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The acoustic tensor determines the plane waves of a locally uniform incremental elastic medium. For a unit material direction $\nu$, its eigenvalues are ${\rho }_0{c}^2$ and its eigenvectors are displacement polarizations. Major symmetry of the incremental elastic moduli makes $Q$ a real symmetric matrix. Positivity on every rank-one matrix $m\otimes \nu$ makes all its wave speeds real and nonzero.

An acoustic wavelength measurement makes an invisible spatial pattern readable with a ruler. The authored guide Interesting and inexpensive experiments for high school physics by Joe Wolfe places a loudspeaker near a pipe's open end, drives it with an oscillator, and slides a microphone attached to a metre rule through the pipe. An oscilloscope or computer displays the microphone signal. Its oscillation amplitude rises and falls with position; adjacent minima locate pressure nodes of a standing wave a distance $d=\lambda /2$ apart. A cap gives a pressure antinode at the closed end; an open end has a pressure node just outside it. These are pressure patterns, distinct from displacement patterns. The guide advises wavelengths longer than the pipe diameter, sufficient pipe length to locate both a node and an antinode, and frequency spectrum analysis to distinguish the intended tone from other sounds.^[ref]

The inference follows from superposition of two oppositely directed acoustic plane waves. In an ideal uniform, lossless pipe, taking equal pressure amplitudes $P$ and a rigid cap at $x=0$ givesThe time-varying pressure perturbation $p^{\prime }$ has local oscillation amplitude $2|P\cos (kx)|$. Its zeros are separated by $\pi /|k|=\lambda /2$. Unequal incident and reflected amplitudes leave nonzero minima, but their spacing remains $\lambda /2$ in this ideal model. A microphone detects that pressure variation through a changing electrical signal, rather than watching a wave crest move. Record the tone's frequency too, then infer phase speed $v_p=2fd$.

For proposed controls, keep the drive frequency and amplitude fixed during a scan, verify the frequency on the recorded signal, and repeat the scan in both directions. Fit several minimum positions $x_j=a+jd$ using linear regression to reduce dependence on one pair of positions. Check the ruler scale and use one consistent reference point on the microphone. Compare with the source switched off to locate background measurement noise; use frequency spectrum analysis to check for additional tones, and reduce drive if the trace ceases to resemble a sinusoid. Record air temperature and avoid strong air motion. These are suggested controls, not additional procedures or observations attributed to the guide.

The pipe must support approximately plane waves over the measured region: transverse structure at shorter wavelengths, the microphone's finite size, and disturbance by the ruler can compromise the interpretation. Weak reflection gives shallow minima, while damping and other reflections can shift the extrema of the simple two-wave model. Measure interior spacings rather than assuming the first minimum is exactly at the pipe lip. A short scan may not include enough minima to estimate $d$ reliably, and drift during sequential scanning can imitate spatial variation.

For an explicitly hypothetical calculation, $f=1000\mathrm{Hz}$ and $d=0.1715\mathrm{m}$ give $\lambda =0.3430\mathrm{m}$, $K\simeq 18.3183\mathrm{rad}{\mathrm{m}}^{-1}$, and $v_p=343.0\mathrm{m}{\mathrm{s}}^{-1}$. No such measurements are claimed here. If independent small measurement errors in $f$ and fitted $d$ have standard deviations ${\sigma }_f$ and ${\sigma }_d$, first-order error propagation gives ${\sigma }_{v_p}/v_p\simeq \sqrt{({\sigma }_f/f{)}^2+({\sigma }_d/d{)}^2}$. A systematic scale error or changing air conditions requires separate assessment.

The Airy disk is the central bright region, up to the first dark ring, in the scalar diffraction pattern of a uniformly illuminated, unobscured circular aperture. The whole pattern includes the weaker surrounding rings; its radius is distinct from its full width at half maximum.

For a circular optical pupil of diameter $D$ under normal, uniform monochromatic plane wave illumination, Fraunhofer diffraction givesHere $I_0$ is the on-axis light intensity, $\theta$ is the angle from the optical axis, $\lambda$ is the wavelength in the propagation medium, and $J_1$ is the order-one Bessel function of the first kind. The paraxial approximation replaces $\sin \theta$ by $\theta$, measured in radians. The first positive zero $j_{1,1}=3.831705970\dots$ givesThis is the angular radius of the Airy disk; the angular diameter is twice as large. The Airy disk derivation explains the origin of the Bessel function of the first kind and of the numerical coefficient.^[ref]

The image and both radial curves show the same calculated light intensity, divided by its peak. The image's logarithmic scale makes the weak rings visible; its colour is a display choice. The linear curve shows how small the ring peaks are compared with the centre. A dark ring marks a zero of wave amplitude, produced by cancellation across the optical pupil; it is not the edge of all transmitted light. The first Airy disk contains about $83.78\%$ of the total radiant flux, as calculated in Airy disk normalization and encircled energy.

For an original design example, take an ideal optical lens in air with focal length $f=100\mathrm{mm}$, circular optical pupil diameter $D=5.00\mathrm{mm}$ and wavelength $\lambda =550\mathrm{nm}$. In the focal plane, the radial coordinate $r\simeq f\theta$ givesThe first-zero diameter is $26.8327\mu \mathrm{m}$. Solving $P(u_{1/2})=1/2$ gives $u_{1/2}=1.616339948\dots$, so the diameter measured by full width at half maximum is onlyThus a quoted “spot diameter” is incomplete without its definition. Doubling $D$ halves both widths at fixed wavelength and focal length; increasing magnification alone changes the image scale without improving object-space discrimination.

The Airy disk supplies a calculable reference for a point spread function, allowing diffraction to be distinguished from defocus and optical aberrations. Its beauty is that one circular boundary determines an entire sequence of bright and dark rings through a Fourier transform. The ideal formula assumes an unobscured pupil, uniform illumination, ideal optical focus, a single wavelength, and a scalar description with small output angles. A blocked centre, nonuniform pupil transmission or a significant pupil wave phase error changes the response. High numerical aperture may require polarization of an electromagnetic wave and a vector field description. The circular-aperture diffraction laboratory experiment tests the far-field pattern directly; the Rayleigh criterion uses its first zero to define a two-source convention.

Take nonrelativistic quantum mechanics, a short-range interaction, a fixed heavy target, and a common projectile mass $m$. An incoming plane wave in atomic state $a$ has unit coefficient. In an open scattering channel $i$, the target finishes in a normalized atomic state ${\chi }_i$, and the projectile has positive velocity magnitude $v_i=\hslash k_i/m$. At large distance,Here $\Omega$ denotes direction, $d\Omega$ is a solid angle element, and ${\epsilon }_i$ is an internal atomic energy. Different ${\chi }_i$ are orthogonal. The scattering amplitude $f_{ia}$ has units of length. A closed scattering channel has no propagating outgoing probability current at this total energy.

The incoming probability current density is $j_{\mathrm{in}}=v_a$. The outgoing radial probability current density in channel $i$ is $j_{r,i}\sim v_i|f_{ia}{|}^2/r^2$. Multiplying by the area $r^{2}d\Omega$ and dividing by $j_{\mathrm{in}}$ gives the differential scattering cross-section:The $r^{-2}$ dilution cancels the detector's area growth. The velocity ratio remains: a squared wave coefficient alone is not generally a flux fraction. A differential scattering cross-section has units of area per solid angle, rather than being a normalized directional probability density. For selected scattering events in acceptance $D$, an ideal detector with equal efficiency in all retained channels instead givesThe denominator specifies the event selection; it need not equal one square unit of area. Absolute rates require incident flux, target number and detector efficiency in addition to these relative weights. The next finite-mode model normalizes outgoing probability current to one incoming event and predicts event fractions directly.

Antennas with effective areas $A_{e,1}$ and $A_{e,2}$ receiving the same polarization-matched plane wave have available powers in that ratio. If their matched-load resistances are equal, terminal voltage magnitudes have ratio $\sqrt{A_{e,1}/A_{e,2}}$. With unequal matched resistances the ratio becomes $\sqrt{A_{e,1}{R}_1/(A_{e,2}{R}_2)}$. Open-circuit voltages additionally depend on each antenna's source impedance. Geometry alone therefore fixes available-power estimates more directly than unspecified terminal voltages.

In an isothermal atmosphere with constant density scale height $H$, the equation $\delta {\rho }_{tt}-c^2\delta {\rho }_{zz}-(c^2/H)\delta {\rho }_z=0$ loses its first spatial derivative under $\delta \rho =e^{-z/(2H)}\Psi$. Direct differentiation gives ${\Psi }_{tt}-c^2{\Psi }_{zz}+c^2\Psi /(4H^2)=0$. Thus a plane wave has dispersion relation ${\omega }^2=c^2{k}^2+{\omega }_c^2$, with real acoustic cutoff frequency ${\omega }_c=c/(2H)$. The cutoff term is positive; a minus sign would instead predict growing sufficiently long spatial modes. Vertical propagation requires $|\omega |\ge {\omega }_c$ within this ideal constant-coefficient model.

Using $V=k_0^2(n^2-1)$, the scalar Lippmann-Schwinger equation is $\psi ={\psi }_i+{\int }_D{G}_{k_0}V\psi$. The Born approximation replaces the total field inside the integral by the known incident field:For a unit plane wave, its far-field pattern is $(4\pi {)}^{-1}{\int }_{D}V({\mathbf{r}}^{\prime })e^{i{k}_0({\hat{\mathbf{r}}}_0-\hat{\mathbf{r}})\cdot {\mathbf{r}}^{\prime }}d{\mathbf{r}}^{\prime }$. Thus each incident direction samples a shifted sphere in the Fourier transform of the potential.

For a real linearly polarized monochromatic vacuum plane wave, the local Poynting vector is proportional to ${\cos }^2(\mathbf{k}\cdot \mathbf{r}-\omega t)\hat{\mathbf{k}}$. Its flux through a closed surface generally oscillates at $2\omega$, with zero cycle average, although particular volume geometries can cancel it identically. For a circularly polarized plane wave of constant amplitude the Poynting vector is spatially and temporally constant, so the net flux through every closed surface is exactly zero. A nonzero constant local energy flux must not be confused with the net closed-surface flux.

In a homogeneous medium driven by a plane wave, coherent scattering amplitudes carry phases $e^{i({\mathbf{k}}_i-{\mathbf{k}}_s)\cdot \mathbf{r}}$. An infinite uniform volume cancels arbitrary nonforward components. Finite shapes, boundaries and fluctuations determine any side signal; summing independent-particle intensities is not the continuous-medium limit.

Linearize the Collisionless Boltzmann equation and the perturbation Poisson equation about a homogeneous velocity distribution $f_0$. A plane wave gives $f_1={\varphi }_1\mathbf{k}\cdot {\nabla }_v{f}_0/(\mathbf{k}\cdot \mathbf{v}-\omega )$. Integrating this density and using $-k^2{\varphi }_1=4\pi G\int f_1{d}^{3}v$ yields $1+(4\pi G/k^2)\int [\mathbf{k}\cdot {\nabla }_v{f}_0/(\mathbf{k}\cdot \mathbf{v}-\omega )]d^{3}v=0$. Growing modes have $\mathrm{Im}\omega >0$ and no real-velocity pole; real and damped frequencies require the causal continuation prescription. The Newtonian homogeneous background requires the Jeans swindle.

Positive compression $\alpha$ contributes negative quadratic potential energy $-\alpha {\varphi }_x^2/2$, while positive bending stiffness $\beta$ contributes $\beta {\varphi }_{xx}^2/2$. A plane wave has ${\omega }^2=(\beta k^4-\alpha k^2)/\rho$, so sufficiently long waves grow exponentially. For displacement-fixed, moment-free endpoints at $0,l$, the natural boundary conditions for a free endpoint on the slope give ${\varphi }_{xx}=0$ as well as $\varphi =0$. The modes are $\sin (n\pi x/l)$, and the first Euler buckling of an elastic filament threshold is ${\alpha }_c=\beta {\pi }^2/l^2$. For truly clamped ends, $\varphi ={\varphi }_x=0$, the threshold is instead $4{\pi }^2\beta /l^2$, with first static mode $1-\cos (2\pi x/l)$. More generally,with the infimum taken over the specified endpoint class. At threshold there is a static zero mode and a marginal linear-in-time partner in the linear dynamics; above it the straight configuration is unstable.

Two distinct normalized plane waves on a periodic one-dimensional ring of length $L$ give the displayed projected contact interaction. Momentum conservation excludes pair conversion between these two modes. Four mixed operator orderings give the coefficient of $n_1{n}_2$. At fixed particle number $N$, the energy is $g[N(N-1)+2n_1(N-n_1)]/(2L)$. For repulsive coupling, its minima are the two endpoint Fock states with all particles in one mode; a one-particle energy difference selects the lower mode. This ground-state statement concerns the two-mode projection.

In the plane wave basis of a ring, $V(x)=-\lambda \cos (2\pi x/L)$ has matrix elements $V_{mn}=-(\lambda /2)({\delta }_{m,n+1}+{\delta }_{m,n-1})$. Near $\alpha =1/2$, degenerate perturbation theory on $n=0,1$ givesup to corrections from remote states. Thus the avoided crossing gap is $|\lambda |$ to first order when $|\lambda |\ll E_L$.

For $|\lambda |\gg E_L$, a parabolic approximation about a potential minimum instead gives a local quantum harmonic oscillator spacing $\sqrt{2|\lambda |E_L}$. A local gauge transformation removes the flux from this approximation. Actual ring levels retain exponentially small flux dependence through quantum tunnelling, while nonquadratic corrections to their mean energies are algebraic in $\sqrt{E_L/|\lambda |}$.

The change in incident and outgoing electron wavefunctions caused by the target Coulomb potential. It can invalidate a plane wave treatment in the Born approximation, especially for a large target atomic number.

The de Broglie wavelength is the distance over which a matter wave accumulates one full spatial quantum phase cycle in its propagation direction:Here $h$ is the Planck constant, $\mathbf{p}$ is the momentum, and $\mathbf{k}$ is the angular wavevector. The scalar $p$ in $\lambda =h/p$ means the positive momentum magnitude, not a signed component. Reversing a momentum reverses the wavevector without making the wavelength negative. At zero momentum there is no finite spatial phase period: the limiting wavelength diverges.

This relation makes microscopic diffraction accessible with particles: increasing the momentum shortens the wavelength until it is comparable with the spacings in a crystal scattering problem. The electron accelerating-voltage wavelength gives a practical way to tune this scale, while the Davisson–Germer experiment tests it by comparing electron diffraction directions with independently known crystal geometry. The same relation applies to neutral particles in a neutron interferometer; electric charge is not a requirement for a matter wave.

The wavelength belongs to the wavefunction, not to a sinusoidal path of a particle. A free-particle momentum convention makes the distinction especially clear: a plane wave has a rotating complex quantum phase but a constant probability density. A localized wave packet contains a range of momenta and therefore generally has a range of wavelengths, rather than one exact spatial period.

If a unit incident plane wave has scattered field ${\psi }_s=e^{ikR}{f}_{\infty }(\hat{\mathbf{r}},{\hat{\mathbf{r}}}_0)/R+O(R^{-2})$, the differential wave scattering cross-section isIt is the far-field limit of $R^2$ times the scattered-to-incident energy flux ratio. The common physical amplitude normalization cancels when the incident and scattered waves are in the same background medium. The formula assumes that the $1/(4\pi )$ Green-function factor, if present, is already included in the far-field pattern.

A diffraction grating is a periodic optical structure whose repeated slits or grooves direct light into diffraction orders through wave interference. Its beauty is that many individually broad contributions cooperate to select narrow directions: the spacing determines where the principal peaks occur, while the number of illuminated elements determines how sharply they are selected. This makes a diffraction grating useful in a spectrometer, where separating wavelengths spatially lets us examine an optical spectrum. ^[ref].

Consider a transmission diffraction grating in air, illuminated at normal incidence by a plane wave. Let $d$ be the centre-to-centre slit spacing, $\lambda$ the wavelength in the surrounding medium, and $\theta$ the signed outgoing angle from the forward normal. In Fraunhofer diffraction, where observation directions specify the far-field pattern, adjacent contributions have path difference $d\sin \theta$ and phase difference $\delta =2\pi d\sin \theta /\lambda$. Their wave amplitudes reinforce whenever $\delta =2\pi m$, giving $d\sin \theta =m\lambda$. This normal-incidence geometry should be kept distinct from the incident and reflected angle convention in the grating equation below. ^[ref].

Angular colour separation follows immediately. At fixed nonzero diffraction order, increasing the wavelength moves the peak farther from the normal. With $d=2\mu \mathrm{m}$, first-order light at $450\mathrm{nm}$ leaves at $13.003^{\circ }$, whereas $650\mathrm{nm}$ leaves at $18.966^{\circ }$: a separation of $5.963^{\circ }$. The zero diffraction order stays at $0^{\circ }$ for both wavelengths. Thus grating dispersion comes from an orderly dependence of wave interference on wavelength, rather than an arbitrary assignment of colours to directions.

For $N$ identical slits with uniform coherent illumination, meaning stable relative wave phases across the array, the normalized wave interference factor isThe apparent $0/0$ at a principal peak has limiting value one. A short explanation uses the geometric series: equal wave phases add $N$ equal complex amplitudes; a small directional change distributes those contributions around the complex plane and allows cancellation. With each slit's contribution held fixed, peak scalar wave intensity scales as $N^2$. The normalized plots remove that height change to make the narrowing visible. Stable relative wave phases, or optical coherence, are essential; a sum of independent intensities would lose this array structure. ^[ref].

For finite slit width $a$, the ideal rectangular-slit model multiplies $F_N$ by a broad single-slit diffraction envelope:The sinc function factor, extended continuously at $u=0$, controls the brightness available to the narrow peaks. For example, $a=0.5\mu \mathrm{m}$ gives envelope values about $0.811$, $0.405$, and $0.090$ at diffraction orders $1$, $2$, and $3$ in the $600\mathrm{nm}$ example. If an envelope zero coincides with an allowed diffraction order, that peak disappears. A permitted direction therefore need not produce a visible bright spot. ^[ref].

Historically, Joseph von Fraunhofer used fine parallel wires stretched between threaded rods to make diffraction gratings. Kenyon's historical apparatus account describes both this construction and a later reconstruction with visible light. The example makes the governing idea tangible: regular spacing can turn a simple array of obstacles into an instrument for distinguishing wavelengths. ^[ref].

For Minkowski metric $\mathrm{diag}(1,-1)$, the displayed gamma matrices satisfy the Clifford algebra and ${\gamma }^5={\gamma }^0{\gamma }^1$ has chirality eigenvalues $+1,-1$. Ordering the spinor as $({\psi }_R,{\psi }_L{)}^{\mathsf{T}}$ gives $i({\partial }_t+{\partial }_x){\psi }_R=m{\psi }_L$ and $i({\partial }_t-{\partial }_x){\psi }_L=m{\psi }_R$. A plane wave has the Dirac Hamiltonian $p{\sigma }_3+m{\sigma }_1$, whose square is $(p^2+m^2)1$, proving $E^2=p^2+m^2$.

A dispersion relation specifies which angular frequencies and wavevectors a linear wave model admits. Fix the plane wave conventionHere $a$ is a constant complex amplitude, $\mathbf{k}$ is the wavevector, and $\omega$ is the angular frequency. Substitution into a constant-coefficient linear partial differential equation replaces ${\partial }_t$ by $-i\omega$ and $\nabla$ by $i\mathbf{k}$, turning the wave problem into an algebraic condition. A system can have several roots; $\omega =\omega (\mathbf{k})$ means one chosen branch, rather than a promise that the full dispersion relation is single-valued. For real $\mathbf{k}$, $\mathrm{Im}\omega >0$ means temporal growth in this convention, while $\mathrm{Im}\omega <0$ means damping. The real-branch discussion of phase velocity and group velocity below assumes negligible loss.

For the one-dimensional wave equation $u_{tt}=c^2{u}_{xx}$, the dispersion relation is ${\omega }^2=c^2{k}^2$. The branch $\omega =ck$ carries right-moving travelling waves for $c>0$; both its phase velocity and group velocity are $c$. One may instead use the positive-angular frequency branch $\omega =c|k|$, with right-moving components at $k>0$ and left-moving components at $k<0$. A real wave pairs each $(k,\omega )$ with $(-k,-\omega )$; that conjugate pair represents the same travelling pattern. Branch and sign choices therefore matter when interpreting a dispersion diagram.

A particularly tangible dispersive example is a small-amplitude surface gravity wave on still water of uniform depth $h$. Neglect surface tension and viscosity, assume incompressible flow and irrotational flow, and impose a flat impermeable bottom. With $k>0$ and positive angular frequency,Here $g$ is gravitational acceleration and $\tanh$ is the hyperbolic tangent. Deep water means $kh\gg 1$: $\omega \simeq \sqrt{gk}$ and longer wavelengths have larger phase speed. Shallow water means $kh\ll 1$: $\omega \simeq \sqrt{gh}k$, an approximately nondispersive wave. Thus depth relative to wavelength, rather than depth alone, determines whether crests and wave packets travel together.^[ref]

The dispersion diagram below also includes the ideal deep-water capillary wave limit $\omega =\sqrt{\gamma /\rho }{k}^{3/2}$, where $\gamma$ is surface tension and $\rho$ is mass density. This limit requires the surface tension restoring term to dominate gravity; it reverses the trend of the surface gravity wave, giving larger phase speed at shorter wavelength. The combined gravity-capillary wave has ${\omega }^2=gk+(\gamma /\rho )k^3$.^[ref]

The curves use dimensionless normalizations chosen separately for each model; they are analytical diagrams, not experimental data. Their beauty is that a single dispersion relation encodes several observable behaviours: its value gives the angular frequency, its origin-to-point slope gives the one-dimensional phase velocity, its tangent slope gives the group velocity, and variation of that tangent slope predicts distortion of a wave packet. A straight line through the origin describes a nondispersive wave; an affine branch $\omega =ck+b$ with $b\ne 0$ has constant group velocity but phase velocity $c+b/k$, so those properties must be distinguished.

For bending stiffness $D$ and mass per area $m$, a plane wave of surface displacement $\eta$ satisfies $m{\eta }_{tt}+D{\nabla }_H^4\eta =p$. An inviscid deep-water velocity potential has vertical dependence $e^{kz}$; the kinematic boundary condition and linear Bernoulli equation give $p=({\rho }_w{\omega }^2/k-{\rho }_{w}g)\eta$. Eliminating $p$ gives the displayed dispersion relation. With water depth $d$, replace it by ${\omega }^2=(gk+D{k}^5/{\rho }_w)\tanh (kd)/[1+(m/{\rho }_w)k\tanh (kd)]$.