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For a free three-dimensional worm-like chain with contour length $L$ and persistence length ${\ell }_p$, the unit-tangent correlation is $⟨\mathbf{t}(s)\cdot \mathbf{t}(s^{\prime })⟩=e^{-|s-s^{\prime }|/{\ell }_p}$. Since $\mathbf{R}={\int }_0^L\mathbf{t}(s)ds$, integrating this correlation givesFor $L\gg {\ell }_p$, the leading term is $2{\ell }_{p}L$, the ideal chain law with Kuhn length $2{\ell }_p$. For $L\ll {\ell }_p$, a Taylor series gives $⟨R^2⟩=L^2-L^3/(3{\ell }_p)+O(L^4/{\ell }_p^2)$, approaching a rigid rod. This ideal semiflexible-chain model omits long-range excluded volume; sufficiently long dilute chains in a good solvent can cross over to swollen-chain statistics.

Multiplying a measured signal by a finite-duration window function convolves its Fourier transform with the transform of the window. A spectrally sharp component therefore contributes to nearby frequencies rather than remaining a single ideal line. Smooth tapering can reduce distant sidelobes at the cost of widening the main spectral feature. This finite-record effect is distinct from aliasing, which superposes copies of the spectrum because of time sampling.

Use the equal-component Wien bridge with upper-left parallel arm and lower-left series arm. Let $A$ be the left arm junction, $C$ the upper input node, and $V_i$ the upper-minus-lower input electric potential difference. With $u=\omega RC$, the unloaded transfer ratio $H=(V_A-V_C)/V_i$ is the displayed expression. Its band-pass filter magnitude is $[9+(u-u^{-1}{)}^2{]}^{-1/2}$, maximal at $u=1$ with $H=-1/3$. One continuous wave phase is $\varphi =-\pi -\arctan [(u-u^{-1})/3]$, tending from $-\pi /2$ at low positive frequency to $-3\pi /2$ at high frequency. Reversing the measured output polarity multiplies $H$ by $-1$ and shifts its phase by $\pi$ without changing its magnitude.

A Cassie–Baxter state is a composite wetting state in which a drop contacts solid asperities over area fraction $f_s$ and trapped gas over the remainder. An ideal area-weighted contact-angle model gives the displayed apparent angle from the smooth-solid Young's equation angle ${\theta }_Y$. Reducing solid contact can strongly enhance apparent hydrophobicity. Such a state can be metastable and can collapse when liquid enters the texture; roughness alone does not guarantee it.

The excess interfacial free energy released per newly wetted area when a liquid film replaces a solid-vapour interface is the displayed coefficient. Positive $S$ favors macroscopic spreading in the ideal equilibrium surface-energy model, whereas negative $S$ permits a finite contact angle obeying Young's equation. Surface preparation, chemical composition and adsorption can change all three interfacial energies.

A liquid spreads macroscopically over a solid with zero macroscopic equilibrium contact angle. In the simple interfacial-energy model the threshold is ${\gamma }_{SV}-{\gamma }_{SL}={\gamma }_{LV}$, or zero spreading coefficient; positive coefficient favors a spreading film. Extrapolating a concentration-dependent liquid-vapour surface tension to this threshold assumes the other interfacial energies remain fixed and no intervening adsorption or micellization behavior changes the trend.

For $m=\tanh (m/t)$ and positive temperature, the positive solution exists uniquely for $0<t<1$: the slope at zero is $1/t>1$, while the hyperbolic tangent is concave and below one. For $t\ge 1$ only $m=0$ remains. Expanding to cubic order near $t=1$ gives $m^2\simeq 3t^2(1-t)$ and $m\simeq t\sqrt{3(1-t)}$, with mean-field critical exponent $1/2$. The prefactor $t$ follows from the truncated equation; it is not controlled beyond leading critical order.

Parity violation means that a physical law fails to be invariant under the spatial reflection represented by the parity operator. A process and its fully inverted counterpart can then have different probabilities. For beta-minus decay with polarized parent atomic nuclei, a nonzero correlation between nuclear spin and emitted electron momentum is such a test. Rotational symmetry alone allows this correlation: a pseudoscalar is unchanged by proper rotations but changes sign under spatial reflection.

A decay amplitude changing strangeness by two units requires at least two insertions of the low-energy weak interaction in the Standard Model. Each charged-current insertion can remove only one net strange quark in such a process. Both two-W boson tree topologies and loop contributions can occur; a loop is not required for every exclusive final state. At hadronic energies, the extra power of the Fermi constant, with the needed powers of an energy scale to balance units, strongly suppresses a second-order amplitude relative to a first-order one. This is distinct from Cabibbo suppression within a fixed weak order.

In a low-energy allowed weak three-body decay, negligible final-state masses and a nearly constant effective matrix element give a fifth-power energy-release dependence. For muon decay in natural units, $\Gamma \simeq G_F^2{m}_{\mu }^5/(192{\pi }^3)$. This four-fermion approximation requires momentum transfers small compared with the W boson mass. Extrapolation to a heavy particle able to emit a real W changes the decay kinematics and the mass scaling, so it is not a reliable heavy-particle lifetime prediction.

For equal-mass one-dimensional scattering between regions with real wavenumbers $k_0>k_2>0$, insert a layer of width $a$ and wavenumber $k_1$. For a purely right-moving incident wave of amplitude $A$ and no reflection, continuity at the first edge gives interior amplitudes $C=A(1+k_0/k_1)/2$ and $D=A(1-k_0/k_1)/2$. If $k_{1}a=\pi /2$, continuity at the second edge gives a transmitted edge amplitude $F=i(k_0/k_1)A$ and the matching condition $k_1^2=k_0{k}_2$. Thus $F/A=i\sqrt{k_0/k_2}$ and the transmitted probability current equals the incident current, despite their unequal amplitudes.

For potentials $0,V_1,V_2$ with $E>V_2>V_1>0$, the condition is $(E-V_1{)}^2=E(E-V_2)$, or $E=V_1^2/(2V_1-V_2)$, requiring $V_2/2<V_1<V_2$. It is the quantum-wave counterpart of a quarter-wave impedance transformer: the intermediate wavenumber is the geometric mean and the two reflected amplitudes cancel.

Anderson localization is the confinement of electron wave functions, or other wave modes, by interference between repeated wave scattering paths in static disorder. A localized eigenstate occupies a limited region even in a much larger sample. Disorder can therefore suppress diffusion without absorbing the wave. In 1958, Philip Warren Anderson studied random site energies and intersite coupling, finding a regime without transport. His paper develops a theoretical model; it does not report an Anderson laboratory measurement.^[ref]

The distinction from ordinary multiple scattering is the retention of wave phases: alternative paths contribute probability amplitudes before their squared modulus is taken. Adding path probabilities alone loses the cross terms responsible for interference. This makes Anderson localization useful as a warning about transport models: repeated deflections need not produce unlimited diffusion. It also connects electron transport with wave propagation in a random medium, although the detailed equations and experimental diagnostics differ between electrons, light and acoustic waves.

Philip Warren Anderson's 1977 lecture distinguishes localized magnetic moments from disorder-localized states. It credits Nevill Francis Mott with recognizing the significance of the mobility edge. A Mott insulator instead requires electron interactions and can exist without disorder.^[ref]

John Hasbrouck Van Vleck developed quantum mechanics explanations of magnetic susceptibility, including its temperature dependence; this is a distinct contribution from Anderson localization.^[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.

A strip of membrane with fixed edges $y=0,b$ supports transverse normal modes proportional to $\sin (n\pi y/b)$, $n\ge 1$. Separation of the two-dimensional wave equation gives the displayed dispersion relation. Confinement produces longitudinal wave dispersion even when the membrane's underlying wave speed $v$ is independent of frequency.

For $y_{tt}-y_{xx}=F(t)\delta (x-x_0)$ with zero endpoint displacement, a Fourier sine series gives $q_n^{\prime \prime }+k_n^2{q}_n=2F(t)\sin (k_n{x}_0)/L$. With the $e^{-i\omega t}$ Fourier transform convention, the causal inverse of this oscillator uses the displayed denominator. Its inverse transform is $H(t)\sin (k_{n}t)/k_n$, giving $q_n(t)=2\sin (k_n{x}_0)/(L{k}_n){\int }_{-\infty }^{t}F(s)\sin [k_n(t-s)]ds$ when the past integral exists. The plain denominator $k_n^2-{\omega }^2$ requires a pole prescription: its Cauchy principal value gives the time-symmetric kernel $\sin (k_n|t|)/(2k_n)$ instead. Homogeneous sine and cosine modes are additional freedom unless temporal data or causality are specified.

For real constants $\gamma ,c$, the substitution $v=e^{\gamma t}u$ turns the displayed equation into the wave equation $v_{tt}=c^2{v}_{xx}$. If $u(x,0)=f(x)$ and $u_t(x,0)=g(x)$, then $v(x,0)=f(x)$ and $v_t(x,0)=g(x)+\gamma f(x)$. For $v_0=|c|>0$, D'Alembert's formula consequently givesThe extra $\gamma f$ term is necessary even when the original initial velocity is zero. For $c=0$, take the limit to obtain $u=e^{-\gamma t}[f+t(g+\gamma f)]$. The formula gives classical solutions for suitably smooth data and weak solutions for less regular data. Positive $\gamma$ damps the transformed wave, whereas negative $\gamma$ amplifies it.

A wave equation with a linear damping term and positive wave speed $c$ separates into spatial normal modes and damped harmonic oscillators. For a spatial eigenvalue $k^2$, the time factor obeys $T^{\prime \prime }+\alpha T^{\prime }+c^2{k}^{2}T=0$. Its characteristic roots are $-\alpha /2\pm \sqrt{{\alpha }^2/4-c^2{k}^2}$. Thus underdamping requires $|\alpha |<2c|k|$, critical damping gives a repeated root and the second solution $t{e}^{-\alpha t/2}$, and overdamping gives two real exponential rates. Negative $\alpha$ amplifies rather than damps the motion.

A fixed circular elastic membrane of radius $a$ has radial normal modes satisfying the displayed Sturm-Liouville problem, with weight $r$, $R(a)=0$ and regularity at the origin. The regular solutions are Bessel functions of the first kind $J_0(\sqrt{\lambda }r)$. Integration by parts gives the positive Rayleigh quotient $\lambda ={\int }_0^{a}r|R^{\prime }{|}^{2}dr/{\int }_0^{a}r|R{|}^{2}dr$. Distinct modes are orthogonal in the weighted inner product ${\int }_0^{a}r\overline{R}Sdr$.

A vortex ring is a vortex tube bent into a closed loop, often approximately a circle. Its vorticity runs around the loop, while the induced velocity field passes through the centre and returns outside. For a thin ring of radius $r$ and circulation $\Gamma$, the Biot-Savart law for incompressible vorticity gives an axial velocity $u_z(z)=\Gamma r^2/[2(r^2+z^2{)}^{3/2}]$, measured relative to the ring centre, with its sign set by the circulation orientation. A vortex ring is not a loop of fluid whose velocity everywhere merely follows the circular vortex line.

A vortex filament is the ideal thin-core limit of a vortex tube, concentrating a finite circulation on a curve. In an unbounded incompressible fluid with velocity tending to zero at infinity, its induced velocity field follows the vorticity version of the Biot-Savart law. The ideal field is singular on the filament; finite-core models are needed there. A closed circular filament is a thin vortex ring.