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UNSW School of Physics documents a Doppler effect experiment using a buzzer based on piezoelectricity, powered by a taped-on $9\mathrm{V}$ battery, and a computer's built-in microphone. Audacity records the acoustic wave. First record the stationary source to establish its baseline frequency; then attach it to a bicycle passing the stationary microphone. The published spectrogram shows a rapid decrease in frequency during passage.^[ref]

For numerical readout, select an approaching or receding interval, display its frequency spectrum using Plot Spectrum, and read the interpolated Peak frequency. The protocol recommends repeating with slightly different selections to test stability, and working outdoors to reduce reflections and standing waves. Longer windows refine frequency spacing while averaging over more motion. The baseline controls for the source's actual tone; repeated selections expose analysis-window sensitivity.^[ref]

Under nearly collinear constant-speed approach, infer $v=c(1-f_s/f_o)$ from the classical Doppler effect for sound. For recession the positive speed is $v=c(f_s/f_o-1)$. These inversions need a specified speed of sound. An offset path changes the projection, as in Doppler frequency along a trajectory; neither inversion measures the full speed near closest passage. The protocol's controls cannot remove arbitrary acceleration, background measurement noise or unresolved reflected paths. No new measurements are claimed here.

Dot the two Linear Alfvén wave equations with ${\mathbf{u}}_{\perp }$ and ${\mathbf{b}}_{\perp }/{\mu }_0$, respectively, and add. The result is conservation of energy withA single traveling Alfvén wave has equal kinetic energy and perturbation magnetic energy, and $F_z=\pm v_A\mathcal{E}$. This equality need not hold instantaneously at each point of a superposed standing wave.

As an original ideal-model example, stipulate $B_0=1.00\mathrm{T}$, $\rho =10^4\mathrm{kg}{\mathrm{m}}^{-3}$ and a uniform path $L=0.20\mathrm{m}$. Using ${\mu }_0\simeq 4\pi \times 10^{-7}\mathrm{H}{\mathrm{m}}^{-1}$, the Alfvén speed is $8.92\mathrm{m}{\mathrm{s}}^{-1}$ and the transit time $L/v_A$ is $22.4\mathrm{ms}$. Doubling the magnetic field halves that time. A sinusoidal velocity with peak amplitude $0.010\mathrm{m}{\mathrm{s}}^{-1}$ has magnetic peak amplitude $1.12\mathrm{mT}$ and mean energy flux $⟨F_z⟩=\rho v_A{U}^2/2=4.46\mathrm{W}{\mathrm{m}}^{-2}$ in the positive direction. These are stipulated ideal parameters, not historical measurements or a prediction including vessel walls.

An antinode is a position of maximum oscillation amplitude in a standing wave. For $u_n=A\sin (n\pi x/L)\cos ({\omega }_{n}t+\phi )$, the antinodes occur atwhere the amplitude is $|A|$. They sit halfway between consecutive nodes of a standing wave. For the first normal mode, the sole antinode is $x=L/2$. It reaches zero displacement at a quarter oscillation period while moving fastest, which illustrates why an antinode is defined by oscillation amplitude rather than instantaneous height.

In 1982, Alain Aspect, Jean Dalibard and Gérard Roger used a calcium cascade and an acoustic standing wave in water to route each photon toward one of two fixed polarizers. Switching took about $10\mathrm{ns}$, against $40\mathrm{ns}$ for light to cross between stations. Photomultiplier coincidence circuit rates, with delayed-background subtraction and polarizer-removal normalization, violated their Bell inequality. Supplementary polarizers checked each channel against Malus's law. The timing attacked the locality loophole, but the authors emphasized that switching was quasiperiodic, not random; incomplete switching and inefficient collection also limited the test. Its normalized statistic differs from the CHSH inequality convention used below.^[ref]

A concrete quantum quench experiment loaded a Bose-Einstein condensate of ${}^{87}\mathrm{Rb}$ chemical atoms from a magnetic field trap into an optical lattice: three orthogonal $838\mathrm{nm}$ laser standing waves, with approximately $125\mu \mathrm{m}$ beam waists. An $80\mathrm{ms}$ loading ramp preceded a $50\mu \mathrm{s}$ increase in lattice depth. This suppressed quantum tunnelling while retaining each well's vibrational ground state. The switch preserved atom-number populations rather than freezing every motional degree of freedom.^[ref]

After variable hold times, releasing the traps and taking photon absorption images after $16\mathrm{ms}$ of flight revealed disappearance and return of interference peaks. Background-subtracted peak counts supplied a coherent fraction. Varying final depth tested the predicted revival period $h/U$; varying initial depth tested the collapse's sensitivity to atom-number statistics. Peak broadening outside counting windows and spatial variation of $U$ limited later signals.^[ref]

Here $U$ is the on-site interaction energy. The approximate isolated-well Hamiltonian operator $H=U\hat{n}(\hat{n}-1)/2$, with number operator $\hat{n}$, gives different quantum phases to its number states, which later rephase. This is a many-level realization of the distinction between fixed populations and changing quantum phases.^[ref]

For the ideal uniform wave equation on a string, the wave speed is $c=\sqrt{\tau /\mu }$, where $\tau$ is constant tension and $\mu$ is linear mass density. Assume small transverse slopes, negligible bending stiffness and no damping. Fix both ends at zero displacement, giving the homogeneous Dirichlet boundary conditions $y(0,t)=y(L,t)=0$. Our standing wave already satisfies the first. For a nonzero amplitude, the second requires $\sin (kL)=0$, henceThe value $n=0$ gives zero displacement in this sine family, not a vibrating normal mode. Each positive $n$ fits $n$ half-wavelengths into the string: it has $n$ antinodes, $n-1$ interior nodes of a standing wave, and two endpoint nodes of a standing wave. For $L=1\mathrm{m}$ and $c=20\mathrm{m}{\mathrm{s}}^{-1}$, $f_n=10n\mathrm{Hz}$; the third normal mode has $f_3=30\mathrm{Hz}$ and wavelength $2/3\mathrm{m}$.

The equally spaced eigenfrequencies follow from this string's uniform wave speed and fixed-end boundary conditions. They are not a universal spectrum for every vibrating object. Changing an endpoint constraint changes the allowed mode shapes; changing tension changes wave speed. At fixed $L$ and $\mu$, quadrupling $\tau$ doubles every $f_n$. The more general construction of arbitrary initial motion belongs to normal mode and wave equation on a string.

A physicist.^[ref]

Lippmann photography records colour through the depth structure of an exposed emulsion. Its account connects a standing wave to a reflected optical spectrum.

The guided wavelength is the distance for a $2\pi$ change of axial wave phase in a propagating normal mode. For the vacuum-filled rectangular electromagnetic waveguide,for ${\mathrm{TE}}_{10}$, with ${\lambda }_0=c/f$ and $f>f_c$. It exceeds the vacuum wavelength at the same frequency. This longer axial spacing is consistent with $v_{\mathrm{ph}}>c$: multiplying ${\lambda }_g$ by $f$ gives the phase velocity, not the group velocity.

Two oppositely traveling fields at the same frequency form a standing wave. For a perfect electric short at $z=0$, the total electric field is proportional to $\sin (\beta z)\sin (\pi x/a)$, so successive axial electric nodes are separated byThus neighboring minima in a slotted line scan give half a guided wavelength. In the original $a=30\mathrm{mm}$, $f=1.25f_c$ example, $d=40\mathrm{mm}$ implies ${\lambda }_g=80\mathrm{mm}$, not $40\mathrm{mm}$. Combined with the chosen $f$, the inferred broad dimension is $a={\lambda }_0/[2\sqrt{1-({\lambda }_0/{\lambda }_g{)}^2}]=30\mathrm{mm}$. Below cutoff, $\beta$ is imaginary and $2\pi /\beta$ is not a physical periodic spacing.

A hard wall at the origin and an attractive delta potential at distance $a$ form a leaky quantum cavity. For unit incident amplitude from outside, the interior standing wave has coefficient $A=-i\lambda x/[1-e^{2ix}+i\lambda x]$, where $x=ka$ and $\lambda ={\hslash }^2/(mqa)>0$. Its modulus squared is the title expression. A resonance pole $p_n=x_n-i{\gamma }_n$ produces an enhancement with a locally Lorentzian line shape, proportional to $[(x-x_n{)}^2+{\gamma }_n^2{]}^{-1}$. At fixed low $n$ and small $\lambda$, $x_n\simeq n\pi (1+\lambda /2+{\lambda }^2/4)$ and ${\gamma }_n\simeq (\lambda n\pi /2{)}^2$.

The coefficient can exceed one because it measures a standing-wave amplitude, while the net interior probability current is zero. Conservation of probability current gives unit reflected flux: no current passes through the hard wall. Near zero, $|A{|}^2\to {\lambda }^2/(2-\lambda {)}^2$ when $\lambda \ne 2$; the threshold case $\lambda =2$ requires a separate limit.

In his 1888 reflection experiments, Heinrich Hertz excited a straight conductor with a central spark gap driven by an induction coil. The receiver was a wire circle of $35\mathrm{cm}$ radius with an adjustable small spark gap; tiny sparks were observed in darkness. There was no connecting transmission wire. ^[ref]

He removed hanging metal fixtures, installed a $4\mathrm{m}$ by $2\mathrm{m}$ zinc reflecting sheet, and moved and rotated the receiver. Spark strength alternated with position and gap orientation, supporting specular reflection and wave interference. He checked observations from farther away because the observer's body perturbed the response. Changing oscillator dimensions changed the spatial pattern. ^[ref]

These tests supported propagating electromagnetic waves, rather than mere detection of nearby induction. They did not establish unlimited reception distance. His wire-versus-air wave interference experiment addressed finite velocity; he later judged its quantitative velocity comparison doubtful because of room effects. ^[ref]

The physical inference combines transmission with a spatially structured response to a reflected disturbance. In an ideal monochromatic incident-and-reflected model, standing wave nodes are separated by $\lambda /2$; combining a reliable wavelength with independently known frequency gives phase velocity $v=\nu \lambda$. A spark is a threshold event, however: its presence or length is not a calibrated wave intensity or a direct arrival-time measurement. The actual loop also responds to its orientation and finite size, so the simple standing-wave model is an explanation of the test's logic, rather than a reconstruction of Hertz's measured points. Poynting theorem explains how source energy can reach and drive the receiver's discharge.

A Kundt's tube turns an invisible standing wave into a measurable spatial pattern. A driven air column supports a longitudinal wave; measuring its frequency and the spacing of repeated nodes of a standing wave gives the speed of sound. The same drive can compare the wavelengths in a gas and a vibrating solid. August Kundt's 1866 paper introduced acoustic dust figures; its footnote explicitly counts successive nodes of a standing wave as half a wavelength.^[ref]

The attractive feature is the separation of measurements: a clock or frequency counter measures repetition in time, while a ruler measures repetition in space. Their product supplies a propagation speed. The pressure pattern and the fluid-motion pattern are different, however; identifying which one a detector responds to prevents a quarter-wavelength mistake.

In his 14 December 1908 Nobel lecture, Gabriel Lippmann described a photographic plate bearing a transparent, even, extremely fine sensitive layer. A holder put Mercury against that layer as a mirror during exposure. Incident and reflected light formed a standing wave within the photographic emulsion; after photographic processing and drying, white light revealed colour by selective reflection. The reflector served during recording; the developed structure supplied the later colour selection.^[ref]

The lecture's control was to project a spectrum photograph, wet it, and project it again. Its colours disappeared as the gelatin swelled, then returned during drying, red before the shorter wavelengths. This tested a spacing-dependent explanation. Lippmann also reported roughly one-minute sunlight exposures and identified portrait exposure time as a limitation.^[ref]

A reversible colour change supports a structural explanation, but does not independently measure every fringe spacing or the refractive index. For a conceptual comparison, record a narrow band and compare reflected optical spectra at fixed illumination and viewing angles; an unexposed processed plate tests background. A second comparison changes only viewing geometry. A third compares the dry state before and after a reversible spacing change. These are proposed discriminating controls, not additional experiments attributed to Lippmann. Background surface reflection must be separated from the selective signal. A photograph's appearance also depends on the illuminating optical spectrum; viewing under an arbitrary lamp is not a calibrated spectral measurement.

At normal incidence, let $z$ measure distance from the reflector into a uniform transparent layer with refractive index $n$. For a monochromatic wave of vacuum wavelength ${\lambda }_0$, the layer wavenumber is $k=2\pi n/{\lambda }_0$. Represent the incident and returning electric fields by complex wave amplitudes with time factor $e^{-i\omega t}$ suppressed:The time-averaged squared electric field, normalized to the incident contribution, isThis electric exposure pattern has maxima separated byThe wave phase of the reflector shifts the maxima's positions but leaves their spacing unchanged in this uniform-layer model. The electric pattern, rather than the zero net energy flux of an ideal standing wave, is the relevant exposure proxy. An imperfect mirror reduces its contrast: $(I_{\max }-I_{\min })/(I_{\max }+I_{\min })=2|r|/(1+|r{|}^2)$, where $I$ denotes this exposure proxy.

Choose ${\lambda }_0=600\mathrm{nm}$ and $n=1.5$. Then the medium wavelength is $400\mathrm{nm}$ and $d=200\mathrm{nm}$. For an ideal phase-reversing reflector, $r=-1$ and the normalized pattern is $4{\sin }^2(kz)$: its first two maxima are at $z=100$ and $300\mathrm{nm}$. These are chosen calculations, not measured properties of a historical photographic emulsion or the optical response of Mercury.

For multiple mutually incoherent wavelengths, time averaging adds exposure contributions rather than their instantaneous electric fields. In a linearized recording model,Here $S$ is incident spectral exposure density, $q$ a wavelength-dependent recording sensitivity and $D$ a model exposure response. An optical spectrum therefore writes overlapping depth oscillations, not necessarily visibly separated sheets. This is a useful connection to the Fourier transform: approximately, depth is paired with optical spatial frequency $2n/{\lambda }_0$. Finite recording depth, varying sensitivity and nonlinear photographic processing prevent treating this expression as an exact inversion rule for an actual plate.

A Lippmann plate records colour as a depth-dependent optical structure in a photographic emulsion. A standing wave writes a spacing; wave interference during later illumination selects the reflected wavelengths. This makes a photographic plate a spatial record of an optical spectrum. The striking idea is that information about colour can live in geometry.

Gabriel Lippmann presented his colour process in 1891 and received the 1908 Nobel Prize in Physics. His career belongs principally to physics: he held chairs in mathematical and experimental physics.^{[ref] [ref]}

Stig Lundquist, the author printed as S. Lundquist, reported an early laboratory test in 1949. The apparatus used mercury in a stainless-steel cylinder, a vibrating disk, and a floating mirror to indicate the motion at the liquid surface. The reproduced apparatus drawing also shows a scale; the accompanying account gives an applied magnetic field of $1.3\mathrm{T}$.^[ref]

The imposed magnetic field ran along the cylinder, while the bottom disk oscillated to excite azimuthal motion. Varying the drive frequency sought a standing wave resonance. A later authored experimental account describes this as an early hint: the resonance peak was not reached, and magnetic damping was substantial.^[ref] The experiment therefore tested a bounded azimuthal shear disturbance, related to a torsional Alfvén wave, rather than an infinite plane wave or a compression-driven magnetosonic wave.

The magnetic field provides the controlled physical parameter: the ideal Alfvén wave prediction makes a propagation speed proportional to its strength, while a resonance also depends on the vessel's boundary conditions. A quantitative test must compare the driven response with finite-electrical conductivity theory and account for mechanical coupling and wall losses. An observed surface motion alone cannot distinguish all these mechanisms. The reproduced apparatus account gives neither a numerical drive frequency nor a field-off comparison. Lundquist's paper itself calls the experiment preliminary and proposes further investigations.^[ref]

For the wave equation ${\psi }_{tt}=c^2{\psi }_{xx}$ on $0<x<L$, impose homogeneous Neumann boundary conditions ${\psi }_x(0,t)={\psi }_x(L,t)=0$, where $c,L>0$. Separation of variables gives $X^{\prime \prime }+\lambda X=0$ and $T^{\prime \prime }+c^2\lambda T=0$. Integration by parts gives ${\int }_0^L(X^{\prime }{)}^{2}dx=\lambda {\int }_0^L{X}^{2}dx$, so every nontrivial spatial eigenfunction has $\lambda \ge 0$.

For $\lambda =0$, $X$ is constant and $T=A_0+B_{0}t$. For $\lambda >0$, the boundary conditions select ${\lambda }_n=(n\pi /L{)}^2$ and $X_n=\cos (n\pi x/L)$. The superposition principle gives the displayed expansion when the initial data have sufficient regularity. The constant zero mode must be retained: homogeneous Neumann boundary conditions do not force the spatial average of the displacement to vanish.

With zero initial velocity, all $B_n$ vanish, including $B_0$. Initial displacement $a+b\cos (n\pi x/L)$ then evolves as $a+b\cos (cn\pi t/L)\cos (n\pi x/L)$, a standing wave above a constant offset.

Gabriel Lippmann received the award for a photographic colour-recording method based on interference.^[ref] The Lippmann colour-recording experiment stores a standing wave pattern in a photographic layer.

A node of an ideal standing wave is a position whose oscillation amplitude is zero: its displacement stays zero for all time. For a nontrivial normal mode $u(x,t)=\varphi (x)q(t)$, this means $\varphi (x)=0$. A momentary zero caused by $q(t)=0$ does not make every position a node. In more dimensions the zeros of a mode shape can form curves or surfaces.

For the $n$th normal mode of a uniform fixed-end wave equation on a string, the nodes of a standing wave areThere are $n+1$ nodes of a standing wave including the two endpoints, and $n-1$ interior ones. For $n=3$ and $L=1\mathrm{m}$, they lie at $0$, $1/3$, $2/3$, and $1\mathrm{m}$. Each remains stationary through every snapshot in the normal mode diagram; an antinode instead marks a maximum oscillation amplitude. ^[ref]

At normal incidence, the reflected electric amplitude is the negative of the incident amplitude at the surface, while the reflected magnetic amplitude has the same sign. Their superposition forms a standing wave with an electric node and a magnetic antinode at the conductor.

Consider a noninteracting electron in a finite tight-binding model with open ends:The vectors $|n⟩$ form an orthonormal basis of site states. The hopping energy is $t$; it is not the time coordinate. Draw each ${ϵ}_n$ independently from the continuous uniform distribution on $[-W/2,W/2]$ and then hold it fixed. The disorder strength $W/t$ is dimensionless. Open ends mean no hopping bond connects sites $N$ and $1$; equivalently set ${\psi }_0={\psi }_{N+1}=0$. There are no electron interactions, phonons or wave absorption in this model.

The eigenvalue equation in the interior isAs a simple local calculation, if a state's peak is at $n_0$, a neighbouring site with $|E-{ϵ}_n|\gg t$ receives a small leading contribution ${\psi }_n\simeq -t{\psi }_{n_0}/(E-{ϵ}_n)$. Several successive large mismatches multiply small factors. This explains why large disorder can restrict spreading, but is not a proof of Anderson localization: near-resonant denominators and alternative interference paths prevent treating this expansion as uniformly valid.

For the first original calculation, take $N=80$, $t=1$, $W/t=6$ and the fixed random seed $20261015$. Diagonalizing the resulting real symmetric matrix with numpy.linalg.eigh and selecting the eigenvalue closest to zero gives $E/t\simeq -0.05962$, a probability peak at site $29$, and $P\simeq 6.0031$. The selected eigenvector has total probability one. This is one chosen realization, not a disorder average or a measured specimen.

The clean comparison needs care. With $W=0$, the open chain has standing wavesFor even $N$, the identity ${\sin }^{4}x=(3-4\cos 2x+\cos 4x)/8$ and finite trigonometric sums give ${\sum }_n|{\psi }_n^{(m)}{|}^4=3/[2(N+1)]$, hence $P=2(N+1)/3$. Thus the clean $80$-site comparison has $P=54$, although it extends across the chain. Extended does not mean uniform probability. For odd $N$, the special central mode $m=(N+1)/2$ has zeros on alternating sites and instead $P=(N+1)/2$; the even-size assumption avoids that exception.

The second original calculation compares $N=40$ and $80$ over $W/t=0,0.5,1,2,3,4,6,8$. For each size, draw $24$ independent site patterns using seed $20261016$, with the size included in the random-seed construction. Scale each fixed pattern across disorder strengths. At every point, average $P$ over the four energy eigenstates closest to $E=0$. The solid line is the median across these sample averages; shading spans their $10$th–$90$th percentiles. The selected states can change as $W/t$ varies: lines connect one sample's selections rather than tracking a single mode.

At $W/t=0$ the two clean values are $27\frac{1}{3}$ and $54$. At $W/t=8$, the computed median sample averages are approximately $3.17$ and $3.01$. In the latter regime, doubling $N$ hardly changes the typical occupied site count, so $P/N$ falls. At weak disorder the larger chain permits more spatial extent. Neither a single curve nor the overlap of finite-size bands proves the infinite-system statement under mobility edge; a long localization length can hide confinement in either size.