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For a slender tube of length $L$ supplied from a large reservoir with excess pressure $P$, negligible gravity and entrance inertia, the Unsteady Bernoulli equation gives $L\dot{U}+U^2/2=P/\rho$. A stationary initial liquid column has the displayed speed and volume flux $\pi a^{2}U$ for tube radius $a$. The inertial term controls the early linear acceleration; the kinetic-energy term sets the terminal inviscid speed. This is a leading slender-tube model while the reservoir remains large and the tube full.

An involute is the curve traced by the end of a taut string as it is unwound from a fixed curve. For a circle of radius $R$, one parametrization is $\mathbf{x}(\theta )=R(\cos \theta +\theta \sin \theta ,\sin \theta -\theta \cos \theta )$. The straight free length is $R\theta$, and differentiation shows that the traced velocity is perpendicular to the taut string. Reversing the motion winds the string back around the circle.

In a one-dimensional Ohmic approximation, a linear conical frustum of length $L$, end radii $a,b>0$ and ionic resistivity $\rho$ has $r(z)=a+(b-a)z/L$. Conservation of electric current gives $d\mathcal{R}=\rho dz/[\pi r(z{)}^2]$. Integration yields the displayed resistance. Two identical opposed frusta, each of length $l$, have resistance $2\rho l/(\pi ab)$, whether their central junction is the wider or narrower end. Their local field profiles nevertheless differ. The approximation neglects surface conduction and resistance outside the defined pore, and is not an exact three-dimensional field solution for arbitrary wall slope.

For a harmonic coastal Kelvin wave, $v=(c/H)\eta$, $c=\sqrt{gH}$ and the offshore decay length is $c/|f|$. Integrating the period-mean pressure-work flux over the offshore direction gives $\mathcal{F}=\rho g^{2}H{A}_0^2/(4|f|)$ for peak coastal amplitude $A_0$. Slow changes with negligible dissipation or scattering conserve this flux, giving $A_0\propto \sqrt{|f|/H}$. The changing mode width explains why depth scaling differs from fixed-width wave shoaling. The approximation is not uniform at the equator.

A beam's transverse intensity changes its own refractive index. For positive nonlinear index and a bright centre, the accumulated phase has the curvature of a converging lens. For $I(r)=I_0{e}^{-2r^2/w^2}$ over a short length $L$, the in-medium thin-lens approximation gives $1/f=4n_2{I}_{0}L/(n_0{w}^2)$. Negative $n_2$ instead produces self-defocusing.

For a parabolic profile $u(s)=U_{\max }(1-s^2/a^2)$ in a pipe of radius $a$, the mean speed is $U_{\max }/2$ and $P_{\mathrm{kin}}=\frac{1}{2}\rho \int u^{3}dA=\pi \rho a^2{U}_{\max }^3/8$. With pressure drop $\Delta p$ over length $\ell$ and kinematic viscosity $\nu$, $U_{\max }=\Delta p{a}^2/(4\rho \nu \ell )$, so $P_{\mathrm{kin}}=\pi \Delta p^3{a}^8/(512{\rho }^2{\nu }^3{\ell }^3)$. This is not the pressure-work dissipation $\Delta pQ$.

A magnetized long-wavelength plasma description with a common leading perpendicular flow, magnetic flux freezing and a gyrotropic pressure tensor. The kinetic magnetohydrodynamic momentum equation couples to a drift-kinetic equation for each species; parallel plasma pressure and perpendicular plasma pressure are calculated as velocity moments, rather than prescribed by an isotropic adiabatic closure. Relative species drifts enter at higher order in the ratio of gyroradius to macroscopic length.

The delta-comb Kronig-Penney potential is

The Kuhn length is the link length of an equivalent long ideal chain with the same contour length and mean squared end-to-end distance. A three-dimensional worm-like chain satisfies $⟨R^2⟩\sim 2l_{p}L$ for $L\gg l_p$, hence $b_K=2l_p$.

The largest radius of the normalized Cornu spiral occurs on its first outer turn at $w_{\ast }\simeq 1.20938$, where $\mathrm{Re}[F(w{)}^{\ast }{F}^{\prime }(w)]=0$. Oddness makes the chord from $F(-w_{\ast })$ to $F(w_{\ast })$ attain twice that radius, while the triangle inequality bounds every other chord by the same length. The interval is therefore the strongest freely selectable connected aperture at that observation point, with normalized point power about1.80142. Signed physical limits follow from $x=w\sqrt{\lambda z/2}$. Existing openings can constrain this optimization.

The electrostatic bisector force is laterally anti-restoring, with the positive stiffness coefficient on the right. A spring of tension $T$ and length $\ell$ adds transverse restoring stiffness $T/\ell$ when its fixed support is directly above the charge. The displayed inequality therefore distinguishes actual transverse stability from the saddle of a model with only a vertical spring force. Axial spring stiffness alone does not specify the transverse verdict. At fixed finite support geometry and positive gravitational load, sufficiently small electric charge preserves lateral stability: the destabilizing coefficient tends to zero as $q^2$, while the tension term approaches $mg/\ell >0$. The vertical equilibrium must separately satisfy its own positive-stiffness condition.

The Lehnert number compares an Alfvén speed to a rotation speed on length scale $L$. With $\mathrm{Le}\ll 1$, magnetic-wave frequencies are small compared with the rotation frequency, allowing nearly geostrophic flow. Ideal weakly damped magnetic waves also require $\eta /L^2$ and $\nu /L^2$ to be small compared with their wave frequency; a small Lehnert number alone does not guarantee weak diffusion.

The length of a piecewise smooth curve is its arc length: $L(\alpha )=\int \Vert {\alpha }^{\prime }(t)\Vert dt$, using either the Euclidean space norm or the norm from a specified Riemannian metric. A straight segment from $p$ to $q$ in Euclidean space has length $\Vert q-p\Vert$, while a route that bends or retraces can be longer. On a Riemannian manifold, comparing these lengths over all routes defines Riemannian distance.

A reference realizing a specified length, under specified conditions. The temperature and mechanical constraints belong to the specification when thermal expansion changes the reference's dimensions.

In the small-amplitude slender-body theory approximation, an undulating swimmer accelerates an added mass of fluid per unit length $m(x)$. With $\mathcal{D}={\partial }_t+U{\partial }_x$ and transverse displacement $h$, the lateral force on the body is $-\mathcal{D}(m\mathcal{D}h)$. Its reaction on the fluid has the opposite sign. For a periodic stroke and the usual negligible leading-edge added-mass contribution, the mean trailing-edge thrust is $m⟨h_t^2-U^2{h}_x^2⟩/2$. This is a reactive inertial model, distinct from viscous resistive-force theory.

Each LIGO instrument is a Michelson interferometer with approximately $4\mathrm{km}$ arms containing Fabry–Pérot interferometers. A stabilized $1064\mathrm{nm}$ laser illuminates suspended $40\mathrm{kg}$ fused-silica mirrors in vacuum. Four-stage suspensions and active isolation suppress ground motion; low-loss materials reduce thermal noise. Power recycling increases circulating power; signal recycling shapes the response to the gravitational wave.^[ref]

Why measure two perpendicular arms? In the long-wavelength limit, take a gravitational wave with plus polarization normally incident on ideal perpendicular arms. Its transverse-traceless tensor has $h_{xx}=h_{+}$ and $h_{yy}=-h_{+}$. Expanding each proper length to first order givesHere $h_{+}$ is dimensionless, whereas $\Delta L$ is a length. A stipulated $h_{+}=10^{-21}$ and $L=4000\mathrm{m}$ imply $\Delta L=4\times 10^{-18}\mathrm{m}$, with individual arm changes of opposite sign and magnitude $2\times 10^{-18}\mathrm{m}$. These are equivalent differential arm-length changes, not a photograph of mirrors moving against a rigid external ruler. The observable compares returning light phases; coordinate systems can distribute the description between mirror coordinates and light propagation.

For unit arm directions $\mathbf{u},\mathbf{v}$, define the detector tensor $D^{ij}=(u^i{u}^j-v^i{v}^j)/2$. If $e_{ij}^{+},e_{ij}^{\times }$ are the transverse polarization tensors, thenThe dimensionless factors $F_A$ encode sky direction and the polarization basis. Thus even identical sources can have different observed amplitudes and wave phases. Motion changing both arms equally cancels in this ideal differential observable, although real asymmetries can couple common disturbances into it.

Suspension supplies an inertial reference above its resonances. For example, a single ideal harmonic oscillator with support motion $y$ obeys $\ddot{x}+{\omega }_0^2(x-y)=0$. Its response at angular frequency $\omega$ is $x/y={\omega }_0^2/({\omega }_0^2-{\omega }^2)$, which decreases in magnitude as $({\omega }_0/\omega {)}^2$ well above resonance. The mirror follows slow support drift but approximately free motion in the high-frequency limit. Real multistage suspensions also require damping and alignment control; resonant frequencies are not regions of good isolation.

Rainer Weiss's 1972 report proposed suspended masses, multiple-pass optical delay lines, and a servo holding an interference fringe. Its noise analysis separately treats photon-counting fluctuations, laser frequency instability, mechanical thermal noise, radiation pressure and ground motion. Those delay lines should be distinguished from the resonant arm cavities of Advanced LIGO.^[ref]

An ideal single-round-trip Michelson interferometer converts differential length into phase difference $\delta \varphi =4\pi \Delta L/\lambda$. If its dark-port power scale is $P_0$ and its static phase offset is ${\varphi }_0$, thenExactly at a dark fringe, ${\varphi }_0=0$, power is quadratic in small displacement and loses its sign. A deliberate offset supplies an optical reference field: interference with the small signal field produces a linear term. Equivalently, $|E_{\mathrm{ref}}+E_{\mathrm{sig}}{|}^2$ contains $2\mathrm{Re}(E_{\mathrm{ref}}^{\ast }{E}_{\mathrm{sig}})$. The photodetector measures power, not a signed displacement directly.

Advanced LIGO uses this DC homodyne principle with a small differential arm offset and an output mode cleaner. Its feedback control maintains optical resonance and alignment, using actuation distributed over the suspension stages.^[ref] The simple formula above omits resonant storage: arm and recycling cavities give a frequency-dependent optical gain and delay, so no constant pass-count multiplier describes the entire observing band.

To see why a locked detector can still measure a disturbance, use a linear feedback control model. Let $x_{\mathrm{ext}}$ be the external differential displacement, $d_{\mathrm{err}}$ the residual readout, $d_{\mathrm{ctrl}}$ the recorded control command, $C(f)$ the sensing response and $A(f)$ the actuation response. With the convention $x_{\mathrm{res}}=x_{\mathrm{ext}}-A{d}_{\mathrm{ctrl}}$, reconstruction givesHere tildes denote Fourier transforms; $C$ has readout units per metre and $A$ metres per command unit. Feedback suppresses residual motion but records the compensating command. Omitting that command would suppress the reconstructed low-frequency signal too. In the actual GW150914 experiment, a modulated auxiliary laser exerted calibrated radiation pressure on a mirror to measure the response; periodic excitations monitored calibration.^[ref]

The linear attenuation coefficient is the infinitesimal probability of removal from the uncollided beam per unit path length. Its units are inverse length. In the homogeneous monoenergetic model, $\mu =-d\ln I/dx$; $1/\mu$ is the thickness leaving fraction $e^{-1}$ uncollided. The mass attenuation coefficient is $\mu /\rho$ for material mass density $\rho$. Removing a photon by scattering of light does not imply deposition of all its energy at that point; attenuation and energy absorption are different quantities.^[ref]