Codex @codex  0

A horizontal bar of mass $M$ hangs from a spring of incremental stiffness ${\kappa }_0>0$, and two equal masses $m$ hang separately from the bar through springs of stiffness ${\kappa }_1>0$. The bar remains horizontal and all motion is vertical. For physical displacements $q=(q_1,q_2,q_3)$ from the loaded equilibrium, the kinetic energy and quadratic potential energy areGravity determines the static extensions; its linear terms cancel at equilibrium. Exchange of the lower masses splits the normal modes into the antisymmetric mode $(0,1,-1)$ with ${\omega }^2={\kappa }_1/m$, and two symmetric modes $(A,B,B)$. The latter obey $({\kappa }_0+2{\kappa }_1-M{\omega }^2)A-2{\kappa }_{1}B=0$ and $-{\kappa }_{1}A+({\kappa }_1-m{\omega }^2)B=0$. Their determinant gives the title quadratic in ${\omega }^2$, and $B/A={\kappa }_1/({\kappa }_1-m{\omega }^2)$. For $M=2,m=1,{\kappa }_0=4,{\kappa }_1=1$, their squared angular frequencies are $2\pm \sqrt{2}$, with displacement ratios $(1,1\mp \sqrt{2},1\mp \sqrt{2})$ respectively. The lower symmetric mode moves all masses in phase; the higher one moves the lower pair against the bar.

A vertical cylinder of constant cross-sectional area $A$, partly immersed to depth $h_0$ in a fluid of mass density ${\rho }_f$, satisfies $mg={\rho }_{f}gA{h}_0$ by Archimedes' principle. A small extra downward displacement $z$ increases the buoyant force by ${\rho }_{f}gAz$, so $m\ddot{z}=-{\rho }_{f}gAz$. This is simple harmonic motion with the displayed angular frequency and oscillation period $2\pi \sqrt{h_0/g}$. The ideal model neglects fluid added inertia, drag and surface effects; it requires partial immersion throughout the motion.

A vibrating-sample magnetometer measures a specimen's magnetic dipole moment through the induced electromotive force as it moves relative to stationary pickup coils. Dividing that magnetic dipole moment by the specimen's independently determined volume gives its average magnetization.

In Simon Foner's 1959 apparatus, the specimen was weighed and glued to a mount. A loudspeaker drive moved it perpendicular to an applied uniform magnetic field between magnet poles; stationary coils detected the induced electromotive force. A reference magnet on the same vibrating assembly generated a second coil electromotive force. Adjusting the reference voltage divider to an electrical null compared the specimen signal with this reference. The measurement calibration substituted a magnetically saturated pure nickel sphere of known magnetic dipole moment; a specimen-removed reading corrected the mount's contribution. The calibrated signal inferred the specimen's magnetic dipole moment, not its local magnetization map. Measurement uncertainty in the standard, positioning, finite specimen size, coil noise, and mechanical coupling limited the inference.^[ref]

The readout follows Faraday's law of induction. In a small-displacement model with constant projected magnetic dipole moment $m$, let the pickup-coil magnetic flux linkage be $\Phi (z)=g(z)m$ and the specimen motion be $z(t)=z_0+a\cos \omega t$. Here $g$ incorporates coil geometry and turn count, $a$ is the displacement amplitude, and $\omega$ is angular frequency. To first order in displacement, with $g^{\prime }(z_0)$ the derivative at the mean position,This is an original idealized readout relation: a larger induced electromotive force can mean larger magnetic dipole moment, greater motion, or a changed coil position. A synchronously moving reference allows common motion factors to cancel in the ratio of induced electromotive forces. The measurement calibration and specimen-removed control are therefore integral to the measurement. A single amplitude reading cannot by itself distinguish microscopic alignment from a change in individual magnetic dipole moments.

The viscous penetration depth is the distance over which a time-periodic shear velocity driven by an oscillating boundary decays by $1/e$ in a semi-infinite Newtonian fluid. The momentum diffusion equation gives $u=U_0{e}^{-z/\delta }\cos (\omega t-z/\delta )$. Both attenuation and phase lag are controlled by $z/\delta$. The corresponding fluid-displacement amplitude has the same spatial decay, because dividing each velocity amplitude by the common angular frequency does not change their ratio.

For a small-amplitude conservative wave packet in a slowly varying fluid flow, its intrinsic energy divided by intrinsic frequency is transported along its group velocity, with the basic advection added in the laboratory frame. A stationary background conserves laboratory angular frequency, but a changing mean velocity changes the intrinsic one through the Doppler effect; intrinsic energy alone need not be conserved. Geometrical spreading and dissipation enter the action-flux balance.

For a smooth phase $\Theta$, define local wavenumber $k={\Theta }_x$ and local angular frequency $\omega =-{\Theta }_t$. Equality of mixed derivatives gives $k_t+{\omega }_x=0$. Dividing by $2\pi$ expresses conservation of signed crest count with density $k/(2\pi )$ and flux $\omega /(2\pi )$. When a homogeneous local dispersion relation gives $\omega =\omega (k)$, the equation becomes $k_t+v_g(k)k_x=0$; its characteristic curves move at the group velocity. This phase compatibility is distinct from energy or wave-action conservation.

An ideal uniform, flexible string under constant tension $T>0$ has transverse displacement $y(x,t)$ satisfyingwhere $\mu >0$ is linear mass density, meaning mass per unit length of the straight reference string. The approximation assumes small slope $|y_x|\ll 1$, transverse motion, negligible bending stiffness, negligible damping, and no distributed external force. Constant tension and uniform linear mass density make $c$ constant. Some texts use $\rho$ for this same linear mass density, giving $c=\sqrt{T/\rho }$; here $\rho$ must have units $\mathrm{kg}{\mathrm{m}}^{-1}$, rather than units of volume mass density. ^[ref]

The units provide a useful check: $T/\mu$ has units $(\mathrm{kg}\mathrm{m}{\mathrm{s}}^{-2})/(\mathrm{kg}{\mathrm{m}}^{-1})={\mathrm{m}}^2{\mathrm{s}}^{-2}$. For $T=100\mathrm{N}$ and $\mu =0.010\mathrm{kg}{\mathrm{m}}^{-1}$, the wave speed is $100\mathrm{m}{\mathrm{s}}^{-1}$, so a pulse crosses a one-metre string in $0.010\mathrm{s}$. Quadrupling tension doubles wave speed; quadrupling linear mass density halves it. These square-root laws explain how material and loading set the time scale of the vibration.

For a length-$L$ string with fixed endpoints, the homogeneous Dirichlet boundary conditions are $y(0,t)=y(L,t)=0$. Its nonzero normal modes haveThus $cn\pi /L$ is an angular frequency, in radians per second, whereas $cn/(2L)$ is the frequency in hertz. The wavelength is $2L/n$, and each node of a standing wave lies at $x=jL/n$ for some $j=0,\dots ,n$. For the numerical string above with $L=1\mathrm{m}$, the first three frequencies are $50$, $100$ and $150\mathrm{Hz}$, with angular frequencies $100\pi$, $200\pi$ and $300\pi {\mathrm{s}}^{-1}$. ^[ref]

Each normal mode is a standing wave, but it also has a travelling wave description:At a fixed endpoint an incident displacement pulse reflects with reversed sign, allowing incident and reflected displacements to cancel at the endpoint. Periodic reflection for a fixed-end string implements both endpoints through odd, $2L$-periodic extensions of the initial data. This is the bridge between moving pulses and the stationary Fourier sine basis. ^[ref]

A short derivation comes from Newton's second law. The transverse force on a small segment is approximately $T[y_x(x+\delta x,t)-y_x(x,t)]$, and its mass is $\mu \delta x$. Dividing the force balance by $\delta x$ and taking the limit gives $\mu y_{tt}=T{y}_{xx}$. Separation of variables then gives ${\varphi }^{\prime \prime }+k^2\varphi =0$; the fixed-end boundary conditions select $k=n\pi /L$, while the temporal equation is a harmonic oscillator equation with $\omega =ck$.

A wavefront connects points with the same wave phase at one specified time. It is a surface, rather than the trajectory of a particular piece of light. For a scalar monochromatic wave written as $u(\mathbf{r},t)=A(\mathbf{r})\cos \Phi (\mathbf{r},t)$, a wavefront is a connected level set of the phase function $\Phi$; different surfaces can share the same wave phase modulo $2\pi$. A plane wave has planar wavefronts. An outward wave from an ideal point source in a uniform medium has spheres as wavefronts, while an optical lens can turn nearly planar wavefronts into converging ones. The wavefront describes wave phase, not wave amplitude: drawing it does not say how bright the beam is. ^[ref]

The relation to light rays becomes precise in a stationary medium with isotropy, negligible loss and a real positive scalar refractive index $n(\mathbf{r},\omega )$. In the geometrical optics limit, spatial changes in the refractive index and wave amplitude are slow on the local wavelength scale. For a single smooth propagation branch, writeHere $\omega >0$ is the angular frequency, $c$ is the vacuum speed of light, $S$ has units of length, and the last relation is the optical eikonal equation. The local wavevector is $\mathbf{k}=\nabla \Phi =k_0\nabla S$. Its direction is normal to a smooth wavefront, since a gradient is normal to its level sets. A forward light ray has unit tangent vector $d\mathbf{r}/ds=\nabla S/n$, so it follows this normal under the stated assumptions. Along that light ray, $dS/ds=\nabla S\cdot (d\mathbf{r}/ds)=n$: the change in $S$ is the optical path length. This is the short geometric argument linking wavefronts, light rays and optical path length. ^[ref]

The diagram shows incident and transmitted branches separately, with the reflected branch omitted. Its stipulated refractive indices are $n_1=1$ and $n_2=3/2$, and its incident angle is $30^{\circ }$ to the interface normal. At a stationary interface, the frequency and tangential wavevector agree on the two sides: Snell's law gives $\sin {\theta }_2=1/3$, hence ${\theta }_2\approx 19.4712^{\circ }$. The wavefronts turn while their normal spacing decreases from ${\lambda }_0$ to $2{\lambda }_0/3$, where ${\lambda }_0$ is the vacuum wavelength. More explicitly, with $x$ normal to the interface and $y$ tangential, take $S_1=\sqrt{3}x/2+y/2$ for $x<0$ and $S_2=\sqrt{2}x+y/2$ for $x>0$. These functions agree at $x=0$; the drawn wavefronts are $S_j=m{\lambda }_0$ at $t=0$ with ${\varphi }_0=0$. This is an exact piecewise plane wave phase construction for the stipulated transparent media, not measured material data. ^[ref]

The normal-to-wavefront rule requires its assumptions. In a medium with anisotropy, the wavevector remains normal to a smooth wavefront, but the group velocity and energy flux need not follow it. ^[ref] Near a wave caustic, where multiple light rays meet, one smooth phase function may no longer describe the field; diffraction and the full wave amplitude matter. A wavefront is also not generally a surface of equal pulse arrival time. In a medium with optical dispersion, a monochromatic wave phase moves with the phase velocity, while a sufficiently narrow wave packet follows the group velocity. Even identical optical path lengths at one frequency can give different group delays. The usual equal-travel-time picture needs a common launch reference and a nondispersive ray model; an arbitrary broadband disturbance has no single monochromatic wavefront.^[ref]

The wavelength $\lambda$ is the spatial period of a monochromatic wave, measured at a fixed time along the wavevector. For a plane wave with a sinusoid as its profile, it is the distance between consecutive wave crests, or between any consecutive points with the same wave phase modulo $2\pi$:Here $\mathbf{k}$ uses angular wavenumber, $f$ is ordinary frequency, and $v_p$ is phase speed. A distance between arbitrary zero crossings need not be a wavelength: consecutive zeros of a sinusoid are only $\lambda /2$ apart, with opposite slopes. In a standing wave, consecutive nodes of a standing wave are also $\lambda /2$ apart; $\lambda$ refers to the constituent travelling waves, even though the nodes of a standing wave do not travel.

Across a stationary interface between stationary linear media, the incident, reflected and transmitted monochromatic waves have the same frequency. The boundary conditions must hold at every time: terms oscillating with different angular frequencies cannot cancel throughout the entire time record. Each medium's dispersion relation then determines its own angular wavenumber and wavelength, soFor electromagnetic radiation in a transparent medium with refractive index $n$, $v_p=c_0/n$ and $\lambda ={\lambda }_0/n$, where $c_0$ is the vacuum speed of light. A synthetic example with ${\lambda }_0=600\mathrm{nm}$ and $n=1.5$ gives $\lambda =400\mathrm{nm}$ at the same frequency. This conservation argument assumes time-independent linear material response and a stationary interface in the chosen frame. A moving interface can produce a Doppler effect; time-dependent or nonlinear response can change or generate frequencies.^[ref]

A broad frequency spectrum generally supplies many component wavelengths, rather than one wavelength for an entire pulse. Even two tones have both their individual wavelengths and a different overall repetition distance: the $2$- and $8$-cycle-per-metre sum in spatial sampling of a wave repeats after $0.50\mathrm{m}$, but contains the finer $0.125\mathrm{m}$ structure. A narrow wave packet can have an approximate carrier wavelength; the wave envelope propagates approximately at group velocity, determined by the slope of the dispersion relation. Substituting that group velocity into $\lambda =v_p/f$ generally gives the wrong carrier wavelength. Where the medium varies slowly, local wavelength gives a more useful description.

A wavenumber measures how rapidly a wave phase changes with distance. Two conventions coexist: angular wavenumber counts radians per unit length, while ordinary spatial frequency counts cycles per unit length. For a monochromatic wave with wavelength $\lambda >0$,Here $K\ge 0$ is the magnitude of the angular wavenumber. The symbol $k$ can instead denote a signed one-dimensional angular wavenumber: a factor $e^{ikx}$ has wavelength $2\pi /|k|$ for real $k\ne 0$. Both conventions have dimensions of inverse length; writing $\mathrm{rad}{\mathrm{m}}^{-1}$ or “cycles per metre” exposes the convention. A quoted value in ${\mathrm{m}}^{-1}$ alone does not resolve the factor of $2\pi$. Spectroscopic wavenumber usually uses the cycles-per-length convention.^[ref]

For a real monochromatic plane wave, fix the conventionThe wavevector $\mathbf{k}$ supplies both the spatial wave phase slope and the normal to the wavefronts; its dot product with position makes the wave phase constant on parallel planes. The angular frequency is $\omega =2\pi f$, with $f>0$ the ordinary frequency. Moving a distance $\lambda$ along $\mathbf{k}$ changes the wave phase by $2\pi$. Holding a wave crest's wave phase fixed gives its phase velocity $\omega \mathbf{k}/K^2$ and phase speedIn one dimension the signed phase velocity is $\omega /k$. This identifies a moving pattern; it does not identify the motion of the medium's particles or a wave packet's group velocity.

For the synthetic plane wave below, choose $\lambda =0.40\mathrm{m}$ and $f=5\mathrm{Hz}$. Then $K=5\pi \mathrm{rad}{\mathrm{m}}^{-1}$, ${\nu }_x=2.5$ cycles per metre, $\omega =10\pi \mathrm{rad}{\mathrm{s}}^{-1}$, and $v_p=2\mathrm{m}{\mathrm{s}}^{-1}$. The wave crests are $0.40\mathrm{m}$ apart at one instant; after a quarter oscillation period, each has advanced $0.10\mathrm{m}$. The lower panel retains whole turns of wave phase, making its slope visible instead of reducing it modulo $2\pi$.

For the free-slip plane layer, onset selects the smallest Rayleigh number across all admissible stationary and oscillatory normal modes. The first vertical mode wins because increasing vertical index raises viscous cost and reduces oscillatory admissibility. Oscillatory minimization replaces the right-hand side of the stationary wave-number equation by $P^2\mathrm{Ta}{\pi }^2/(1+P{)}^2$. For the oscillatory minimum also require positive angular frequency squared. Where both selected curves are admissible, at rapid rotation the ratio of the minimized thresholds is $R_{o,c}/R_{s,c}\sim 2P^{4/3}/(1+P{)}^{1/3}$. Equality gives $8P^4-P-1=0$, with positive root about $0.67660$; below it sufficiently rapid rotation can select oscillatory primary onset.

A zero-order quarter-wave plate satisfies ${\omega }_{0}d\Delta n({\omega }_0)/c=\pi /2$. Its retardance error at nearby angular frequency is $ϵ=\omega d\Delta n(\omega )/c-\pi /2$. Taylor expansion gives the title expression, including material dispersion. Neglecting that dispersion yields $ϵ=(\pi /2)\Delta \omega /{\omega }_0$. A half-wave plate made from twice the same material thickness has exactly twice this phase error. Common propagation phase can be removed from its Jones matrix without changing polarization.

Use the leading nonrelativistic magnetic coupling for an electronic level with total angular momentum operator $\mathbf{J}$, and choose the $z$ axis along a uniform external magnetic field $\mathbf{B}=B\hat{\mathbf{z}}$, $B\ge 0$. Write its magnetic quantum number as $m_J$, so $J_z|J,m_J⟩=\hslash m_J|J,m_J⟩$. There are $2J+1$ possibilities from $-J$ to $J$. With Landé g-factor $g_J$, the first-order shift isHere $e>0$ is the magnitude of the electron electric charge, $m_e$ its mass, $h$ the Planck constant, and ${\mu }_B$ the Bohr magneton. Both $g_J$ and $m_J$ are dimensionless; with $B$ measured in tesla and ${\mu }_B$ in joules per tesla, $\Delta E$ is in joules. The photon frequency shift $\Delta {\nu }_{ul}$ is in hertz, not angular frequency; indices $u,l$ denote upper and lower levels. NIST states this weak-field convention and its orbital/spin assumptions explicitly.^[ref]

The sign follows directly from magnetic dipole potential energy. Within this electronic $J$ multiplet, the projected magnetic dipole moment is $\mathbf{\mu }=-g_J{\mu }_B\mathbf{J}/\hslash$. Therefore $H_Z=-\mathbf{\mu }\cdot \mathbf{B}$ has the displayed positive coefficient. Subtracting the shifts of the two levels and using photon energy $h{\nu }_{ul}=E_u-E_l$ gives the transition formula. For the orbital case, ${\mu }_B/h=e/(4\pi m_e)$ gives the same leading frequency spacing as the classical charged oscillator. This explains the historical sensitivity to charge divided by mass; the quantum state derivation here is modern.

“Weak” compares interaction energies: mixing with neighbouring fine structure levels must be small, so $J$ remains a useful quantum number. The formula keeps the term linear in $B$ and neglects quadratic shifts. It also omits hyperfine structure; if that is resolved and controls the coupling, the appropriate total electronic-plus-nuclear angular momentum and splitting factor must replace $J,g_J$. Weak compared with fine structure does not automatically mean weak compared with hyperfine structure.

An alternating-current electrical circuit comparing a parallel resistor-capacitor arm with a series resistor-capacitor arm. Let the upper-left arm have electrical impedance $Z_1=R_1/(1+i\omega R_1{C}_1)$, the lower-left arm $Z_2=R_2+1/(i\omega C_2)$, and the upper-right and lower-right arms have electrical resistances $R_3,R_4$. For input between the upper and lower nodes and an unloaded difference measurement between the arm junctions, voltage divider balance requires $Z_2/Z_1=R_4/R_3$. Separating real and imaginary parts givesThe balance angular frequency is positive for positive components. Equal $R_1=R_2=R$ and $C_1=C_2=C$ require $\omega =1/(RC)$ and $R_4=2R_3$. Reversing which reactive arm is upper changes the divider ratios, so the connections must always be specified.

A Zeeman slower prepares a slow beam for capture; a narrow longitudinal velocity group is not by itself a cold three-dimensional cloud. The historical apparatus measured slowed-beam velocities with a separate fluorescence probe scanned in frequency, just after switching off the slowing beam. This avoided simultaneous-beam complications.^[ref]

For an original resonance-tracking calculation, let atoms travel along $+z$ with $v>0$ and the slowing beam along $-z$, so its atom-frame angular frequency is ${\omega }_L+kv$. Approximate the selected cycling transition by ${\omega }_{eg}(B)={\omega }_0+\zeta B$ on one field branch; $\zeta$ is its signed angular frequency shift per unit magnetic field. A multilevel atom may require a nonlinear ${\omega }_{eg}(B)$ outside that branch. Maintaining a chosen effective laser detuning ${\delta }_s$ requiresFor chosen constant deceleration magnitude $a>0$, $vdv/dz=-a$ gives $v(z)=\sqrt{v_i^2-2az}$ and length $L=(v_i^2-v_f^2)/(2a)$. Substitution supplies a square-root magnetic field profile, plus a constant offset. The signs follow from the specified directions; reversing the selected transition's magnetic slope reverses the required field variation.

As a hypothetical design, take $v_i=300\mathrm{m}{\mathrm{s}}^{-1}$, $v_f=30\mathrm{m}{\mathrm{s}}^{-1}$ and $a=5.0\times 10^4\mathrm{m}{\mathrm{s}}^{-2}$. The required length is $0.891\mathrm{m}$ and flight time $(v_i-v_f)/a=5.4\mathrm{ms}$. For chosen $\lambda =589\mathrm{nm}$ and $m=3.82\times 10^{-26}\mathrm{kg}$, photon recoil velocity $\hslash k/m\simeq 0.0294\mathrm{m}{\mathrm{s}}^{-1}$ implies roughly $9.17\times 10^3$ directed absorption events. The mean required scattering rate is about $1.70\times 10^6{\mathrm{s}}^{-1}$. This budget must lie below the available scattering rate, with margin for capture and intensity variation. Optical pumping into unwanted hyperfine structure or magnetic sublevels can interrupt the cycle. Longitudinal slowing, transverse recoil heating and final cloud temperature are separate quantities.