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A slender axisymmetric column of a Newtonian fluid, with radius $a(z,t)$, axial velocity $w$, external pressure $p_e$ and axial body-force density $f$, obeys $(a^2{)}_t+(a^{2}w{)}_z=0$ and $3\mu (a^2{w}_z{)}_z/a^2=p_{e,z}-f$ when inertia, surface tension and leading external shear are negligible. The factor three is the Trouton ratio. Radial mass conservation gives $u_r=-r{w}_z/2$; the Newtonian fluid stress tensor gives $p=p_e-\mu w_z$ and axial excess stress $3\mu w_z$. This reduces a three-dimensional Stokes flow to one-dimensional evolving geometry.

For two resting incompressible inviscid layers between impermeable horizontal walls, the upper layer has density ${\rho }_1$ and depth $L_1$, and the lower layer has ${\rho }_2<{\rho }_1$ and depth $L_2$. Solving the Laplace equation for the velocity potential and imposing the linearized kinematic and capillary pressure conditions gives the displayed dispersion relation. With positive surface tension $\gamma$, exponential Rayleigh-Taylor instability occurs for $0<k<k_c$, where $k_c=\sqrt{g({\rho }_1-{\rho }_2)/\gamma }$. The depths change the growth rate, but not this cutoff.

A freely floating body at rest has no supporting contact or applied holding force. Neglecting surface tension and the buoyancy of air, its upward buoyancy balances its downward force from gravity:For a body of uniform mass density ${\rho }_{\mathrm{body}}$ and total volume $V_{\mathrm{body}}$, its mass is $m={\rho }_{\mathrm{body}}{V}_{\mathrm{body}}$. Substitution into Archimedes' principle givesPartial immersion is possible when ${\rho }_{\mathrm{body}}<{\rho }_f$. A nonuniform body uses its mean mass density $m/V_{\mathrm{body}}$ instead; for a hollow body this volume must count the enclosed space that excludes water. This explains how the amount of displaced water can support a vessel whose walls alone are denser than water. ^[ref]

For the $600\mathrm{kg}{\mathrm{m}}^{-3}$ cube in water of mass density $1000\mathrm{kg}{\mathrm{m}}^{-3}$, the displaced fraction is $600/1000=0.60$: sixty percent of its volume lies below the surface. With a horizontal top and bottom, the immersed depth $d$ satisfies $V_{\mathrm{displaced}}=Ad$, soIts remaining height is $0.040\mathrm{m}$, and the buoyancy is $1000\times 9.81\times 0.000600=5.886\mathrm{N}$, exactly balancing its downward force from gravity.

A gravity-capillary wave has restoring contributions from both gravity and surface tension. Consider an initially resting, horizontally unbounded liquid layer of constant depth $h$, with uniform mass density $\rho$, incompressible flow, inviscid flow and irrotational flow. The upper gas exerts constant pressure and has negligible inertia. Assume a clean interface with constant surface tension $\gamma$: no surfactant stresses, imposed current or wind. The wave amplitude $a$ must satisfy $ka\ll 1$ and $a\ll h$, so linearized boundary conditions apply. Deep water means $kh\gg 1$, rather than a particular depth in metres. ^[ref]

Let $z=0$ be the undisturbed free surface, with liquid below it. Potential flow gives a velocity potential $\varphi$ with $\mathbf{u}=\nabla \varphi$, and incompressible flow gives the Laplace equation. The linearized kinematic boundary condition for a free-surface graph and the Unsteady Bernoulli equation, including gravity and the Young–Laplace equation, giveHere $\eta (x,t)$ is the interface displacement; the signed small-slope pressure jump is $p_{\mathrm{liquid}}-p_{\mathrm{gas}}=-\gamma {\eta }_{xx}$. A wave crest has ${\eta }_{xx}<0$, so its capillary pressure is positive. Choose a right-travelling Fourier mode of positive wavenumber $k$ and angular frequency $\omega$:The factor $e^{kz}$ decays into the deep liquid, satisfies the Laplace equation, and makes ${\varphi }_z={\eta }_t$ at $z=0$. Substituting in the dynamic boundary condition leavesThis derivation displays the two restoring contributions with the same sign. Dimensional analysis also checks the result: $\gamma /\rho$ has units ${\mathrm{m}}^3{\mathrm{s}}^{-2}$, so both $gk$ and $(\gamma /\rho )k^3$ have units ${\mathrm{s}}^{-2}$. ^[ref]

At finite depth, a flat impermeable bottom at $z=-h$ requires ${\varphi }_z=0$. Replacing $e^{kz}$ by the normalized vertical factor $\cosh (k(z+h))/\cosh (kh)$ makes the surface ratio ${\varphi }_z/\varphi =k\tanh (kh)$. Eliminating the mode amplitudes then givesThus the same model supplies its own depth check: use the deep-water formula only when $\tanh (kh)$ is sufficiently close to one for the required accuracy. For example, $kh=3$ reduces $\omega$ by about $0.248\%$ relative to deep water, while $kh=1$ reduces it by about $12.7\%$. These percentages are calculations from this factor. ^[ref]

In deep water the phase velocity magnitude follows a wave crest, while the group velocity magnitude follows the wave envelope of a narrow wave packet. Differentiating the positive-frequency dispersion relation givesFor $k{\ell }_c\ll 1$, the surface gravity wave limit has $c_g\simeq c_p/2$. For $k{\ell }_c\gg 1$, the deep-water capillary-wave dispersion relation has $c_g\simeq 3c_p/2$: a wave crest slips backward relative to the faster wave envelope, even while both advance. Neither speed is the oscillating fluid element's own velocity. ^[ref]

There is a particularly neat coincidence at the crossover. Differentiate $c_p^2=g/k+\gamma k/\rho$: its unique minimum is at $k_c=\sqrt{\rho g/\gamma }=1/{\ell }_c$, exactly where the restoring terms are equal. ThereforeFor the stipulated parameters in capillary length, the minimum is $0.231\mathrm{m}{\mathrm{s}}^{-1}$. At $\lambda =3\mathrm{mm}$, direct substitution gives $f=\omega /(2\pi )\simeq 131\mathrm{Hz}$, $c_p\simeq 0.394\mathrm{m}{\mathrm{s}}^{-1}$ and $c_g\simeq 0.580\mathrm{m}{\mathrm{s}}^{-1}$. This short wavelength is capillary dominated, but its group velocity is still slightly below the pure-capillary ratio $3c_p/2$ because gravity has not vanished.

A wide, shallow drop with cross-sectional area $A$ has the gravity-driven volume flux $q=-g{h}^3{h}_x/(3\nu )$ when substrate no-slip and zero surface shear are imposed. Mass conservation yields a porous medium equation, $h_t=[g/(3\nu )](h^3{h}_x{)}_x$. Its symmetric source-type similarity solution has half-width $\ell$ in the display and height $h=[9\nu ({\ell }^2-x^2)/(10gt){]}_{+}^{1/3}$, where $I={\int }_0^{\pi /2}{\cos }^{5/3}\theta d\theta$. Integrating the profile recovers $A$. The contact-edge slope diverges, so the bulk lubrication approximation requires an inner edge regularization if contact-line microphysics or surface tension is retained.

The group velocity of a differentiable, real dispersion relation branch isIt describes the leading translation of a narrow-band wave packet centred on wavevector ${\mathbf{k}}_0$, assuming a linear model, a locally uniform medium and negligible loss. Narrow-band means that the appreciably excited wavevectors occupy a small neighbourhood on which one smooth branch can be linearized. Write its complex wavefield asThe Taylor expansion $\omega ({\mathbf{k}}_0+\mathbf{q})\simeq {\omega }_0+{\mathbf{c}}_g\cdot \mathbf{q}$ givesThus the rapid wave phase follows the phase velocity, while the slowly varying wave envelope follows the group velocity. This Fourier analysis argument is an outline of the approximation, rather than a statement that every wave packet translates rigidly forever. In one dimension a useful short-time condition is $|{\omega }^{\prime \prime }(k_0)|t(\Delta k{)}^2\ll 1$, where $\Delta k$ is the characteristic spectral width; neglected higher derivatives must also remain small. Curvature of the dispersion relation eventually changes the wave envelope.^[ref]

Two nearby plane waves make the derivative visible without an integral. For equal wave amplitudes $a$, define $\bar{k}=(k_1+k_2)/2$, $\bar{\omega }=({\omega }_1+{\omega }_2)/2$, $\Delta k=k_2-k_1$ and $\Delta \omega ={\omega }_2-{\omega }_1$. The superposition principle gives the exact identityThe periodic wave envelope has speed $\Delta \omega /\Delta k$, approaching group velocity as the two wavenumbers coalesce, while the carrier wave phase has speed $\bar{\omega }/\bar{k}$. Two tones alone form an infinite beat train, not a localized wave packet. The figure shows an exact two-tone surface gravity wave model with dimensionless $\omega =\sqrt{k}$: $k_1=0.9$, $k_2=1.1$. Its carrier phase velocity is approximately $0.99875$, its finite-difference envelope velocity approximately $0.50063$, and ${\omega }^{\prime }(1)=0.5$ is the limiting group velocity. The marked carrier phase line need not coincide exactly with a maximum of the full modulated wave.

The deep-water surface gravity wave therefore has $c_g=c_p/2$, whereas the deep-water capillary wave limit has $c_g=3c_p/2$: crests can drift either forward or backward relative to a wave envelope. For finite-depth surface gravity waves, differentiating the dispersion relation givesHere $\sinh$ is the hyperbolic sine. The shallow-water limit has $c_g\simeq c_p\simeq \sqrt{gh}$; the deep-water limit has $c_g\simeq c_p/2$.^[ref]

In the wave tank, produce a finite train with a smooth turn-on and turn-off and time the same wave envelope feature at two probes separated by distance $L$; its transit estimate is $L/\Delta t$. Independently follow wave crests through that train. Techet's 2003 lab asks students to observe a deep-water train's foremost crest disappearing as the group moves forward, and to compare deep- and shallow-water cases. The interpretation is that a wave crest overtakes the slower wave envelope, dwindling at its front while other crests become visible at its rear. This is a qualitative expected observation in the teaching protocol, not a published set of measured curves.^[ref]

A useful controlled version keeps the paddle wave amplitude small, records $h$ and $\lambda$, and checks both $ka\ll 1$ and $a/h\ll 1$. Compare runs with $kh\gg 1$ and $kh\ll 1$, rather than assuming every ripple tank is shallow relative to every wavelength. Increase train duration at fixed central frequency to narrow its spectrum, and check that the envelope transit estimate stabilizes. Calibrate both probes, sample fast enough to resolve the carrier, and compare records before reflected waves return; an absorbing beach reduces this contamination. A current changes laboratory velocities, large wave amplitudes introduce nonlinear effects, and very short wavelengths require surface tension. Strong damping or noticeable reshaping makes the chosen threshold or peak an unreliable marker of simple group velocity. These are proposed controls and model limits, not claims that the cited lab measured every correction.

Three velocities should consequently be kept separate. The group velocity predicts narrow-band wave envelope motion. An energy-transport velocity is energy flux divided by energy density, using the appropriate time average; it agrees with group velocity in standard lossless linear wave models when all stored energy is included.^[ref] A signal-front velocity concerns the earliest response to a new disturbance and requires a causal model over its full spectrum. A strictly band-limited wave packet has no sharply bounded onset, and a mathematical two-tone envelope cannot test front arrival. In strongly absorbing or broadband propagation, a single group velocity need not describe the observed peak, energy transport, or first detectable response. A complex $\omega$ or $k$ also prevents interpreting its derivative as an ordinary real transport velocity without further analysis.

In particular, an unusually large or negative group velocity from pulse reshaping does not establish superluminal information transmission. For the causal dielectric model analysed by Fitzpatrick, the field ahead of the front vanishes for $t<x/c$, even when its phase velocity exceeds the vacuum speed of light $c$. That result concerns the electromagnetic model's causality; the ideal water-wave diagrams above contain no relativistic signal-front claim.^[ref]

Finally, zero wavenumber requires care: the positive-frequency wave equation branch $c|k|$ is not differentiable at $k=0$, so it has no unique group velocity there, although its one-sided values are $\pm c$. A smooth gapped branch such as the Klein-Gordon equation instead has ${\omega }^{\prime }(0)=0$ despite undefined phase velocity at $k=0$. The same signed branch must be used in both derivatives and ratios; a negative group velocity means an envelope moving in the negative coordinate direction, not a negative elapsed time. The practical value of group velocity is precisely this translation between a local slope on a dispersion diagram and a measurable packet transit, with its assumptions made explicit.

The Gaussian envelope in a Klein-Gordon medium supplies a quantitative next step: the first derivative translates the wave envelope, the second derivative spreads it, and higher derivatives limit that approximation. At finite spectral width, a peak's measured speed need not equal the group velocity at the central wavenumber. Neither a peak nor a fixed fraction of its height generally identifies front velocity.

Under single-length dynamical scaling of binary-fluid coarsening, mass density $\rho$, dynamic viscosity $\eta$ and surface tension $\sigma$ determine the displayed unique length and time scales by dimensional analysis. Writing $L=L_{0}f(t/t_0)$ reduces the force-density estimate to $\alpha f^{\prime \prime }+\beta f^{\prime 2}/f=\gamma f^{\prime }/f^2+\delta /f^2$. Resistive and driving terms need opposite signed geometric coefficients. Viscous hydrodynamic coarsening has $f\sim u$ and inertial hydrodynamic coarsening has $f\sim u^{2/3}$. These follow respectively from density-independent and viscosity-independent powers in $L\propto {\eta }^{2-3p}{\rho }^{p-1}{\sigma }^{2p-1}{t}^p$. They require thin interfaces, statistically self-similar bicontinuous domains, dominant advection and no additional relevant macroscopic scales. A degeneracy in the signed inertial coefficients can invalidate the pure-power force balance without invalidating the dimensional calculation.

An icicle ripple is a corrugation of the ice-water interface produced while a thin flowing water film freezes. Coupling lubrication theory, surface tension, thermal conduction, and the Stefan condition determines its growth and migration.

At a material interface without surface tension, continuity of displacement and of the pressure evaluated on the displaced interface givesThe first follows from the kinematic boundary condition $ik(U-c)\hat{\eta }=\hat{w}$. The second uses $\hat{p}={\rho }_0[(U-c){\hat{w}}^{\prime }-U^{\prime }\hat{w}]/(ik)$ and hydrostatic displacement $[\hat{p}-g\rho \hat{\eta }]=0$. These conditions apply to density, velocity and vorticity jumps within the Boussinesq approximation.

Two deep layers of incompressible flow with upper mass density ${\rho }_1$, lower mass density ${\rho }_2$, and tangential velocities $U_1,U_2$ have this dispersion relation when surface tension is absent. Decaying velocity potentials and the linearized kinematic boundary condition determine their amplitudes; the dynamic boundary condition for an inviscid interface gives the displayed relation. For $k>0$ the squared departure of the phase velocity $c$ from the density-weighted mean velocity is $g({\rho }_2-{\rho }_1)/[k({\rho }_1+{\rho }_2)]-{\rho }_1{\rho }_2(U_1-U_2{)}^2/({\rho }_1+{\rho }_2{)}^2$. A negative value gives Kelvin-Helmholtz instability despite stable density stratification.

Zhu, Miao, Xu and Cao's 2007 experiment drove a triangular metal frame electromagnetically with a sinusoidal generator. In a roughly $3\mathrm{cm}$-deep, temperature-regulated pool, it produced nearly plane capillary waves. A $632.8\mathrm{nm}$ laser illuminated the surface obliquely; the moving ripples acted as a reflection diffraction grating. One split beam monitored laser stability, while a CCD recorded diffraction orders on a screen $6.76\mathrm{m}$ away. An air-supported optical table reduced ambient vibration. ^[ref]

The frequency sweep was $100$–$290\mathrm{Hz}$ in $10\mathrm{Hz}$ steps, using distilled water and kerosene at $20{}^{\circ }\mathrm{C}$. For each setting they averaged at least three measurements of first-order spot separation. Known optical geometry and the grating equation yielded $k=2\pi /\lambda$; generator readings yielded $\omega =2\pi f$. Agreement with the predicted log–log slope $3/2$ tests the capillary law, and its intercept gives surface tension if mass density is known. The authors identify surface contamination as a major error source; noncontact optical detection reduces additional disturbance. ^[ref]

The optical measurement can be understood from the grating equation. Let ${\lambda }_L$ be the laser wavelength, $\alpha$ the incident grazing angle above the mean surface, and ${\beta }_m$ the outgoing grazing angle of diffraction order $m$, with the sign of $m$ chosen so thatThus calibrated spot positions determine outgoing angles, which determine the ripple wavelength $\lambda$, independently of the mechanical dispersion relation. This optical geometry treats the surface as a reflection diffraction grating at the laser wavelength. ^[ref]

The inferred surface tension in the full deep-water model isThis formula makes useful experimental controls explicit. Measure mass density at the sample's temperature; check depth with $kh$; keep the driven wave amplitude small enough that $ka\ll 1$; and distinguish $f$ from angular frequency $\omega$. For the stipulated parameters, the $100$–$290\mathrm{Hz}$ window corresponds to calculated wavelengths about $3.62$–$1.76\mathrm{mm}$ and $kh\simeq 52$–$107$ at $h=30\mathrm{mm}$. Depth corrections are negligible here, but the gravity contribution is about $4.5\%$–$1.1\%$ of the capillary contribution. These are ideal predictions, not digitized results from the experiment.

Omitting gravity would infer ${\gamma }_{\mathrm{cap}}=\rho {\omega }^2/k^3=\gamma +\rho g/k^2$, biasing surface tension upward. In the capillary limit, the differential estimate $d\gamma /\gamma \simeq d\rho /\rho +2d\omega /\omega -3dk/k$ also shows why careful spot-spacing calibration matters. A clean surface and weak viscosity are modeling assumptions: appreciable damping, surfactant stresses or large wave amplitudes call for additional physics. The experiment tests the short-wave regime; its drive window does not establish the crossover or the long-wave surface gravity wave limit.

For symmetric finite-mass Marangoni spreading on a liquid film, the leading outer profiles are $h=2h_0|x|/x_N$ and $C=(M/x_N)(1-|x|/x_N)$ inside $|x|<x_N$, where $x_N=(6AM{h}_{0}t/\mu {)}^{1/3}$. Outside, $h=h_0$ and $C=0$. The similarity solution follows from $H{\Gamma }^{\prime }=-\eta /3$, $\eta H^{\prime }=H$, $2\int \Gamma d\eta =1$ and $\int (H-1)d\eta =0$. The film reaches height $2h_0$ immediately behind the front, leaving an outer height jump that gravity or surface tension must smooth. These are ideal leading outer profiles, not a description of the microscopic central dry region or the smoothed edge.

Line tension is energy per unit length of a boundary, such as the edge between two phases of a lipid bilayer. Positive line tension tends to minimize the perimeter at fixed domain area. Its dimensions and geometry differ from surface tension, which is energy per unit area.

A membrane osmometer connects osmosis to a visible liquid level or a measured balancing pressure. In a teaching version, a solute-containing pouch is joined to a narrow vertical tube and submerged in external water. Entry increases the enclosed volume and raises the meniscus. Both the column and any stretched pouch can contribute an opposing pressure.

The University of Toronto's Osmosis (Brine) by University of Toronto Department of Physics demonstration (activity Me-012b, Description, steps 1–14) specifies soaking a dialysis membrane tube, tying one end closed and fastening the other around a thistle tube's narrow end. A stand holds the tube. Colored brine is cooled to room temperature and injected using a syringe to reduce bubbles; the pouch is completely immersed in a $1$ or $2\mathrm{L}$ beaker of distilled water. A taped marker records the initial level; subsequent meniscus height is the readout. The protocol proposes another pouch containing distilled water as a comparison, and its older instructions explicitly check seals for leaks.^[ref]

A rise is consistent with net water entry. To infer osmotic pressure quantitatively, use a retained solute and establish a stable equilibrium after accounting for dilution. If both free surfaces are at the same atmospheric pressure, the liquids have approximately equal mass densities $\rho$, and the pouch adds negligible mechanical load, the hydrostatic pressure difference at membrane height is approximately $\Delta P=\rho gh$, where $h$ is the meniscus height above the external surface and $g$ is gravitational acceleration. This is the liquid-column inference, not a pressure calibration supplied by the Toronto demonstration. For unequal mass densities, compare the two column pressures separately.

Useful additional controls are matched initial levels and geometry, constant temperature, replicate runs, and monitoring external solute to test retention. These are proposed checks, not reported Toronto results. Leaks, trapped bubbles, surface tension in the narrow tube, evaporation and membrane stretching can alter the level. In particular, a dialysis membrane may pass dissolved salt: the brine setup illustrates transient movement but cannot be assumed to retain every dissolved ion. Color improves visibility without proving salt retention. A stationary level can reflect weakening concentration differences or leakage rather than osmotic thermodynamic equilibrium; a rising level alone supplies neither $n$ nor a verified value of $nRT/V$.

The component of traction normal to a surface with unit normal $n$. For an incompressible Newtonian fluid, $\sigma =-pI+2\mu e$, so ${\sigma }_{nn}=-p+2\mu n\cdot en$. A free-interface normal-stress jump includes the curvature force from surface tension.

For a broad, thin oil lens below a flat ice-water interface, define the effective interfacial energy $\Gamma ={\gamma }_{io}+{\gamma }_{ow}-{\gamma }_{iw}$ from the relevant surface tensions. At fixed oil volume $\mathcal{V}=Ah$, the energy is $E=\Gamma \mathcal{V}/h+\frac{1}{2}({\rho }_w-{\rho }_o)g\mathcal{V}h$. Setting its derivative to zero gives $h^2=2\Gamma /[({\rho }_w-{\rho }_o)g]$ for $\Gamma >0$, and the positive second derivative proves a minimum. A contact angle measured through the oil gives $\Gamma ={\gamma }_{ow}(1-\cos \vartheta )$. Oil-ice tension alone does not determine this effective quantity. Complete wetting or rough underside pockets invalidate this finite, flat-lens approximation. With densities $1025$ and $850\mathrm{kg}{\mathrm{m}}^{-3}$, a lens below $1\mathrm{cm}$ requires $0<\Gamma <0.08584\mathrm{N}{\mathrm{m}}^{-1}$.