Equations |
|---|---|
| Fringe order | $m = 0, 1, 2, … \newline n = 1, 2, 3, …$ |
| Constructive interference | $d\sin\theta = \pm m\lambda$ |
| Destructive interference | $d\sin\theta = \pm(n-\frac{1}{2})\lambda$ |
| Distance between adjacent maxima | $D = \dfrac{L\lambda}{d}$ |
Multiple slit interference
| Description | Equations |
|---|---|
| Variables | $m = 0, 1, 2, …$ $\newline N =$ number of slits $\newline k =$ integer not multiple of $N$ |
| Constructive interference (principal maxima) | $d\sin\theta = \pm m\lambda$ |
| Destructive interference (minima) | $d\sin\theta = \pm\dfrac{k}{N}\lambda$ |
| Minima adjacent to principle maxima | $d\sin\theta = \pm \dfrac{mN+1}{N}\lambda$ |
| Phasors | $\delta\varphi = 2\pi\dfrac{\delta s}{\lambda}$ |
Thin-film interference
Note: constructive and destructive interference refers to reflected light, not transmitted light.
| Description | Equations |
|---|---|
| Thin-film interference $t$ - thickness $m = 0, 1, 2, …$ a - incident medium b - thin film medium c - transmitted medium |
$\phi = \dfrac{4\pi n_b t\cos\theta_b}{\lambda_a} + \phi_{r2} - \phi_{r1} \newline$ $\phi = \begin{cases} 2m\pi & \small\text{constructive} \cr (2m+1)\pi & \small\text{destructive} \end{cases} \newline$ $\phi_{r} = \begin{cases} 0 & n_i>n_f \cr \pi & n_i<n_f \end{cases}$ |
| Constructive interference if in phase (If $\pi$-shifted, use destructive condition) |
$2t = m\lambda_b = m\dfrac{n_a}{n_b}\lambda_a$ |
| Destructive interference if in phase (If $\pi$-shifted, use constructive condition) |
$2t = (m+\frac{1}{2})\lambda_b = (m+\frac{1}{2})\dfrac{n_a}{n_b}\lambda_a$ |
Circular aperture
| Description | Equations |
|---|---|
| Variables | $d$ - diameter of the circular aperture $f$ - focal distance of the lens |
| First minimum with circular aperture | $\sin\theta = 1.22\dfrac{\lambda}{d}$ |
| Angular resolution Rayleigh’s criterion of resolution angle |
$\theta \approx \sin\theta = 1.22\dfrac{\lambda}{d}$ |
| Linear resolution Radius of the first minimum by a lens |
$y = 1.22\dfrac{\lambda f}{d}$ |
Wave-particle duality
| Description | Equations |
|---|---|
| Bragg’s condition with Bragg angle X-ray diffraction |
$2d\sin\alpha = m\lambda \newline (2d\cos\theta = m\lambda, \alpha = 90^\circ - \theta)$ |
| Energy of a photon | $E = h\nu$ |
| Momentum of a photon | $p = \dfrac{h\nu}{c}$ |
| Wavelength of a particle | $\lambda = \dfrac{h}{p} = \dfrac{h}{mv}$ |
| Photoelectric effect | $E_k = h\nu - \Phi = eV_{\text{stop}}$ |
| Stopping potential | $V_{\text{stop}} = \dfrac{h\nu}{e} - \dfrac{\Phi}{e}$ |
| Intensity of light | $I \propto$ rate of electron emitted from the metal |
Fluid Mechanics
| Quantity | Unit | Definition |
|---|---|---|
| Density | $\mathrm{kg/m^3}$ | $\rho = \dfrac{m}{V}$ |
| Pressure | $\mathrm{Pa} \newline \mathrm{N/m^2}$ | $P = \dfrac{dF}{dA}$ |
| Volumetric flow rate | $\mathrm{m^3/s}$ | $Q = \dot{V} = \dfrac{dV}{dt}$ |
| Description | Equations |
|---|---|
| Pressure of stationary fluid | $P = P_{\text{surf}} + \rho gh$ |
| Pressure of stationary fluid | $P_1 + \rho gy_1 = P_2 + \rho gy_2$ |
| Buoyant force | $F_b = F_{\text{bottom}} - F_{\text{top}}$ |
| Archimedes' principle | $F_b = \rho_f gV_{\text{disp}}$ |
| Volume of displaced fluid of floating object | $V_{\text{disp}} = \dfrac{\rho_o}{\rho_f}V_o$ |
| Condition of object buoyancy o - object; f - fluid |
$\begin{cases}\rho_o < \rho_f & \text{float} \cr \rho_o = \rho_f & \text{hang} \cr \rho_o > \rho_f & \text{sink}\end{cases}$ |
| Absolute pressure and gauge pressure | $P_{\text{abs}} = P_{\text{atm}} + P_g$ |
| Hydraulic system | $P = \dfrac{F_1}{A_1} = \dfrac{F_2}{A_2}$ |
| Continuity equation Laminar flow of nonviscous fluid |
$\begin{aligned}\dot{m}_1 &= \dot{m}_2 \cr \rho_1 Q_1 &= \rho_2 Q_2 \cr \rho_1 A_1 v_1 &= \rho_2 A_2 v_2 \end{aligned}$ |
| Bernoulli’s equation Laminar flow of incompressible nonviscous fluid |
$P_1 + \rho gy_1 + \frac{1}{2}\rho v_1^2 = P_2 + \rho gy_2 + \frac{1}{2}\rho v_2^2$ |
Entropy
| Description | Equations |
|---|---|
| Partition | $M = \dfrac{V}{\delta V}$ |
| Microstate (basic state) | $\Omega = M^N$ |
| Entropy | $S = \ln\Omega = \ln M^N$ |
| Linearity of entropy | $S = S_A + S_B \newline \Omega = \Omega_A\Omega_B$ |
| Constant temperature change in entropy | $\Delta S = N\ln\left(\dfrac{V_f}{V_i}\right)$ |
| Second law of thermodynamics in closed system |
$\Delta S \begin{cases} >0 & \footnotesize\text{toward equilibrium} \cr = 0 & \footnotesize\text{at equilibrium} \end{cases}$ |
| Equipartition of space | $\dfrac{N_A}{V_A} = \dfrac{N_B}{V_B}$ |
| Root-mean-square (rms) speed | $v_{\text{rms}} = \sqrt{\overline{v^2}}$ |
| Absolute temperature | $\dfrac{1}{k_BT} = \dfrac{dS}{dE_{\text{th}}}$ |
| Equipartition of energy | $\dfrac{E_{\text{th}, A}}{N_A} = \dfrac{E_{\text{th}, B}}{N_B}$ |
Monoatomic ideal gas
| Description | Equations |
|---|---|
| Thermal energy | $E_{\text{th}} = N\overline{K} = \dfrac{1}{2}Nmv_{\text{rms}}^2$ |
| Pressure | $P = \dfrac{2}{3}\dfrac{E_{\text{th}}}{V}$ |
| Thermal energy | $E_{\text{th}} = \dfrac{3}{2}Nk_BT$ |
| Average kinetic energy | $\overline{K} = \dfrac{3}{2}k_BT$ |
| rms speed | $v_{\text{rms}} = \sqrt{\dfrac{3k_BT}{m}}$ |
| Ideal gas law | $PV = nRT \newline PV = Nk_BT$ |
| Constant volume change in entropy | $\Delta S = \dfrac{3}{2}N\ln\left(\dfrac{T_f}{T_i}\right)$ |
| Total change in entropy | $\Delta S = \dfrac{3}{2}N\ln\left(\dfrac{T_f}{T_i}\right) + N\ln\left(\dfrac{V_f}{V_i}\right)$ |
Thermodynamic Processes
| Description | Equations |
|---|---|
| General energy balance | $\Delta E = W + Q$ |
| Energy balance of ideal gas | $\Delta E_{\text{th}} = W + Q$ |
| Change in thermal energy of ideal gas | $\Delta E_{\text{th}} = \dfrac{d}{2} Nk_B \Delta T$ |
| PV Work | $W = \displaystyle\int_{V_i}^{V_f} P \ dV$ |
| Entropy change | $\Delta S = N \ln\left(\dfrac{V_f}{V_i}\right) + \dfrac{d}{2}N \ln\left(\dfrac{T_f}{T_i}\right)$ |
| Constant volume heat capacity | $C_V = \dfrac{Q}{N\Delta T} = \dfrac{d}{2}k_B$ |
| Constant pressure heat capacity | $C_P = \dfrac{Q}{N\Delta T} = \left(\dfrac{d}{2}+1\right) k_B$ |
| Heat capacity relationship | $C_P = C_V + k_B$ |
| Heat capacity ratio | $\gamma = \dfrac{C_P}{C_V} = 1+\dfrac{2}{d}$ |
Isochoric process
| Description | Equations |
|---|---|
| Isochoric process | $\Delta V = 0$ |
| Work | $W = 0$ |
| Thermal energy and heat | $\Delta E_{\text{th}} = Q = \dfrac{d}{2}Nk_B T = NC_V \Delta T$ |
| Entropy | $\Delta S = \dfrac{d}{2}N \ln\left(\dfrac{T_f}{T_i}\right) = \dfrac{NC_V}{k_B}\ln\left(\dfrac{T_f}{T_i}\right)$ |
Isentropic process
| Description | Equations |
|---|---|
| Isobaric process (quasistatic adiabatic) | $\Delta S = 0$ |
| Heat | $Q = 0$ |
| Thermal energy and work | $\Delta E_{\text{th}} = W = NC_V \Delta T$ |
| PVT relationship | $P_1 V_1^\gamma = P_2 V_2^\gamma \newline T_1 V_1^{\gamma-1} = T_2 V_2^{\gamma-1} \newline P_1^{(1/\gamma)-1} T_1 = P_2^{(1/\gamma)-1} T_2$ |
Isobaric process
| Description | Equations |
|---|---|
| Isobaric process | $\Delta P = 0$ |
| Work | $W = -P\Delta V = -Nk_B\Delta T$ |
| Heat | $Q = NC_P\Delta T$ |
| Thermal energy | $\Delta E_{\text{th}} = NC_V\Delta T$ |
| Entropy | $\Delta S = \dfrac{NC_P}{k_B}\ln\left(\dfrac{T_f}{T_i}\right)$ |
Isothermal process
| Description | Equations |
|---|---|
| Isothermal process | $\Delta T = 0$ |
| Thermal energy | $\Delta E_{\text{th}} = 0$ |
| Work and Heat | $Q=-W$ |
| Work | $W = -Nk_BT\ln\left(\dfrac{V_f}{V_i}\right)$ |
| Heat | $Q = Nk_BT\ln\left(\dfrac{V_f}{V_i}\right)$ |
| Entropy | $\Delta S = N\ln\left(\dfrac{V_f}{V_i}\right) = \dfrac{Q}{k_B T}$ |
Degradation of Energy
| Description | Equations |
|---|---|
| Complete cycle of steady device | $\Delta E = 0 \newline W = Q_{\text{out}} - Q_{\text{in}} \newline \Delta S_{\text{sys}} = 0 \newline \Delta S_{\text{surr}} \ge 0$ |
| Steady device thermally transferring energy to lower temperature | $\Delta S_{\text{surr}} = \dfrac{Q_{\text{out}}}{k_B}\left(\dfrac{1}{T_{\text{out}}} - \dfrac{1}{T_{\text{in}}}\right)$ |
| Steady device thermally transferring energy to higher temperature | $\Delta S_{\text{surr}} = - \dfrac{Q_{\text{out}}}{k_B}\left(\dfrac{1}{T_{\text{out}}} - \dfrac{1}{T_{\text{in}}}\right)$ |
| Steady device converting mechanical energy to thermal energy | $\Delta S_{\text{surr}} = \dfrac{Q_{\text{out}}}{k_B T_{\text{out}}}$ |
| Reversible heat engine | $\dfrac{Q_{\text{out}}}{Q_{\text{in}}} = \dfrac{T_{\text{out}}}{T_{\text{in}}} = \dfrac{T_{\text{low}}}{T_{\text{high}}}$ |
| Energy balance | $Q_{\text{in}} + W_{\text{in}} = Q_{\text{out}} + W_{\text{out}}$ |
| Efficiency of heat engine | $\eta = -\dfrac{W_{\text{out}}}{Q_{\text{in}}} = 1-\dfrac{Q_{\text{out}}}{Q_{\text{in}}}$ |
| Maximum efficiency of reversible heat engine | $\eta_{\mathrm{max}} = 1-\dfrac{T_{\text{low}}}{T_{\text{high}}}$ |
| Coefficient of performance of heating | $\mathrm{COP_{heating}} = \dfrac{Q_{\text{out}}}{W} = \dfrac{1}{1-Q_{\text{in}}/Q_{\text{out}}}$ |
| Maximum coefficient of performance of heating (reversible heat pump) | $\mathrm{COP_{heating, max}} = \dfrac{1}{1-T_{\text{in}}/T_{\text{out}}}$ |
| Coefficient of performance of cooling | $\mathrm{COP_{cooling}} = \dfrac{Q_{\text{in}}}{W} = \mathrm{COP_{heating}} - 1$ |
| Maximum coefficient of performance of cooling (reversible heat pump) | $\mathrm{COP_{cooling, max}} = \dfrac{T_{\text{in}}}{T_{\text{out}}-T_{\text{in}}}$ |