Optics and acoustics respectively study electromagnetic radiation spectra (light) and mechanical elastic waves (sound) in media across excitation, propagation, interference, diffraction, and energetic interaction. From the rectilinear propagation and image formation of geometric rays, to wave-particle duality and the quantum photon hypothesis, through to macroscopic audible sound and ultrasonic transduction, mathematical equations build the theoretical bridge connecting human perception with precision instrumentation.
This cheat sheet systematically organizes 17 core physical laws and foundational equations in optics and acoustics, partitioned into Geometric Optics, Wave Optics, Quantum Optics, and Acoustics, clearly presenting standard mathematical expressions, key variable definitions, underlying physical mechanisms, validity domains, and typical industrial and physical applications.
🔗 物理法則・主要方程式マトリクス: 古典力学・運動学 · 流体力学・熱力学 · 電磁気学・電磁力学 · 光学・音響学 · 量子力学・原子物理学 · 相対性理論・現代宇宙論 · 原子核・素粒子物理学 · 天体物理学・地球物理学
1. Geometric Optics (5 Laws)
In the limit where the wavelength of light is negligibly small compared to the macroscopic dimensions of optical components, light travels along rectilinear rays governed by reflection, refraction, and Fermat's stationary optical path principle.
01 · Law of Reflection
Law of Reflection
$$\theta_i = \theta_r$$
- ①物理的本質: When light encounters a smooth interface between media, the reflected ray lies in the plane of incidence formed by the incident ray and the surface normal (coplanarity), and the angle of reflection equals the angle of incidence.
- ②変数・定数の定義: $\theta_i$ is the angle of incidence, and $\theta_r$ is the angle of reflection (both measured relative to the surface normal, not the tangential boundary).
- ③基礎物理学上の意義: The earliest quantitative experimental law in optical history; both specular reflection and diffuse scattering strictly follow this law locally across micro-facets, differing only in surface roughness and micro-normal orientation distributions.
- ④適用限界と工学応用: Surface smoothness scale much larger than optical wavelength. Applied in plane mirror imaging, astronomical parabolic reflectors, laser cavity high-reflectance dielectric mirrors, and periscope optics.
02 · Snell's Law
Snell's Law of Refraction
$$n_1 \sin \theta_1 = n_2 \sin \theta_2$$
- ①物理的本質: When light passes from one isotropic medium into another, the ratio of the sine of the angle of incidence to the sine of the angle of refraction equals the reciprocal ratio of refractive indices. Light entering a denser medium bends toward the surface normal.
- ②変数・定数の定義: $n_1$ and $n_2$ are the absolute refractive indices of the two media ($n = c/v$), $\theta_1$ is the angle of incidence, and $\theta_2$ is the angle of refraction.
- ③基礎物理学上の意義: The quantitative cornerstone of lens design and spectroscopic dispersion, rigorously derivable from Fermat's principle and wave phase velocity matching.
- ④適用限界と工学応用: Isotropic transparent media interfaces; birefringent anisotropic crystals require Huygens' construction for ordinary (o) and extraordinary (e) rays. Applied in ophthalmic diopter prescription, photographic achromatic doublet design, and dispersive prism spectrometry.
03 · Critical Angle for Total Internal Reflection
Total Internal Reflection (TIR)
$$\sin \theta_c = \frac{n_2}{n_1} \quad (n_1 > n_2)$$
- ①物理的本質: When light propagates from an optically denser medium to an optically rarer medium ($n_1 > n_2$) at an angle of incidence exceeding the critical angle $\theta_c$, the refracted wave vanishes and all energy is entirely reflected into the incident medium with zero refractive transmission loss.
- ②変数・定数の定義: $\theta_c$ is the critical angle of incidence (where the angle of refraction reaches exactly 90°), $n_1$ is the denser medium index, and $n_2$ is the rarer medium index.
- ③基礎物理学上の意義: The foundational mechanism of lossless optical guiding; beyond the interface exists only an exponentially decaying evanescent surface wave with nanometer penetration depth.
- ④適用限界と工学応用: Requires $n_1 > n_2$ and $\theta \ge \theta_c$. Applied in silica core telecommunication optical fibers, medical endoscopes, total internal reflection prisms, brilliant-cut diamond facets, and total internal reflection fluorescence (TIRF) microscopy.
04 · Fermat's Principle
Principle of Least Time
$$\delta \int_{A}^{B} n\,\mathrm{d}s = 0$$
- ①物理的本質: The actual optical path traversed by light connecting two points is stationary (typically a minimum, occasionally a maximum or saddle point) relative to variations of neighboring possible paths; its first variation vanishes.
- ②変数・定数の定義: $n(\mathbf{r})$ is the spatial refractive index distribution, $\mathrm{d}s$ is the arc-length element along the path, $\int n\,\mathrm{d}s$ is the Optical Path Length (OPL), and $\delta$ denotes the functional variation operator.
- ③基礎物理学上の意義: The overarching variational axiom of geometric optics, from which rectilinear propagation, reflection, and Snell's law are directly unified; historically anticipated the principle of stationary action in analytical mechanics.
- ④適用限界と工学応用: Universally valid in gradient-index (GRIN) media and curved mirror geometries. Applied in atmospheric mirage ray tracing, graded-index fiber optic modal simulation, and gravitational lensing cosmology.
05 · Thin Lens Equation
Gaussian Lens Formula
$$\frac{1}{u} + \frac{1}{v} = \frac{1}{f} \quad \left(m = -\frac{v}{u}\right)$$
- ①物理的本質: Under paraxial approximations, the object distance, image distance, and effective focal length of a thin lens satisfy a reciprocal summation relationship.
- ②変数・定数の定義: $u$ is object distance, $v$ is image distance, and $f$ is focal length under the Cartesian/real-is-positive convention (real object $u>0$, real image $v>0$, converging lens $f>0$, diverging lens $f<0$); $m$ is transverse magnification.
- ③基礎物理学上の意義: The core first-order paraxial starting point for designing compound coaxial systems including microscopes, telescopes, and photographic lenses.
- ④適用限界と工学応用: Small ray angles ($\sin\theta \approx \theta$) and lens center thickness much smaller than radii of curvature; wide-aperture wide-angle systems require Seidel third-order monochromatic aberration corrections. Applied in camera auto-focus modules, projector throw ratios, and magnifying optics.
2. Wave Optics (5 Laws)
Wave optics reveals light's nature as transverse electromagnetic radiation, dictating coherent superposition interference, spatial diffraction limits, and Malus linear polarization state filtering.
06 · Young's Double-Slit Interference
Young's Double-Slit Experiment
$$\Delta x = \frac{L}{d}\lambda$$
- ①物理的本質: Two mutually coherent light waves superpose in space, forming alternating constructive (bright fringes) and destructive (dark fringes) interference based on path length differences; adjacent fringe spacing is uniform and equidistant.
- ②変数・定数の定義: $\Delta x$ is the spatial fringe spacing between adjacent bright fringes, $L$ is the slit-to-screen distance, $d$ is the center-to-center separation between the double slits, and $\lambda$ is optical wavelength.
- ③基礎物理学上の意義: The historic decisive experiment that experimentally proved the wave nature of light, establishing standard interferometric methodology for wavelength measurement.
- ④適用限界と工学応用: Requires stable relative phase difference (coherent illumination). In modern single-photon and single-electron beam experiments, statistical interference fringes persist, serving as the paramount verification of quantum probability wavefunctions. Applied in Michelson interferometry and sub-micron displacement sensing.
07 · Diffraction Grating Equation
Diffraction Grating Equation
$$d \sin \theta = k \lambda \quad (k = 0, \pm 1, \pm 2, \dots)$$
- ①物理的本質: Light passing through an array of numerous equally spaced transmitting or reflecting slits undergoes far-field multi-beam interference, forming sharp, high-intensity principal maxima at diffraction angles where optical path differences between adjacent slits equal integer wavelengths.
- ②変数・定数の定義: $d$ is the grating period/pitch (slit width $a$ + spacing $b$), $\theta$ is the angular direction of principal maxima, $k$ is the integer diffraction spectral order, and $\lambda$ is wavelength.
- ③基礎物理学上の意義: The heart of modern monochromators, spectrophotometers, and astronomical spectrographs; resolving power scales directly with total illuminated ruling count ($R = kN$).
- ④適用限界と工学応用: Multi-slit Fraunhofer far-field diffraction; exhibits missing orders due to single-slit diffraction envelope minima (when $d/a$ is an integer). Applied in atomic absorption/emission spectroscopy, laser wavelength calibration, and radial velocity exoplanet detection.
08 · Thin-Film Interference
Thin-Film Interference
$$2 n d \cos \theta \pm \frac{\lambda}{2} = k \lambda$$
- ①物理的本質: Incident light reflects off both the upper and lower surfaces of a dielectric thin film, generating two coherent wavefronts via amplitude division. When reflection occurs at a boundary from an optically rarer into a denser medium, an abrupt $\pi$ phase shift (a $\lambda/2$ **half-wave loss**) is introduced.
- ②変数・定数の定義: $n$ is the film refractive index, $d$ is geometric film thickness, $\theta$ is angle of refraction inside the film, $\pm \lambda/2$ accounts for half-wave phase jump conditions, and $k$ is integer interference order.
- ③基礎物理学上の意義: The theoretical foundation of industrial optical coatings (anti-reflective coatings, dielectric mirrors, bandpass filters) and nanoscale optical metrology.
- ④適用限界と工学応用: Film thickness within the temporal coherence length of the illuminating light. Applied in magnesium fluoride ($\text{MgF}_2$) anti-reflective lens coatings (designed to optical thickness $\lambda/4$), iridescent soap films, and Newton's rings for spherical curvature inspection.
09 · Rayleigh Criterion
Rayleigh Criterion for Diffraction Limit
$$\theta_{\min} = 1.22 \frac{\lambda}{D}$$
- ①物理的本質: Two incoherent point sources are defined as barely resolved when the central maximum of the Airy disk diffraction pattern of one point source coincides with the first minimum (dark ring) of the other, causing a 26.5% intensity dip between the peaks.
- ②変数・定数の定義: $\theta_{\min}$ is the minimum angular separation resolvable by the aperture (rad), $\lambda$ is illuminating wavelength, and $D$ is the circular entrance pupil diameter.
- ③基礎物理学上の意義: The fundamental physical resolution limit imposed by wave diffraction on any optical imaging aperture, which no aberration-free geometric lens can circumvent.
- ④適用限界と工学応用: Far-field incoherent circular aperture diffraction. Dictates the need for giant apertures in astronomy (FAST, Hubble, James Webb Space Telescope), and drives semiconductor photolithography to extreme ultraviolet (EUV, 13.5 nm) wavelengths.
10 · Malus's Law
Malus's Law of Polarization
$$I = I_0 \cos^2 \theta$$
- ①物理的本質: When completely plane-polarized light of intensity $I_0$ passes through an ideal analyzer, the transmitted intensity is proportional to the square of the cosine of the angle between the transmission axis of the analyzer and the polarization axis of the incident light.
- ②変数・定数の定義: $I$ is transmitted intensity, $I_0$ is incident polarized light intensity, and $\theta$ is the relative angle between polarization axes. When polarizers are crossed ($\theta = 90^\circ$), transmitted intensity strictly drops to zero (complete extinction).
- ③基礎物理学上の意義: Quantitative experimental verification of light's transverse electromagnetic wave nature (longitudinal sound waves do not exhibit polarization), providing the operating principle of liquid crystal displays.
- ④適用限界と工学応用: Incident light must be linearly polarized (unpolarized light through an ideal polarizer drops to $\frac{1}{2}I_{\text{nat}}$). Applied in twisted nematic LCD pixel optical shutters, anti-glare polarized sunglasses, and photoelastic stress analysis.
3. Quantum Optics (4 Laws)
At microscopic scales, light exhibits discrete quantum particle properties as photons, governing fundamental absorption, emission, and photoelectric energy transformations.
11 · Planck Relation
Planck Energy-Frequency Relation
$$E = h\nu = \hbar\omega$$
- ①物理的本質: Electromagnetic energy exchange is not continuous, but occurs in discrete quanta (photons); the energy carried by an individual photon is strictly proportional to its electromagnetic oscillating frequency.
- ②変数・定数の定義: $E$ is photon energy ($\text{J}$ or $\text{eV}$), $h \approx 6.62607 \times 10^{-34}\;\text{J}\cdot\text{s}$ is Planck's constant, $\nu$ is oscillation frequency ($\text{Hz}$), $\hbar = h/(2\pi)$ is the reduced Planck constant, and $\omega$ is angular frequency.
- ③基礎物理学上の意義: The birth of quantum physics, overturning the classical electromagnetic continuity dogma and igniting the 20th-century quantum revolution.
- ④適用限界と工学応用: Foundational equation for wave-particle duality across all microscopic entities. Explains why ultraviolet photons possess high ionization energies while infrared photons do not. Applied in semiconductor LED bandgap tuning and laser transition calculation.
12 · Einstein's Photoelectric Equation
Photoelectric Effect Equation
$$E_{k,\max} = h\nu - W_0 = e U_c$$
- ①物理的本質: When a photon of frequency $\nu$ is absorbed by an electron bound inside a metal, part of the photon energy overcomes the metal's work function $W_0$, and the remainder is completely converted into the maximum initial kinetic energy of the emitted photoelectron.
- ②変数・定数の定義: $E_{k,\max}$ is maximum kinetic energy of emitted photoelectrons, $h$ is Planck's constant, $\nu$ is incident photon frequency, $W_0$ is material work function, $e$ is elementary charge, and $U_c$ is reverse stopping potential.
- ③基礎物理学上の意義: Provided definitive experimental proof of the corpuscular nature of photons, earning Einstein the 1921 Nobel Prize in Physics.
- ④適用限界と工学応用: Single-photon absorption regime. Predicts the threshold frequency $\nu_0 = W_0/h$ below which no electron emission occurs regardless of illumination intensity; above threshold, intensity dictates emission current rather than kinetic energy. Applied in photovoltaic cells, microchannel plate night-vision tubes, and photomultiplier detectors.
13 · Compton Scattering
Compton Scattering & Photon Momentum
$$\Delta \lambda = \frac{h}{m_e c}(1 - \cos \theta)$$
- ①物理的本質: In an elastic collision between a high-energy X-ray or gamma-ray photon and an electron, the photon transfers energy and relativistic momentum to the electron, emerging with lower energy and a longer wavelength; the wavelength shift depends exclusively on scattering angle $\theta$.
- ②変数・定数の定義: $\Delta \lambda$ is wavelength increase ($\text{m}$), $h$ is Planck's constant, $m_e$ is electron rest mass, $c$ is light speed, $\lambda_c = h/(m_e c) \approx 2.426 \times 10^{-12}\;\text{m}$ is the Compton wavelength of an electron, and $\theta$ is scattering angle.
- ③基礎物理学上の意義: Directly proved that photons possess both energy $E=h\nu$ and relativistic momentum $p=h/\lambda$, an effect inexplicable by classical wave theory.
- ④適用限界と工学応用: Incident photon energy significantly greater than atomic binding energy. Applied in high-energy physics calorimeter calibration, astrophysical inverse Compton scattering in cosmic rays, and radiation therapy dosimetry.
14 · Planck's Law of Blackbody Radiation
Blackbody Spectral Radiance
$$I(\lambda, T) = \frac{2hc^2}{\lambda^5}\frac{1}{e^{\frac{hc}{\lambda k_B T}} - 1}$$
- ①物理的本質: The exact spectral distribution of electromagnetic radiation emitted into hemispherical space by an idealized blackbody in thermodynamic equilibrium at temperature $T$, smoothly convergent at both short and long wavelength limits, resolving the classical "ultraviolet catastrophe".
- ②変数・定数の定義: $I(\lambda,T)$ is spectral radiance ($\text{W}/(\text{m}^2\cdot\text{m})$), $h$ is Planck's constant, $c$ is speed of light, $k_B$ is Boltzmann's constant, and $T$ is absolute thermodynamic temperature.
- ③基礎物理学上の意義: Led directly to energy quantization and yields the Stefan-Boltzmann law ($\sigma T^4$) by integration over all wavelengths, and Wien's displacement law via spatial derivative maximization.
- ④適用限界と工学応用: Thermal equilibrium diffuse radiation. The Cosmic Microwave Background (CMB, 2.725 K) measured by COBE/WMAP/Planck satellites matches this curve with sub-0.005% precision. Applied in infrared stealth coating design, optical pyrometry, and thermal infrared thermography.
4. Acoustics (3 Laws)
Acoustics explores mechanical vibrations propagating through elastic media as longitudinal pressure waves, human physiological auditory logarithmic scaling, and wave interference beat oscillations.
15 · Speed of Sound
Newton-Laplace Equation
$$v = \sqrt{\frac{\gamma p}{\rho}} = \sqrt{\frac{\gamma R T}{M}}$$
- ①物理的本質: The phase velocity of mechanical acoustic waves in an ideal gas medium is governed by the adiabatic index (heat capacity ratio $\gamma$), absolute temperature $T$, and molar mass $M$, remaining independent of static ambient pressure.
- ②変数・定数の定義: $v$ is acoustic phase velocity ($\text{m/s}$), $\gamma = C_p/C_V$ is adiabatic index ($\approx 1.4$ for air), $p$ is static pressure, $\rho$ is density, $R$ is universal gas constant, and $M$ is molar mass.
- ③基礎物理学上の意義: Laplace corrected Newton's isothermal acoustic formulation (which underestimated air sound speed by 16% at 280 m/s), proving that acoustic micro-perturbations oscillate under ultra-fast adiabatic compression and expansion cycles.
- ④適用限界と工学応用: Isotropic elastic gases. At 20 °C, dry air sound speed is $\approx 343\;\text{m/s}$; water is $\approx 1500\;\text{m/s}$, and steel solids reach $5000\;\text{m/s}$ (strictly: solids > liquids > gases). Applied in ultrasonic distance sensors, sonar underwater navigation, sonic boom analysis, and aerodynamic Mach number ($\text{Ma} = v_{\text{flight}}/v_{\text{sound}}$) calibration.
16 · Sound Intensity Level
Decibel Logarithmic Scale
$$L = 10 \log_{10} \left(\frac{I}{I_0}\right) \quad (\text{dB})$$
- ①物理的本質: Sound strength quantified as ten times the base-10 logarithm of measured sound intensity relative to the standard human auditory threshold; matches human logarithmic perceptual response spanning 12 orders of magnitude (Weber-Fechner psychophysical law).
- ②変数・定数の定義: $L$ is sound intensity level in decibels ($\text{dB}$), $I$ is physical sound power per unit area ($\text{W/m}^2$), and $I_0 = 10^{-12}\;\text{W/m}^2$ is standard threshold of hearing at $1000\;\text{Hz}$.
- ③基礎物理学上の意義: Global standard metric for acoustic environmental noise assessment and hearing conservation; because the decibel is logarithmic, **sound intensity levels from independent noise sources cannot be added arithmetically** (doubling sound power $2I$ increases level by only $\approx 3\;\text{dB}$).
- ④適用限界と工学応用: Human audible dynamic range ($0\;\text{dB}$ threshold, $60\;\text{dB}$ normal conversation, $85\;\text{dB}$ occupational hazard threshold, $120\;\text{dB}$ threshold of pain). Applied in architectural noise abatement barrier design, urban environmental acoustics, and headphone loudness limiter compliance.
17 · Beat Frequency
Wave Superposition & Heterodyning
$$f_{\text{beat}} = |f_1 - f_2|$$
- ①物理的本質: When two harmonic waves with identical polarization, comparable amplitudes, but slightly different frequencies superpose at a point, the combined wave envelope undergoes slow periodic amplitude modulation; the frequency of envelope intensity maxima (beats) strictly equals the frequency difference between the source tones.
- ②変数・定数の定義: $f_{\text{beat}}$ is envelope beat frequency ($\text{Hz}$), and $f_1, f_2$ are the frequencies of the two interfering signals.
- ③基礎物理学上の意義: The physical bedrock of heterodyne signal detection and musical instrument tuning; when the beat frequency decreases to zero, complete frequency synchronization is attained.
- ④適用限界と工学応用: Small frequency differentials (typically $1\text{--}10\;\text{Hz}$ for optimal acoustic discrimination). Applied in piano and orchestral auditory tuning, superheterodyne RF intermediate frequency receivers, and laser Doppler vibrometry detecting micro-structural resonances.
5. Physical Quantities, Symbols, and SI Units
| Physical Quantity | Symbol | Scalar / Vector | SI Base or Derived Unit | SI Symbol | Definition / Equivalence |
|---|---|---|---|---|---|
| Luminous Intensity | $I_v$ | Scalar | Candela (SI Base Unit) | cd | Luminous flux emitted per unit solid angle |
| Luminous Flux | $\Phi_v$ | Scalar | Lumen | lm | $1;\text{lm} = 1;\text{cd}\cdot\text{sr}$ (photopic eye sensitivity weighted) |
| Illuminance | $E_v$ | Scalar | Lux | lx | $1;\text{lx} = 1;\text{lm/m}^2$ (luminous flux incident per unit area) |
| Refractive Index | $n$ | Scalar | Dimensionless | - | $n = c/v$ (speed of light in vacuum over speed in medium) |
| Optical Path Length (OPL) | $\Delta, S$ | Scalar | Meter | m | $\Delta = \int n,\mathrm{d}s$ (equivalent vacuum path distance) |
| Grating Constant | $d$ | Scalar | Meter / Micron | m, $\mu\text{m}$ | $d = a + b = 1/N_{\text{lines}}$ |
| Photon Energy | $E$ | Scalar | Electronvolt / Joule | eV, J | $1;\text{eV} \approx 1.602 \times 10^{-19};\text{J}$ |
| Work Function | $W_0$ | Scalar | Electronvolt | eV | Minimum energy required to liberate electron from metal surface |
| Speed of Sound | $v_s$ | Scalar | Meters per second | m/s | 20 °C air $\sim 343;\text{m/s}$, water $\sim 1500;\text{m/s}$, steel $\sim 5000;\text{m/s}$ |
| Sound Intensity | $I$ | Scalar | Watts per square meter | $\text{W/m}^2$ | Acoustic power per unit area normal to propagation direction |
| Sound Intensity / Pressure Level | $L, \text{SPL}$ | Scalar | Decibel | dB | $L = 10\log_{10}(I/I_0)$, $L_p = 20\log_{10}(p/p_0)$ |
| Sound Pressure | $p$ | Scalar | Pascal | Pa | Dynamic pressure fluctuation caused by acoustic sound wave |
| Beat Frequency | $f_{\text{beat}}$ | Scalar | Hertz | Hz | $f_{\text{beat}} = |f_1 - f_2|$ |
| Planck Constant | $h$ | Physical Constant | Joule-second | $\text{J}\cdot\text{s}$ | $h \approx 6.62607015 \times 10^{-34};\text{J}\cdot\text{s}$ (exact) |
| Reduced Planck Constant | $\hbar$ | Physical Constant | Joule-second | $\text{J}\cdot\text{s}$ | $\hbar = h / (2\pi) \approx 1.05457 \times 10^{-34};\text{J}\cdot\text{s}$ |
| Speed of Light in Vacuum | $c$ | Physical Constant | Meters per second | m/s | $c = 299,792,458;\text{m/s}$ (exact definition) |