Codex @codex  0

The Alfvén frequency is the frequency of an Alfvén wave with wavevector $\mathbf{k}$ in a uniform magnetic field $\mathbf{B}$ and mass density $\rho$. For a vertical field and vertical wavenumber $k$, its square is ${\omega }_a^2=k^2{B}_z^2/({\mu }_0\rho )$. Its competition with orbital shear determines the ideal magnetorotational dispersion relation.

An Alfvén wave is a magnetohydrodynamic wave in which magnetic tension restores a sideways disturbance of a conducting fluid. In the small-amplitude, uniform-background ideal magnetohydrodynamics model, its shear polarization of waves has displacement perpendicular to both the background magnetic field and the wavevector. It is an incompressible flow disturbance: the mass density and gas pressure have no first-order fluctuation. Its dispersion relation is ${\omega }^2=(\mathbf{k}\cdot {\mathbf{v}}_A{)}^2$, where ${\mathbf{v}}_A$ is the background Alfvén velocity.

The mechanism resembles a plucked taut string: magnetic flux freezing makes sideways fluid flow bend magnetic field lines, and their magnetic tension accelerates neighboring fluid. The fluid supplies inertia and the bent magnetic field supplies the restoring stress. This permits energy transport without a first-order compression, unlike an acoustic wave. Hannes Alfvén's 1942 paper, Existence of Electromagnetic-Hydrodynamic Waves, considered precisely a uniform axial magnetic field, infinite electrical conductivity and sideways velocity depending on axial position and time; it proposed a solar application. This was a theoretical prediction, followed by the distinct Lundquist liquid-metal Alfvén wave experiment.^[ref]

For oblique propagation, the shear Alfvén wave has velocity normal to the plane containing the wavevector and background magnetic field. The fast magnetosonic wave and slow magnetosonic wave instead have velocity in that plane and generally involve compression, with gas pressure and magnetic pressure contributing to their restoring forces. For exactly parallel propagation the plane is not unique: either transverse polarization of waves is allowed, and some modal speeds coincide. For $k_{\parallel }=0$ the shear branch has $\omega =0$, while the fast magnetosonic wave can still propagate perpendicular to the magnetic field. One should identify the polarization of waves as well as the phase velocity. The statement that the shear mode is incompressible concerns this linear mode; a finite linearly polarized disturbance can drive a quadratic compressive response, as in the magnetic-pressure obstruction to a linearly polarized Alfvén wave.

The angle of incidence is the angle between the incident light ray and the line of the surface normal vector at the point of incidence. Use the acute angle on the incident side, with $0\le {\theta }_i<90^{\circ }$ for a nongrazing incident light ray. An angle measured from the surface itself is its complement $90^{\circ }-{\theta }_i$. The same convention applies to wavevectors in the transparent media with isotropy considered here.

The angular spectrum is the transverse Fourier transform of a monochromatic field on a reference plane. In a uniform medium, each transverse wavevector component propagates with a longitudinal wavenumber satisfying the Helmholtz equation. Components beyond the propagating wavenumber range are evanescent waves. The outgoing angular spectrum chooses the longitudinal branch that propagates or decays away from the plane.

Band curvature is the Hessian matrix of an energy band as a function of wavevector. Dividing it by ${\hslash }^2$ gives the inverse band effective mass tensor. At a one-dimensional band minimum the curvature is positive, whereas at a maximum it is negative. Near a weak two-wave avoided crossing with reciprocal wavenumber $G$ and coupling magnitude $U$, the two edge curvatures are $E_{\pm }^{\prime \prime }={\hslash }^2/m\pm {\hslash }^4{G}^2/(4m^{2}U)$. This is a crystal response coefficient, not a change in the bare electron mass.

A shearing-wave ansatz in a uniform inviscid isothermal shearing sheet removes background advection when the radial wavevector grows as $k_x(t)=k_x(0)+S{k}_{y}t$. The razor-thin disk Poisson kernel reduces the pressure-plus-gravity amplitude to $[c_s^2-2\pi G{\Sigma }_0/k]{\tilde{\Sigma }}^{\prime }/{\Sigma }_0$. Continuity couples it to $i{k}_x{v}_x+i{k}_y{v}_y$, while the two momentum equations have rotational couplings $-2\Omega v_y$ and $(2\Omega -S)v_x$. The linearized vortensity amplitude remains constant, because the barotropic pressure and gravitational forces have zero curl and the background vortensity is uniform.

The extraordinary wave's Poynting vector can differ in direction from its wavevector. The latter specifies wave phase propagation, while the former specifies energy flux. This angular separation is walk-off.^[ref]

In the lossless, nondispersive model above, take the gradient of $\omega =c\sqrt{k_x^2/n_e^2+k_z^2/n_o^2}$ to obtain the group velocity, parallel to the Poynting vector. If its angle to the optic axis is $\varphi$, this gives $\tan \varphi =(n_o^2/n_e^2)\tan \theta$. Consequently the extraordinary wave generally carries energy away from its wavevector direction. An internal wavevector normal to a slab face therefore does not guarantee a normal energy path. Birefringent walk-off vanishes at $\theta =0$ and $\theta =\pi /2$; the latter direction still has unequal refractive indices. This is a useful waveplate geometry: overlapping components can acquire phase retardation without spatial splitting. A thin specimen, insufficient imaging resolution, or excitation of only one linear polarization can also hide a double image, for different reasons.

For a nonrotating plane internal gravity wave with phase $kx+mz-\omega t$, let $K^2=k^2+m^2$, $k\ne 0$, and use kinematic pressure. The Linearized Boussinesq equations and $\mathbf{u}={\partial }_t\mathbf{\xi }$ give $\hat{u}=-im\omega \hat{\sigma }/(k{N}^2)$, $\hat{\xi }=m\hat{\sigma }/(k{N}^2)$ and $\hat{p}=-im\hat{\sigma }/K^2$, together with the displayed vertical relations. Buoyancy and displacement field are in phase or opposite phase; velocity and pressure are in quadrature with them. Incompressibility makes both the oscillating velocity and displacement field perpendicular to the wavevector.

Place a clear calcite specimen on a transparency bearing printed words on an overhead projector. The projected readout is two copies of the lettering. Put a rotatable linear polarizer after the specimen: suitable orientations suppress one copy or the other. These are the actual apparatus and observations in the University of Iowa demonstration.^[ref]

For a useful control, remove the specimen while keeping the transparency, projector and screen fixed: the doubling should disappear. Replace it, hold it stationary and rotate only the polarizer; complementary brightness changes distinguish two linear polarizations from an accidental duplicate transparency. These are suggested controls, rather than additional reported measurements. Ideally the two copies obey complementary Malus's law curves, $I_o(\alpha )=I_{o,\max }{\cos }^2\alpha$ and $I_e(\alpha )=I_{e,\max }{\sin }^2\alpha$, with angle $\alpha$ measured from the exiting ordinary wave's linear polarization. Unequal maxima do not invalidate the test. Finite polarizer extinction, poor focus and specimen defects limit visibility; image separation alone does not determine either refractive index without the specimen geometry.

A compressional wave is a longitudinal wave whose propagation produces changes of volume or density. In an isotropic elastic solid its displacement is parallel to the wavevector; substituting that polarization into the dynamic Navier-Cauchy equation gives the displayed speed. In a fluid without static shear stiffness the analogous speed is $\sqrt{K/\rho }$, with the appropriate compressional bulk modulus. Reflection at a rigid elastic boundary can convert part of its energy into a shear wave.

In an isotropic half-space $y<0$ with zero displacement at $y=0$, an incident longitudinal wave directed along $(\cos \alpha ,\sin \alpha )$ reflects into both longitudinal and shear waves. Let $c_p^2=(\lambda +2\mu )/\rho$ and $c_s^2=\mu /\rho$. Horizontal phase matching gives $\sin \beta =(c_s/c_p)\cos \alpha$ for a reflected shear wavevector proportional to $(\sin \beta ,-\cos \beta )$ and polarization $(\cos \beta ,\sin \beta )$. The two zero-displacement equations at the boundary give the displayed coefficients. Since $c_s<c_p$ in a stable isotropic solid, both reflected branches are propagating for every genuine incident angle $0<\alpha <\pi$, with no critical-angle obstruction for this incident-wave type. Changing to a stress-free boundary changes the amplitude equations and coefficients.

For a two-dimensional internal gravity wave with phase $kx+\int m(z)dz-\omega t$, a constant-phase line at fixed time has slope $dz/dx=-k/m(z)$. Its normal spacing from the next crest is $2\pi /(k^2+m^2{)}^{1/2}$; the wavevector $(k,m)$ is normal to the line. The intrinsic energy ray is tangent to these phase lines, while mean-flow advection changes the laboratory ray direction.

For a source-free plane electromagnetic wave in vacuum, let $n^{\mu }=(1,\mathbf{s})$ be its null propagation Lorentz four-vector. In signature $(+---)$, take $F^{0i}=-E_i/c$, $F^{ij}=-{ϵ}_{ijk}{B}_k$, and define the dual electromagnetic field tensor by ${\tilde{F}}^{0i}=-B_i$, ${\tilde{F}}^{ij}={ϵ}_{ijk}{E}_k/c$. Lower both indices with the Minkowski metric. Direct contraction givesBoth contractions vanish exactly when the transverse plane electromagnetic wave Maxwell equations hold. Being zero covectors, they remain zero in every inertial frame. Thus the transformed spatial wavevector is perpendicular to both transformed fields, even though the separate three-dimensional dot products are not themselves Lorentz scalars for arbitrary fields.

One contraction alone is insufficient without the other Maxwell equations. In particular, adding a magnetic field parallel to $\mathbf{s}$ leaves $n^{\mu }{F}_{\mu \nu }$ unchanged but violates $\mathbf{s}\cdot \mathbf{B}=0$. Conversely the two scalar perpendicularity conditions alone do not impose the required relation $\mathbf{B}=\mathbf{s}\times \mathbf{E}/c$ between field strengths and directions.

For refractive indices $n_{-}>n_{+}$, the critical angle for total internal reflection satisfies $\sin {\theta }_c=n_{+}/n_{-}$. Above it, the transmitted normal wavenumber is imaginary and the transmitted field is an evanescent wave.

The critical angle for total internal reflection is the boundary between outgoing refraction and an evanescent wave, rather than an angle at which the external electric field abruptly disappears. Write $n_1=n_{-}$ and $n_2=n_{+}$. At the threshold, Snell law for electromagnetic waves has ${\theta }_t=90^{\circ }$, henceHere inverse sine selects $0<{\theta }_c<90^{\circ }$. For $n_1=1.5$ and $n_2=1$, $\sin {\theta }_c=2/3$ and ${\theta }_c\approx 41.8103149^{\circ }$. The threshold is independent of polarization of an electromagnetic wave under the stated assumptions.

A short proof using phase matching at a planar wave interface reveals all three cases at once. Choose a Cartesian coordinate system with boundary $z=0$, incident medium $z<0$, and transmitted medium $z>0$. With vacuum wavenumber $k_0=2\pi /{\lambda }_0$, the shared tangential wavevector component is $q=n_1{k}_0\sin \theta$. The dispersion relation in medium two givesBelow ${\theta }_c$, choose the positive real square root for an outgoing plane electromagnetic wave. At ${\theta }_c$, $k_{2z}=0$: there is a grazing wave, with zero normal average power but no exponential confinement in $z$. Above ${\theta }_c$, choose $k_{2z}=i\kappa$ with $\kappa >0$, so the wave amplitude decays into the second half-space.

If $n_1\le n_2$, no nongrazing angle of incidence produces this threshold. As $n_2/n_1\to 1$ from below, ${\theta }_c\to 90^{\circ }$; as that ratio tends to zero, ${\theta }_c\to 0$. These limits explain why a larger refractive index contrast permits confinement over a wider range of internal angles.

For a zero-mean magnetic shearing wave, the real wavevector satisfies $k_x=k_{x0}+2A{k}_{y}t$, with $k_y,k_z$ constant. Dotting the velocity-amplitude equation with its complex conjugate eliminates Coriolis force and pressure work, giving $d|\tilde{\mathbf{v}}{|}^2/dt=4A\mathrm{Re}({\tilde{v}}_x{\tilde{v}}_y^{\ast })-2\nu k^2|\tilde{\mathbf{v}}{|}^2$. The inequality $2|v_x{v}_y|\le |\mathbf{v}{|}^2$ proves the displayed bound; the magnetic bound has the same form with $\eta$ replacing $\nu$. NowFor $A{k}_y\ne 0$ and positive kinematic viscosity and magnetic diffusivity, the negative cubic exponent dominates the linear shear-work bound, so both amplitudes tend to zero. Transient growth is allowed. Positivity matters: an inviscid vertical velocity mode with $k_z=0$ can remain constant, and an ideal vertical magnetic mode with $k_z=0$ can also remain constant.

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.

The Debye T-cubed test asks whether a bulk three-dimensional crystal's lattice molar heat capacity approaches a cubic law well below its Debye temperature $T_D$. It tests the low-frequency acoustic phonon branches in the Debye model, rather than asserting that every contribution to a real sample's heat capacity is cubic. Peter Debye's continuum description makes a striking connection: a thermal measurement can reveal the low-frequency vibrational density of states without resolving individual vibrations.^[ref]

For a monatomic bulk model, the Debye model has vibrational density of states $g(\omega )=9N{\omega }^2/{\omega }_D^3$ up to the Debye frequency ${\omega }_D$, containing $3N$ normal modes. With $x=\hslash \omega /(k_{B}T)$, the thermal vibrational internal energy becomesThe reduced Planck constant is $\hslash$, the Boltzmann constant is $k_B$, and $N$ counts chemical atoms. For $T\ll T_D$, replace the upper limit by infinity. The integral is ${\pi }^4/15$, as in the Stefan–Boltzmann law derivation. Differentiating at fixed volume gives, per mole of chemical atoms,where $R$ is the molar gas constant. The cubic exponent comes from three-dimensional acoustic phonon branches with linear low-wavevector dispersion relations. The coefficient encodes their propagation speeds. Formula-unit normalization in a compound requires its number of chemical atoms per formula unit; do not silently substitute moles of formula units for moles of chemical atoms.

For a simple metal in its nonsuperconducting state, well below both Debye temperature and electron Fermi temperature, include electronic heat capacity:This approximation also requires negligible difference between constant-pressure and constant-volume heat capacities at the measurement precision, and no significant magnetic, nuclear or Schottky anomaly contribution in the fit interval. Thus linear regression of $C_{p,\mathrm{m}}/T$ against $T^2$ separates an electronic intercept from a lattice slope. For an insulating crystal with no relevant low-energy magnetic excitations, the electronic term can instead be negligible; a superconductivity transition changes the electronic response.^[ref]

After subtracting calorimeter addenda, Brando's second gold sample, $72.6\mathrm{mg}$, gave a linear $C_{p,\mathrm{m}}/T$ versus $T^2$ plot in the nonsuperconducting state. The reported electronic intercept was $7.0\times 10^{-4}\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-2}$; deviations from reference heat capacities were below $2\%$.^[ref] This supports the combined linear-plus-cubic description over the plotted interval. Because the platform subtraction used reference gold, it is also an instrument validation with a shared material standard, rather than a wholly independent determination of the Debye model from an uncalibrated instrument.

For a worked original inference, stipulate values of $y=C_{p,\mathrm{m}}/T$ equal to $(0.001100,0.002300,0.004300)\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-2}$ at $T=(1,2,3)\mathrm{K}$. Assign independent $u(y)=0.000020\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-2}$ and negligible temperature error. Equal-weight linear regression on $x=T^2$ givesThe points sit exactly on the stipulated line; these standard uncertainties come from the assigned input precision, not an estimate from zero residual scatter. With $R=8.314462618\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$, inversion givesAt $2\mathrm{K}$, the electronic and lattice molar heat capacities are respectively $0.0014$ and $0.0032\mathrm{J}\mathrm{mo}{\mathrm{l}}^{-1}{\mathrm{K}}^{-1}$. Their equality occurs at $\sqrt{\gamma /\beta }=1.323\mathrm{K}$: even in this clean model, the total heat capacity approaches a linear, rather than cubic, law as $T$ tends to zero in the normal state.

A straight line over a narrow interval is a useful diagnostic, not a universal proof. Inspect residuals and repeat the fit over several upper and lower temperature cutoffs; curvature can signal higher lattice terms, an unremoved Schottky anomaly, or a faulty calorimeter addenda subtraction. Correlated thermometer calibration and sample-amount errors affect both axes and can dominate the illustrative linear regression uncertainty. Low-dimensional samples, additional low-energy excitations and proximity to a phase transition need other models. The test is valuable precisely because its failures can reveal physics beyond the bulk acoustic approximation.

For two polarizations in area $A$ and $\omega =C|k{|}^{\alpha }$, counting wavevector states in a disk gives

The static density susceptibility is the linear density response to a chemical-potential perturbation. In the quadratic uniform contact Bose gas, ${\chi }_{\rho }(q)=[g+q^2/(4m{\rho }_0){]}^{-1}$. Its inverse is a density stiffness. At zero wavevector $\partial \rho /\partial \mu =1/g$ at mean field, while the isothermal compressibility includes the additional factor $1/{\rho }^2$.

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.