Formula solvers starting with W — 51 solvers

Wall Penetration and Remaining Life

L = \frac{T - T_r}{P}

Corrosion & ProtectionHow many years of wall are left: the metal above the retirement thickness divided by the penetration rate. The number that turns a corrosion rate into an inspection interval.

Wall Shear Stress in a Capillary

\tau_w = \frac{\Delta P \, R}{2L}

Polymers & Plastics ProcessingA force balance and nothing else: the pressure pushing melt down a round channel has to be carried by shear at the wall, and this is what that shear works out to. Paired with the apparent shear rate it gives one point on a flow curve, and the slope of that curve is the power-law index.

Wall Thinning Rate and Remaining Life from Metal Loss

L = \frac{w}{P}, \qquad P = \frac{\Delta m}{\rho_M \, A \, \Delta t}

Erosion & SedimentA weighed loss of metal, spread over the area it came off and the time it took, is a thinning rate; the wall you can afford to lose divided by that rate is the remaining life. This is the arithmetic that turns an inspection result — a weight on a balance, or two ultrasonic readings — into a date on a maintenance plan.

Warp or Weft Cover Factor from Setting and Count

K_1 = \frac{n}{\sqrt{N_e}}

Textiles & FabricOne thread system's contribution to how much of the cloth is covered: threads per inch divided by the square root of the English cotton count. The square root is there because the yarn's diameter goes as the square root of its linear density, so this ratio is really diameter over spacing in disguise.

Water (Moisture) Content

w = \frac{M_w}{M_s}\times 100

Soil MechanicsStrength of MaterialsGravimetric water content of a soil as the mass of pore water divided by the mass of oven-dry solids, expressed as a percentage.

Water Hammer Surge (Joukowsky Equation)

\Delta P = \rho \, a \, \Delta v

HVAC & HydronicsFluid MechanicsPhysicsPeak pressure surge from a sudden change in flow velocity: fluid density times pressure-wave celerity times the velocity change.

Water Removed in Drying Grain

W_w = W_1 \, \frac{M_1 - M_2}{100 - M_2}

Crop ProductionThe mass of water that has to leave a lot of grain to take it from one moisture to another — the quantity a dryer is actually sized, fuelled and costed against.

Water Resistivity and Conductivity

\rho = \frac{1}{\sigma}

Water TreatmentChemistryRelates the resistivity and conductivity of ultrapure and deionized water as exact reciprocals, the pair of scales used on DI and RO polishing loops.

Water-Cement Ratio

\dfrac{w}{c} = \dfrac{m_w}{m_c}

ConcreteThe mass of free water in a batch divided by the mass of cementitious material in it. Two weights and a division, and it is the single number that governs strength, permeability, durability and very nearly everything else about hardened concrete.

Wave Energy Density

E = \frac{\rho g H^{2}}{8}

Coastal & Ocean WavesThe mean energy in the water column per square metre of sea surface, averaged over a wavelength. Half of it is potential — water lifted into the crests and taken out of the troughs — and half is kinetic, in the orbital motion. It goes as the SQUARE of the height, which is why a sea twice as high is four times the problem.

Wave Speed (v = fλ)

v = f \lambda

Waves & OscillationsPhysicsThe universal wave equation: a wave's speed equals its frequency times its wavelength.

Wave Speed on a String

v = \sqrt{\frac{F}{\mu}}

Waves & OscillationsPhysicsWaves travel faster on a tighter, lighter string: speed is the square root of tension over linear density.

Wave Steepness

S = \frac{H}{L}

Coastal & Ocean WavesHeight divided by wavelength: the one dimensionless number that says whether a wave is a gentle swell or a breaking sea. It sets whether the wave is stable, how hard it hits, and — through the surf-similarity parameter — what kind of breaker it becomes when it reaches the beach.

Weber Number (Inertia against Surface Tension)

We = \frac{\rho v^{2} L}{\sigma}

Fluid MechanicsDisrupting inertia divided by the surface tension holding a drop, a jet or a bubble together. Small We and the interface wins and the drop survives; large We and the flow tears it apart. It is the number behind atomisation, spray nozzles, raindrop size limits and the breakup of a liquid jet.

Webster Uniform Delay per Vehicle

d_1 = \frac{0.5\,C\,(1 - \lambda)^{2}}{1 - \lambda x}

Traffic Flow & SignalsThe first and largest term of Webster's delay equation: the average wait per vehicle when arrivals are perfectly uniform. It falls out of the area of the queue triangle over one cycle, and it is the part of delay that would remain even if traffic arrived like clockwork.

Webster's Optimum Cycle Length

C_o = \frac{1.5L + 5}{1 - Y}

Traffic Flow & SignalsThe cycle length that minimises total delay at a fixed-time signal, from Webster's 1958 Road Research Laboratory paper. Lost time pushes the cycle up, because a longer cycle spends a smaller share of itself on startup and clearance; heavy demand pushes it up much harder, because the denominator collapses as Y approaches one.

Weibull Modulus from Two Points

m = \frac{\ln\!\left[ \dfrac{\ln(1/P_1)}{\ln(1/P_2)} \right]}{\ln(\sigma_1 / \sigma_2)}

Ceramics & GlassThe slope of a Weibull plot taken between two points on it. Plot ln[ln(1/P_s)] against ln σ and the two-parameter Weibull becomes a straight line whose slope is m; this is that slope, worked from any two points that lie on it.

Weibull Strength Size Scaling

\frac{\sigma_1}{\sigma_2} = \left( \frac{V_2}{V_1} \right)^{\! 1/m}

Ceramics & GlassTake one ceramic, make two parts of different size out of it, and the bigger one is weaker. Not weaker in total load — weaker in STRESS, in megapascals, before any design decision is made. This is the most surprising and most consequential result in brittle materials, and it falls straight out of the weakest-link argument.

Weibull Survival Probability

P_s = \exp\!\left[ -\left( \frac{\sigma}{\sigma_0} \right)^{\! m} \right]

Ceramics & GlassThe chance a brittle part survives a stress, given the two numbers a Weibull fit produces: a characteristic strength and a modulus describing the scatter. It is the ceramic engineer's replacement for a single strength value, because a ceramic does not have one.

Weibull Survival with Volume Scaling

P_s = \exp\!\left[ -\frac{V}{V_0} \left( \frac{\sigma}{\sigma_0} \right)^{\! m} \right]

Ceramics & GlassThe survival probability with the specimen's size written into it. This is the form Weibull actually derived in 1939 from the weakest-link argument, and it is the one that explains why a laboratory bend bar and the component it was meant to qualify do not have the same strength.

Weight (W = mg)

W = m g

MechanicsPhysicsWeight of a mass at Earth's surface, using standard gravity g = 9.80665 m/s² (exact by definition).

Weight Component Along an Incline (mg sin θ)

F_{\parallel} = m g \sin\theta

MechanicsPhysicsComponent of an object's weight acting down the slope of an incline — the force that drives it toward the bottom.

Weight Utilisation

U_w = \frac{W}{W_{\max}}

Freight & ShippingThe share of the payload limit a load actually uses. Paired with cube utilisation it settles the only question that matters when a trailer leaves half empty: was it space or weight that stopped you?

Weighted Mean of Two Groups

\bar{x} = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}

StatisticsAlgebraCombines the averages of two groups into one overall mean, weighting each group by how many observations it contains.

Weld Deposition Rate

D = \eta_d \, \rho \, \frac{\pi \, d^{2}}{4} \, w

Welding & JoiningHow much weld metal an arc actually lays down per hour: the cross-section of the wire times how fast it is fed, times the density, times the fraction that survives as deposit rather than leaving as spatter, slag or fume. The number that turns a welding procedure into an estimate.

Weld Metal Volume, Single-V Groove

V_w = L \left( g \, t + t^{2} \tan\frac{\alpha}{2} \right)

Welding & JoiningThe cross-sectional area of a single-V groove — the root gap's rectangle plus the two triangles the bevel opens up — multiplied by the length of the joint. The starting point for every consumable estimate and every argument about bevel angle.

Welding Heat Input

H = \frac{\eta \, V \, I}{S}

Welding & JoiningThe energy the arc delivers into every millimetre of joint: volts times amperes gives the power at the arc, arc efficiency takes off what never reaches the plate, and travel speed divides it over the length welded. The one number every welding procedure specification records, and the one most often mistaken for a verdict on the weld.

Well Efficiency

E = \frac{BQ}{BQ + CQ^{2}} = \frac{B}{B + CQ}

Water & WastewaterThe fraction of a well's drawdown that the aquifer is responsible for, rather than the well's own construction. Everything below 100 per cent is head thrown away in turbulence through the screen and the gravel pack, and unlike the aquifer, that part can be fixed.

Well Screen Entrance Velocity

V_e = \frac{Q}{\pi \, D \, L \, P}

Water & WastewaterVelocity of water entering a well screen: the pumping rate divided by the open area, which is the screen's cylindrical surface multiplied by the fraction of it that is actually slot. The long-standing design limit is 0.03 m/s, or 0.1 ft/s.

Wet Steam Enthalpy from h_f and h_g

h = (1 - x) \, h_f + x \, h_g

ThermodynamicsWater TreatmentEnthalpy of a wet steam mixture as the mass-weighted blend of saturated water and dry saturated steam — the lever rule on the h_f and h_g columns.

Wet-Bulb Temperature (Stull 2011)

T_w = T\,\arctan\!\left[0.151977\sqrt{\mathrm{RH} + 8.313659}\,\right] + \arctan(T + \mathrm{RH}) - \arctan(\mathrm{RH} - 1.676331) + 0.00391838\,\mathrm{RH}^{3/2}\arctan(0.023101\,\mathrm{RH}) - 4.686035

Snow & IceHVAC & HydronicsThermodynamicsWet-bulb temperature from dry-bulb temperature and relative humidity in a single closed-form expression, fitted by Roland Stull in 2011 to replace the iterative psychrometric solve. It is the lowest temperature evaporation alone can reach, which makes it the control variable for cooling towers, evaporative coolers, heat-stress limits and snowmaking alike.

Wien's Displacement Law

\lambda_{max} = \frac{b}{T}

Modern PhysicsThermodynamicsPhysicsThe peak wavelength of thermal radiation, with b = 2.8978 × 10⁻³ m·K.

Wilke–Chang Liquid Diffusivity

D_{AB} = 7.4 \times 10^{-8} \, \frac{(\phi M_B)^{1/2} \, T}{\mu_B V_A^{0.6}}

Chemical EngineeringAn estimate of how fast a solute diffuses through a liquid, from the solvent's molar mass, viscosity and association behaviour plus the solute's molar volume. Wilke and Chang's 1955 correlation, still the default when nobody has measured the real thing.

Wind Chill (2001 North American Formula)

T_{wc} = 13.12 + 0.6215\,T - 11.37\,V^{0.16} + 0.3965\,T\,V^{0.16}

Everyday & HealthThermodynamicsThe wind chill index adopted by Canada and the United States in 2001, from air temperature and wind speed at 10 m.

Wind Correction Angle

\mathrm{WCA} = \arcsin\left(\frac{W\sin\theta}{V}\right)

Navigation & PositionHow far off the desired track to point the nose so the wind carries you along it instead of off it. Only the crosswind component matters: the along-track part of the wind changes the speed, never the heading.

Wind Power Density

\frac{P}{A} = \tfrac{1}{2} \rho v^{3}

Solar & Wind PowerThe power carried past every square metre of open air by a wind of speed v. The figure that ranks one site against another before any turbine is chosen, and the place the cube law first bites.

Wind Shear Power Law

v_2 = v_1 \left( \frac{h_2}{h_1} \right)^{\alpha}

Solar & Wind PowerWind speed at one height from a measurement at another. Ground friction slows the lowest air, so the wind at a 100 m hub is meaningfully stronger than the wind at the 10 m mast that measured it — and the cube law turns that difference into a large one.

Wind Speed at Height (Power Law)

u_2 = u_1 \left(\frac{z_2}{z_1}\right)^{p}

Air Quality & DispersionExtrapolates a wind measurement to another height using the power-law profile. Anemometers sit at 10 m and stacks do not, so this step precedes every plume-rise calculation.

Wind Triangle Ground Speed

V_g = \sqrt{V^{2} - W^{2}\sin^{2}\theta} - W\cos\theta

Navigation & PositionWhat the aircraft is actually making over the ground once the wind has had its say: true airspeed, wind speed, and the angle between the wind and the intended track. The E6B's whole reason for existing.

Wind Turbine Power Output

P = \tfrac{1}{2} \rho A v^{3} C_p

Solar & Wind PowerThe mechanical power a rotor of swept area A takes from a wind of speed v, where the power coefficient Cp is the fraction of the available kinetic flux it actually captures.

Wind Velocity Pressure (qz = 0.613 Kz Kzt Kd V²)

q_z = 0.613 \, K_z K_{zt} K_d V^{2}

Civil & SurveyingVelocity pressure at height z from the ASCE 7 wind provisions, SI form: 0.613 times the exposure, topographic and directionality factors times the basic wind speed squared, in pascals.

Wing Aspect Ratio

AR = \frac{b^{2}}{S}

Aerospace & FlightHow slender a wing is: span squared divided by planform area. For a rectangular wing it is simply span over chord, and for every other wing it is the number that decides how expensive lift will be.

Wing Loading

w = \frac{W}{S}

Aerospace & FlightWeight carried per unit of wing area. One number, and it sets the stall speed, the ride in turbulence, the turn radius and the runway length — which is why it is the first figure quoted about any new design.

Wire Feed Speed and Welding Current (Burn-Off)

w = a \, I + b \, L \, I^{2}

Welding & JoiningThe relation that ties wire feed speed to welding current on a constant-voltage machine: a term proportional to current for the heat delivered at the arc, plus a term in current squared for the resistive heating of the electrode extension. It is why longer stickout gives more wire at the same amps — and less penetration.

Womersley Number (pulsatile flow)

\alpha = R \sqrt{\frac{2\pi f \rho}{\mu}}

Biomedical & ClinicalWhether a pulsing flow has time to develop a proper velocity profile between beats. Small alpha and every instant looks like steady Poiseuille flow; large alpha and the core moves as a plug with viscosity confined to a thin layer at the wall.

Work (W = Fd cos θ)

W = F d \cos\theta

MechanicsPhysicsWork done by a constant force acting at an angle to the displacement.

Work from Force and Displacement Components

W = F_x d_x + F_y d_y

Vectors & MatricesPhysicsMechanicsComputes the work done by a force from the components of the force and the displacement, without needing the angle between them.

Work–Energy Theorem

W = \tfrac{1}{2} m \left(v^{2} - v_0^{2}\right)

MechanicsPhysicsNet work done on an object equals its change in kinetic energy, linking force and distance to a change in speed.

Wright's Learning Curve (Unit Time)

T_n = T_1\,n^{\,\log b / \log 2}

Industrial EngineeringTime to build the nth unit when every doubling of cumulative output cuts the time to a fraction b of what it was: the aircraft-industry curve that still prices first-of-a-kind work.

Wye Line and Phase Voltage

V_{L} = \sqrt{3} \, V_{\varphi}

Electrical TradeElectricity & MagnetismIn a wye (star) connection the line-to-line voltage is √3 times the line-to-neutral phase voltage of each winding.

Wye to Delta Resistance Transformation

R_{ab} = \frac{R_{A} R_{B} + R_{B} R_{C} + R_{C} R_{A}}{R_{C}}

Electricity & MagnetismElectrical TradeThe delta leg between nodes A and B equivalent to a three-arm wye: the sum of the pairwise arm products divided by the arm at the opposite node.