Mean-drift contribution to a turbulence-intensity estimate is the apparent fluctuation produced when a changing mean velocity is included in the deviations used to estimate wind-tunnel turbulence intensity. Subtracting one mean from an entire record removes its constant offset, not its slow evolution.

An original continuous-record model on $0\le t\le T$ isAssume the time mean of $\epsilon$ is zero. Writing $⟨g⟩=T^{-1}{\int }_0^{T}g(t)dt$, the record has mean $U_0$ andThe first term follows by integrating $(t-T/2{)}^2$. The last is a covariance term: it need not vanish in one finite record even when a stochastic model predicts zero in expectation.

With no random fluctuation, take $U_0=30$ m/s, $a=0.02\mathrm{m}{\mathrm{s}}^{-2}$ and $T=60$ s. The drift contributes standard deviation $|a|T/\sqrt{12}\approx 0.3464$ m/s, so an estimate ${\sigma }_u/U_0$ reports about 1.155% despite zero stipulated turbulence. If an uncorrelated fluctuation has standard deviation 0.24 m/s, the combined model gives $\sqrt{0.12+0.0576}/30\approx 1.405\%$, rather than its fluctuation-only value of 0.800%.

Removing a fitted trend changes the measured quantity. It can remove genuine slowly varying turbulence as well as instrumental or operating drift, so a reduced residual variance does not identify the physical origin of the removed signal. Report recording duration and the filtering or trend-removal procedure with the estimate.

The NASA tunnel experiment used hot-wire anemometry with separate mean and fluctuating signal paths and specified their filters.^[ref] Its apparatus illustrates why a reported fluctuation level includes a bandwidth. The linear-drift calculation here is a constructed diagnostic example, not a reinterpretation of those tunnel data.

An empty-tunnel support tare is the wind-on aerodynamic load measured with the support arrangement present but the test model absent. Subtracting that load from the model-plus-support measurement removes a specified support contribution under matched conditions. It does not automatically remove the wind-tunnel support interaction residual created by their combination.

The word empty here means empty of the test model, not empty of its mounting hardware. A wind-off zero or orientation tare addresses other contributions, such as instrument offset or the changing component of gravitational weight. Neither measurement establishes the support's aerodynamic load in moving air.

An original scalar example uses model-plus-support force 12.00 N and matched support-only force 1.08 N. Their difference is 10.92 N. With air density $\rho =1.2\mathrm{kg}{\mathrm{m}}^{-3}$, speed $U=30$ m/s and reference cross-sectional area $A=0.10{\mathrm{m}}^2$, the dynamic pressure and load normalization areIf the channel is drag, the support tare corresponds to drag coefficient increment $1.08/54=0.0200$, and the corrected measured coefficient is $10.92/54\approx 0.20222$. Those numbers describe the subtraction; they do not certify an isolated-model drag coefficient when interference remains.

At the same normalization, a 0.10 N tare error contributes about 0.00185 to the coefficient error. A tare measured at the wrong speed, orientation or support geometry can therefore matter even when the force offset appears small. Document conditions and propagate measurement uncertainty, rather than treating a stored tare as exact for every run.

The variable-density-tunnel support-interference investigation demonstrates why direct support load and interference require separate attention.^[ref] The arithmetic above is an original normalized-load example, not a reported tunnel result.

A wind-tunnel support interaction residual is the part of a measured aerodynamic load caused by placing model and support together that remains after subtracting the support's separately measured load. The support can alter the fluid flow around the model, and the model can alter the fluid flow around the support. Their combined load need not equal the sum of two isolated loads.

For an original four-state measurement model, let $y_{00}$ be the common background, $y_{10}$ the isolated-model channel, $y_{01}$ the support-only channel and $y_{11}$ the combined channel, all at matched conditions. DefineIf $F_M=y_{10}-y_{00}$ is the desired isolated-model load, support-tare subtraction givesThe four-state expression defines an interaction contrast; it does not promise that an isolated-model state is experimentally accessible without some alternative mounting arrangement.

Agreement between two mounting corrections can be misleading. In a constructed example, let the isolated-model force be 20 N. Two supports have isolated loads 3 and 5 N, but each creates an interaction load of 2 N. Combined measurements are 25 and 27 N; after their respective tares both report 22 N. The perfect agreement leaves the same 10% error. This is common-mode support interference in mounting comparisons.

An independent reference, a justified interference model or experiments that constrain the interaction are needed to remove that remaining ambiguity. Changing a support and observing a difference establishes mounting sensitivity; observing no difference does not establish zero common interaction.

The effect has concrete experimental history. The 1939 variable-density-tunnel report found that support interference affected measured airfoil drag coefficients and their apparent dependence on thickness.^[ref] A correction can therefore alter a comparative scientific conclusion, not merely the zero of one load channel.

A rain gauge measures precipitation collected over a known horizontal area. The reported rainfall depth is collected liquid volume divided by collector area: $h=V/A$. A tipping-bucket gauge converts repeated fixed-volume tips into accumulated depth; a weighing gauge measures collected mass.

Wind can deflect drops around the opening, and evaporation, splashing or freezing can bias the result. A gauge samples one location, so its reading need not represent a whole catchment during spatially uneven rain. The instrument geometry and exposure are therefore part of interpreting the observation.

Wavefront curvature describes the change in the normal direction across a surface of equal wave phase. A plane wavefront has zero curvature; a spherical wavefront has curvature magnitude $1/|R|$ for radius $R$. A general wavefront can have different principal curvatures in different transverse directions. The sign convention must be stated when distinguishing converging and diverging waves.^[ref]

In a paraxial transverse plane, choose the convention in which a diverging spherical wave has $R>0$. Its quadratic phase variation is approximatelyThe corresponding local ray angle is approximately $r/R$. For illustrative $R=0.50\mathrm{m}$ and $r=2\mathrm{mm}$, the angle is $0.004$ radians and the optical path difference is $r^2/(2R)=4\mu \mathrm{m}$. At wavelength $500\mathrm{nm}$, that is eight phase cycles. A modest geometrical departure can therefore produce a substantial optical phase difference.

For an ideal Gaussian beam with its waist at $z=0$,The reciprocal form remains finite at the waist: its value is zero, so the wavefront there is locally planar even though the beam radius is minimal. At $z=z_R$, $R=2z_R$. Beam size and curvature are distinct beam parameters.^[ref]

A thin lens changes the curvature according to $1/R^{\prime }=1/R-1/f$ in this convention, while the beam radius at the lens is unchanged in the thin-element approximation.^[ref] Thus a lens can alter convergence without an immediate jump in the beam width.

Experimentally, curvature can be inferred from a measured phase profile or local wavefront slopes. Phase unwrapping and the chosen reference matter because an intensity image in one plane does not itself provide the quadratic phase. A curved wavefront is not necessarily aberrated: the appropriate ideal reference can itself be spherical or Gaussian-beam curvature.