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Formula solvers — S

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S-Wave Velocity from Shear Modulus

Vs=GρV_s = \sqrt{\dfrac{G}{\rho}}

SeismologyHow fast a shear wave travels: shear modulus over density, square-rooted. The slower of the two body waves, the one that does most of the shaking damage, and the single number site classification codes are written around.

Sabine Reverberation Time (RT60)

T60=0.161 VAT_{60} = \frac{0.161\,V}{A}

Acoustics & NoiseHow long a sound takes to fade by 60 decibels after the source stops — the single number that decides whether a room is a concert hall, a classroom or a swimming pool. Big rooms ring; absorptive rooms do not.

Sacrificial Anode Mass for a Required Life

W=I tC uW = \frac{I \, t}{C \, u}

Corrosion & ProtectionHow much anode metal a cathodic protection system has to carry to supply its current for the design life, given the anode alloy's current capacity and the fraction of it that can be consumed before the anode stops working.

Safe Storage Time

t=t0⋅2−ΔM⋅2−ΔT/5.56t = t_0 \cdot 2^{-\Delta M} \cdot 2^{-\Delta T / 5.56}

Farm OperationsHow long grain will keep before spoilage, as a halving rule: storage life halves for every percentage point of moisture above the reference condition, and halves again for every 10 F° of warming.

Safety Stock (Statistical Buffer)

SS=z σd L\mathit{SS} = z\,\sigma_d\,\sqrt{L}

Industrial EngineeringBuffer that absorbs demand variability over the lead time: a service-level z value times the standard deviation of daily demand times the square root of the lead time in days.

Sales Tax and Total Price

T=P (1+r)T = P\,(1 + r)

Money & BusinessTotal payable from a pre-tax price and a tax rate, and the reverse: recovering the pre-tax price from a tax-inclusive total.

Salt Dose per Regeneration

msalt=D Vm_{\text{salt}} = D \, V

Water TreatmentChemistryGives the mass of sodium chloride a softener draws each regeneration from the resin volume and the programmed salt dosage per unit volume.

Sample Size for a Mean

n=(zσE)2n = \left( \frac{z \sigma}{E} \right)^{2}

StatisticsAlgebraHow many observations a study needs to estimate a mean within a target margin of error at a chosen confidence level.

Sample Size for a Proportion

n=z2 p(1−p)E2n = \frac{z^{2} \, p (1 - p)}{E^{2}}

StatisticsAlgebraHow many respondents a survey needs to estimate a percentage within a target margin of error at a chosen confidence level.

Saturated Unit Weight

γsat=(Gs+e) γw1+e\gamma_{sat} = \frac{(G_s + e)\,\gamma_w}{1 + e}

Soil MechanicsStrength of MaterialsUnit weight of a soil whose voids are completely full of water, from the specific gravity of the solids and the void ratio.

Saturation pH (pHs) for Langelier's Index

pHs=(9.3+A+B)−(C+D)\mathrm{pH_s} = (9.3 + A + B) - (C + D)

Water TreatmentChemistryComputes the pH at which water is saturated with calcium carbonate from its TDS, temperature, calcium hardness and total alkalinity as CaCO₃.

Saturation Temperature and Pressure of Steam

Tsat=Ts(psat)psat=ps(Tsat)T_{sat} = T_s(p_{sat}) \qquad p_{sat} = p_s(T_{sat})

ThermodynamicsWater TreatmentHVAC & HydronicsSaturation temperature from pressure and saturation pressure from temperature, on the IAPWS-IF97 Region 4 line the printed steam tables come from.

Saturation Vapour Pressure (Magnus / Alduchov–Eskridge)

pws=610.94exp⁡ ⁣(17.625 tt+243.04)p_{ws} = 610.94 \exp\!\left(\frac{17.625\,t}{t + 243.04}\right)

HVAC & HydronicsThermodynamicsAir Quality & DispersionThe vapour pressure of water at saturation, from temperature alone, by the Alduchov–Eskridge Magnus fit — the ceiling every other psychrometric quantity is measured against.

SAVI — Soil-Adjusted Vegetation Index

SAVI=(ρNIR−ρred)(1+L)ρNIR+ρred+L\mathrm{SAVI} = \frac{(\rho_{NIR} - \rho_{red})(1 + L)}{\rho_{NIR} + \rho_{red} + L}

Photogrammetry & Remote SensingNDVI with a soil correction: a constant L in the denominator, and a matching (1 + L) on top to keep the range. Huete introduced it in Remote Sensing of Environment 25 (1988) 295 after showing that NDVI over partial canopy moves with soil brightness rather than with the plants. It exists because NDVI fails on sparse cover, which is exactly when an agronomist most wants a reading.

Scalar Projection of One Vector onto Another

compb⃗a⃗=a⃗⋅b⃗∣b⃗∣\text{comp}_{\vec{b}}\vec{a} = \frac{\vec{a}\cdot\vec{b}}{|\vec{b}|}

Vectors & MatricesAlgebraPhysicsMeasures how far a vector reaches along the direction of another, the length of the shadow it casts on that second vector.

Scalar Triple Product (Parallelepiped Volume)

V=∣a⃗⋅(b⃗×c⃗)∣V = \left|\vec{a}\cdot(\vec{b}\times\vec{c})\right|

Vectors & MatricesGeometryAlgebraFinds the volume of the parallelepiped spanned by three space vectors, from the nine components, as the size of their scalar triple product.

Scan Time per Layer

ts=Av ht_s = \frac{A}{v \, h}

Additive ManufacturingHow long the beam takes to hatch one layer: the area to be scanned divided by the rate the beam covers area, which is speed times hatch spacing. The per-layer number that the layer count multiplies into a build time.

Scheil–Gulliver Segregation

CS=k C0(1−fS)k−1C_S = k \, C_0 \left( 1 - f_S \right)^{k-1}

Metallurgy & Heat TreatmentWhat actually happens when an alloy freezes at a real rate: the first solid is lean in solute, the rejected solute piles up in the shrinking liquid, and every later layer of solid is richer than the one before. Gulliver set out the argument in 1913 and Scheil put it in this form in 1942, and it is why castings are cored and why homogenisation anneals exist.

Schmidt Number

Sc=μρD\mathrm{Sc} = \frac{\mu}{\rho D}

Chemical EngineeringThe ratio of momentum diffusivity to mass diffusivity — how readily a fluid spreads motion compared with how readily it spreads molecules. The mass-transfer twin of the Prandtl number, and the fluid property every mass-transfer correlation is written around.

Schwarzschild Radius

rs=2GMc2r_s = \frac{2GM}{c^{2}}

Astronomy & GravitationPhysicsRadius of the event horizon of a non-rotating black hole of mass M.

SCS Curve Number Runoff Depth

Q=(P−0.2S)2P+0.8S,S=25400CN−254Q = \frac{(P - 0.2S)^2}{P + 0.8S}, \quad S = \frac{25400}{CN} - 254

Water & WastewaterCivil & SurveyingDirect runoff depth from a storm by the SCS (NRCS) curve-number method, with the potential maximum retention S in millimetres taken from the curve number and the initial abstraction fixed at 0.2S.

SCS Lag Time of Concentration

tc=L0.8 (S+1)0.71140 Y0.5t_c = \frac{L^{0.8} \, (S + 1)^{0.7}}{1140 \, Y^{0.5}}

Water & WastewaterCivil & SurveyingTime of concentration from the SCS watershed lag equation, in which the curve number carries the retardance: lag equals flow length to the 0.8 power times the retention term over 1900 times the root of the percent slope, and the time of concentration is the lag divided by 0.6.

SCS Triangular Unit Hydrograph Peak

qp=2 A Q2.67 Tpq_p = \frac{2 \, A \, Q}{2.67 \, T_p}

Water & WastewaterCivil & SurveyingFluid MechanicsPeak discharge of the SCS synthetic triangular unit hydrograph, from the runoff volume spread over a base length of 2.67 times the time to peak.

Seasonal Heating Energy (Degree-Day Method)

E=Q˙d ΔTm tΔTd ηE = \frac{\dot{Q}_d \, \Delta T_m \, t}{\Delta T_d \, \eta}

HVAC & HydronicsThermodynamicsEstimates seasonal fuel energy by scaling the design heat loss with the average temperature deficit, season length and equipment efficiency.

Second-Order Integrated Rate Law

1[A]=1[A]0+kt\frac{1}{[\mathrm{A}]} = \frac{1}{[\mathrm{A}]_0} + kt

ChemistryGives the concentration remaining in a second-order reaction, where the reciprocal of concentration rises linearly with time.

Seed Drill Calibration from a Catch Test

R=md wR = \frac{m}{d \, w}

Crop ProductionThe rate a drill is actually sowing, from seed caught over a measured distance and a measured width. The only figure that settles what the machine is doing, as against what its chart says it should be doing.

Seeding Rate from Target Plant Population

S=P⋅TKW1000 G ES = \frac{P \cdot TKW}{1000 \, G \, E}

Crop ProductionThe seeding rate in weight per unit area that establishes a target plant population, from the thousand-kernel weight, the tag germination and the field emergence you expect to get.

Seeds per Unit of Seed Weight

n=1000 MTKWn = \frac{1000 \, M}{TKW}

Crop ProductionHow many seeds a given weight of seed contains, from the thousand-kernel weight. The step between a seed order written in kilograms and a planter set in seeds.

Seepage Velocity from Discharge Velocity

vs=vnv_s = \frac{v}{n}

Soil MechanicsFluid MechanicsActual seepage velocity through the pores, obtained by dividing Darcy's fictitious discharge velocity by the porosity of the soil.

Seismic Base Shear (V = Cs W)

V=CsWV = C_s W

Civil & SurveyingTotal lateral force an earthquake is designed to put into a building's base, from the ASCE 7 equivalent lateral force procedure: a seismic response coefficient times the effective seismic weight.

Seismic Moment

M0=μADˉM_0 = \mu A \bar{D}

SeismologyThe physical size of an earthquake: the shear modulus of the rock, times the area of fault that broke, times the average slip across it. The only measure of earthquake size that carries real units, and the quantity every modern magnitude is computed from.

Selection Differential

S=xˉs−xˉS = \bar{x}_s - \bar{x}

Plant Breeding & GeneticsHow much better the selected parents are than the population they were chosen from — the input to the breeder's equation, and the part the breeder actually controls.

Sensible Heat (Q = mcΔT)

Q=mcΔTQ = m c \Delta T

ThermodynamicsPhysicsChemistryHeat needed to change a mass's temperature: mass times specific heat times the temperature change.

Sensible Heat Ratio (SHR)

SHR=Q˙sQ˙s+Q˙l\mathrm{SHR} = \frac{\dot{Q}_s}{\dot{Q}_s + \dot{Q}_l}

HVAC & HydronicsThermodynamicsThe fraction of a cooling coil's total load that is sensible, the number that decides whether a room ends up cool or merely cold and clammy.

Sensitivity (True Positive Rate)

Se=TPTP+FN\mathrm{Se} = \frac{\mathrm{TP}}{\mathrm{TP} + \mathrm{FN}}

Epidemiology & Diagnostic AccuracyThe share of people who genuinely have the condition that the test correctly calls positive — read straight down the disease-present column of a 2×2 table.

Series RL or RC Impedance

Z=R2+X2Z = \sqrt{R^{2} + X^{2}}

Electrical TradeElectricity & MagnetismImpedance magnitude of a resistance in series with a single reactance, whether that reactance is inductive or capacitive.

Series RLC Impedance

Z=R2+(XL−XC)2Z = \sqrt{R^{2} + (X_{L} - X_{C})^{2}}

Electrical TradeElectricity & MagnetismMagnitude of impedance in a series RLC circuit, combining resistance with the net reactance left after XL and XC cancel.

Series System Reliability

Rs=RnR_s = R^{n}

Computer ScienceProbabilityReliability of n identical components that must all work: the individual reliability raised to the nth power. Every dependency you add makes the whole thing worse, never better.

Serum Anion Gap

AG=Na+−(Cl−+HCO3−)\mathrm{AG} = \mathrm{Na^+} - (\mathrm{Cl^-} + \mathrm{HCO_3^-})

Biomedical & ClinicalChemistryThe difference between the measured cation and the measured anions in serum, the standard first split in the workup of a metabolic acidosis.

Set and Drift of the Current

Dr=Δn2+Δe2tD_r = \frac{\sqrt{\Delta n^{2} + \Delta e^{2}}}{t}

Navigation & PositionThe current the water was making, worked backwards from the gap between where dead reckoning said you were and where the fix put you. Drift is its speed; set is the true direction it flows toward.

Sewer Credit for Evaporated Water

Cc=Ve psC_c = V_e \, p_s

Water TreatmentFluid MechanicsWater & WastewaterSewer credit for water a tower evaporates: the volume that never reaches the drain, valued at the municipal sewer rate.

Shaft Diameter from Allowable Torsional Shear

d=16Tπτ3d = \sqrt[3]{\frac{16 T}{\pi \tau}}

Machine DesignThe diameter a solid round shaft needs so that the torsional shear stress at its surface stays within an allowable value. The torsion formula rearranged, and the reason a shaft carrying eight times the torque only has to be twice as thick.

Shaft Torque from Power and Angular Speed

T=PωT = \frac{P}{\omega}

Machine DesignThe torque in a rotating shaft: transmitted power divided by angular speed, with ω in radians per second. The relation that decides how thick every shaft in a machine has to be, because a shaft feels the torque and never the power.

Shallow Box Validity Check for McKee

r=7dZr = \frac{7 d}{Z}

Packaging & Box StrengthMcKee's formula assumes the box fails by its side panels buckling, and a box has to be deep enough for panels to buckle. Below a depth of about the perimeter divided by seven they do not — the panels crush in compression instead, the caliper term stops meaning anything, and the equation is wrong in FORM rather than merely out of calibration. This ratio is one when the box sits exactly on that limit.

Shallow-Water Wave Celerity

c=g dc = \sqrt{g\,d}

Coastal & Ocean WavesOnce the water is shallower than about a twentieth of the wavelength, the period drops out of the celerity entirely and every wave travels at the square root of g times the depth. This is why tsunamis cross oceans at jetliner speed, why a bore runs up an estuary at a fixed pace, and why waves bend to face the beach.

Shannon Diversity Index (H′)

H′=−∑ipiln⁡piH' = -\sum_{i} p_i \ln p_i

Ecology & BiodiversityStatisticsDiversity of a community of up to four species from their proportional abundances, in nats. It rises with both the number of species and the evenness of their shares, and it is Shannon's information entropy with species where the symbols were.

Shannon Entropy of a Binary Source

H=−plog⁡2p−(1−p)log⁡2(1−p)H = -p\log_2 p - (1 - p)\log_2(1 - p)

Computer ScienceProbabilityAverage information carried by each symbol of a two-outcome source, in bits. It peaks at exactly 1 bit for a fair coin and falls to zero as the outcome becomes certain.

Shannon-Hartley Channel Capacity

C=Blog⁡2 ⁣(1+SN)C = B\log_2\!\left(1 + \frac{S}{N}\right)

Computer ScienceHighest error-free bit rate a channel of bandwidth B can carry at a given signal-to-noise power ratio. Bandwidth in hertz gives a capacity in bits per second.

Shear Flow (q = VQ/I)

q=VQIq = \frac{V Q}{I}

Strength of MaterialsCivil & SurveyingMechanicsLongitudinal shear force per unit length that must cross a joint in a built-up beam — the number that sets nail spacing, bolt pitch and weld size.

Shear Modulus (G = τ/γ)

G=τγG = \frac{\tau}{\gamma}

Strength of MaterialsMechanicsPhysicsShear (rigidity) modulus as shear stress divided by shear strain, roughly 0.38 of Young's modulus for common metals.

Shear Stress in a Parallel Key

τ=2Td w L\tau = \frac{2T}{d \, w \, L}

Machine DesignThe average shear stress across a parallel key in a shaft keyway: the tangential force at the shaft surface spread over the key's width times its length. The check that decides whether the key or the shaft gives way first — and it is meant to be the key.

Shear Stress on a Fillet Weld Throat

τ=F0.707 wL\tau = \frac{F}{0.707 \, w L}

Machine DesignShear stress on the effective throat of an equal-leg fillet weld: the load divided by 0.707 times the leg size times the weld length. A fillet weld is always checked in shear on the throat, whichever way the load pulls.

Shear Velocity (Friction Velocity u*)

u∗=τ0ρu_* = \sqrt{\frac{\tau_0}{\rho}}

Erosion & SedimentA shear stress divided by a density has the units of a velocity squared, and the square root of it is the shear velocity — not a velocity anything actually travels at, but the natural velocity scale of a turbulent boundary layer. The log law, the Rouse number and every roughness correlation are written in terms of it.

Sheets or Tiles Needed (with Waste Allowance)

N=A(1+w100)aN = \frac{A\left(1 + \frac{w}{100}\right)}{a}

Trades & ConstructionCounts the sheets, tiles or planks a surface takes: the area to cover, padded by a percent waste allowance for cuts and breakage, divided by the area one unit covers.

Sherman IDF Rainfall Intensity

i=atci = \frac{a}{t^{c}}

Water & WastewaterCivil & SurveyingFluid MechanicsSherman's power-law IDF form: intensity falls as the duration raised to a fitted negative power, with no offset term. It is the straight line an IDF curve becomes on log-log paper, which is why a and c can be read off a fitted plot directly. Both are regional constants for a single return period, and a assumes intensity in mm/h against duration in minutes.

Sherwood Number

Sh=kcLD\mathrm{Sh} = \frac{k_c L}{D}

Chemical EngineeringThe convective mass transfer coefficient made dimensionless: transfer with the flow divided by transfer by diffusion alone. Named for Thomas Sherwood, whose 1934 wetted-wall work with Gilliland set the pattern every correlation since has followed.

Shields Parameter (Dimensionless Shear Stress)

θ=τ(ρs−ρ) g d\theta = \frac{\tau}{(\rho_s - \rho) \, g \, d}

Erosion & SedimentAlbert Shields' 1936 dissertation reduced the whole question of whether a grain moves to one ratio: the shear stress the flow applies to the bed, divided by the submerged weight of a grain spread over its own area. Above about 0.03 to 0.06 the bed is in motion; below it, mostly not. It is the single most used number in sediment transport.

Shipment Density

ρ=mV\rho = \frac{m}{V}

Freight & ShippingMass divided by the space it takes up — the ordinary definition of density, applied to a skid or a shipment and measured in pounds per cubic foot, because that is the number LTL freight classification is built on.

SHM Displacement at Time t

x=Acos⁡(ωt)x = A \cos\left(\omega t\right)

MechanicsPhysicsDisplacement of a simple harmonic oscillator at time t, a cosine of amplitude A and angular frequency ω released from full stretch.

SHM Maximum Acceleration

amax⁡=Aω2a_{\max} = A \omega^{2}

MechanicsPhysicsMaximum acceleration of a simple harmonic oscillator, reached at the turning points where the restoring force is largest.

SHM Maximum Velocity

vmax⁡=Aωv_{\max} = A \omega

MechanicsPhysicsMaximum speed of a simple harmonic oscillator, reached at the equilibrium point, equal to amplitude times angular frequency.

Shoaling Coefficient

Ks=cg0cgK_s = \sqrt{\frac{c_{g0}}{c_g}}

Coastal & Ocean WavesAs a wave moves into shallower water it slows down, and because it keeps delivering the same energy per second, the height has to rise to compensate. The shoaling coefficient is the ratio H/H₀ that results, and it comes straight out of conserving energy flux between deep water and the point of interest.

Shockley Diode Equation

I=Is(eV/(nVT)−1),VT=kTqI = I_s\left(e^{V/(n V_T)} - 1\right), \quad V_T = \frac{kT}{q}

Semiconductors & ChipsCurrent through a p-n junction as a function of the voltage across it. The thermal voltage kT/q is computed from the temperature you enter — 25.85 mV at 300 K — because the temperature dependence is the whole point of the equation.

Shrinkage: Compacted Volume from Bank Volume

VC=VB(1−Sh100)V_C = V_B\left(1 - \frac{S_h}{100}\right)

Civil & SurveyingSoil MechanicsConverts bank volume into the compacted volume it fills in an engineered embankment, using the soil's percent shrinkage.

Simple Interest

I=P r tI = P \, r \, t

AlgebraInterest earned on a principal at a flat rate per period, without compounding.

Simple Payback Period

t=CSt = \frac{C}{S}

Money & BusinessYears for an upgrade to repay its own capital cost out of the money it saves each year, ignoring interest and inflation.

Simple Pendulum Period

T=2πLgT = 2\pi \sqrt{\frac{L}{g}}

Waves & OscillationsMechanicsPhysicsPeriod of a simple pendulum swinging through small angles, with g = 9.80665 m/s² (standard gravity).

Simpson's Index of Diversity (1 − D)

1−D=1−∑ipi21 - D = 1 - \sum_{i} p_i^{2}

Ecology & BiodiversityStatisticsProbability that two individuals drawn at random from a community of up to four species belong to DIFFERENT species. This page returns 1 − D, which rises with diversity; Simpson's own D = Σ pᵢ² is the dominance and runs the opposite way.

Single-Phase Real Power with Power Factor

P=VI PFP = V I \, \text{PF}

Electrical TradeElectricity & MagnetismTrue power of a single-phase AC load: volts times amps times power factor, the fraction of the current doing real work.

Single-Screw Drag Flow

Qd=π2D2NHsin⁡φcos⁡φ2Q_d = \frac{\pi^{2} D^{2} N H \sin\varphi \cos\varphi}{2}

Polymers & Plastics ProcessingRowell and Finlayson's 1928 result, unchanged since: the melt in a screw channel is dragged forward by the barrel moving over it, exactly as fluid is dragged between a moving plate and a fixed one. This is the forward half of a single-screw extruder's output, and it is proportional to screw speed and to channel depth — a straight line through the origin on an output-against-rpm plot.

Single-Screw Pressure Flow

Qp=πDH3ΔPsin⁡2φ12ηLQ_p = \frac{\pi D H^{3} \Delta P \sin^{2}\varphi}{12 \eta L}

Polymers & Plastics ProcessingThe die at the end of an extruder builds pressure, and that pressure drives melt BACK up the screw channel against the drag. This is the same Poiseuille flow that runs a pipe, written for a shallow rectangular channel wrapped round a screw — and it is why an extruder's output falls as the die is made more restrictive.

Skin Depth

δ=2ρωμ\delta = \sqrt{\frac{2\rho}{\omega\mu}}

Electricity & MagnetismElectrical TradeDepth at which alternating current has fallen to 1/e of its surface value — why fat conductors and high frequencies waste copper.

Slenderness Ratio (KL/r)

λ=KLr\lambda = \frac{K L}{r}

Strength of MaterialsMechanicsPhysicsSlenderness ratio KL/r of a compression member, the single number that decides whether a column crushes or buckles.

Slide Distance Before Natural Roll

d=12v0249μgd = \frac{12 v_0^{2}}{49 \mu g}

Billiards & Cue SportsMechanicsHow far a cue ball struck with no spin slides before cloth friction has spun it up into a roll. It is the range over which a stun shot still behaves like a stun shot, and it is the reason the 90° rule works on a short shot and fails on a long one.

Slide Time Before Natural Roll

t=2v07μgt = \frac{2 v_0}{7 \mu g}

Billiards & Cue SportsMechanicsHow long a cue ball struck with no spin slides before it is rolling. The ball decelerates at μg while friction spins it up at 5μg/2R, and the two meet when v = Rω — which happens after 2v₀/7μg, linearly in the starting speed rather than quadratically like the distance.

Slope Between Two Points

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

AlgebraGeometrySteepness of a line as rise over run between two points.

Slope from Standard Form of a Line

m=−ABm = -\frac{A}{B}

AlgebraGeometrySlope of a line written in standard form Ax + By = C, read straight off the two coefficients without rearranging anything.

Slope Ratio (H:V) to Percent Grade

G=100nG = \frac{100}{n}

Civil & SurveyingSoil MechanicsConverts an embankment slope quoted as n horizontal to one vertical into the equivalent percent grade, and back.

Slope-Intercept Form of a Line

y=mx+by = mx + b

AlgebraGeometryEquation of a straight line from its slope and y-intercept.

Sludge Volume from Dry Solids

V=mS ρw (P/100)V = \frac{m}{S \, \rho_w \, (P/100)}

Water & WastewaterWater TreatmentVolume of wet sludge holding a given mass of dry solids, from the solids percentage and the sludge specific gravity, at 1000 kg/m³ water.

Sludge Volume Index (SVI)

SVI=SV30X\text{SVI} = \frac{SV_{30}}{X}

Water & WastewaterWater TreatmentSludge volume index in mL/g: the volume a gram of mixed liquor solids occupies after settling thirty minutes in a litre cylinder.

Small Grain Yield Estimate

Y=H g w10001000Y = \frac{H \, g \, w_{1000}}{1000}

Crop ProductionYield of a standing cereal from three counts: heads per unit area, grains per head, and the thousand-kernel weight of the variety. The small-grain equivalent of the corn ear count.

Snell's Law of Refraction

n1sin⁡θ1=n2sin⁡θ2n_1 \sin\theta_1 = n_2 \sin\theta_2

OpticsPhysicsLight bends at an interface so that n sin θ stays the same on both sides.

Snow Avalanche Alpha-Beta Runout (Lied-Bakkehøi)

α=k β+c\alpha = k\,\beta + c

Soil MechanicsRock MechanicsThe alpha-beta statistical model for extreme snow avalanche runout: the runout angle α from the release area to the distal limit is a regression on the beta angle, the angle to the point where the path first flattens to 10°. The coefficients are regional and must be entered.

Snow Gun Output Rate

V˙s=V˙w ρwρs\dot{V}_s = \dot{V}_w \, \frac{\rho_w}{\rho_s}

Snow & IceHVAC & HydronicsTrades & ConstructionSnow production rate from the water flow a gun is fed and the density of the snow it makes. Guns are specified in gallons or litres per minute of water and hills are planned in cubic metres or acre-feet of snow, and this is the conversion between the two.

Snow Made in a Wet-Bulb Window

Vs=V˙w t (1−f) ρwρsV_s = \dot{V}_w \, t \, (1 - f) \, \frac{\rho_w}{\rho_s}

Snow & IceHVAC & HydronicsTrades & ConstructionHow much snow a given water flow actually puts on the ground over a night, once the water lost to evaporation and drift is taken off. THE FACT THIS PAGE EXISTS TO TEACH: snowmaking is gated on WET-BULB temperature, not on the thermometer. You can make snow at +2 °C if the air is dry enough, and you cannot make it at −1 °C if the air is damp.

Snow Volume from Water Volume

Vs=Vw ρwρsV_s = V_w \, \frac{\rho_w}{\rho_s}

Snow & IceHVAC & HydronicsTrades & ConstructionHow much snow a given volume of water becomes, at whatever density the snow is made to. It is a mass balance and nothing more — the water does not change amount, only how much space it occupies — which is why the snow density on the bottom is the only interesting number in it.

Snow Water Equivalent (SWE)

SWE=d ρsρw\mathrm{SWE} = d \, \frac{\rho_s}{\rho_w}

Snow & IceWater & WastewaterThermodynamicsThe depth of water you would be left with if a snowpack melted where it lies — snow depth scaled by the ratio of snow density to water density. It is the number hydrology, irrigation and flood forecasting actually use, because depth alone says nothing about how much water is standing on the ground.

Snowpack Settlement (Viscous Compaction)

ρs=ρ0 exp⁡ ⁣(σ tη)\rho_s = \rho_0 \, \exp\!\left( \frac{\sigma \, t}{\eta} \right)

Snow & IceSoil MechanicsWater & WastewaterHow much a layer of snow densifies under a constant overburden, treating snow as a Newtonian fluid in compaction — Kojima's model, in the form land-surface schemes have used since Anderson wrote SNTHERM. It is the reason a snowpack is shallower in the morning than the depth board said last night, with the same water still in it.

Softener Resin Volume Required

V=mcapqV = \frac{m_{\text{cap}}}{q}

Water TreatmentChemistrySizes the ion-exchange resin bed from the hardness capacity required per regeneration and the resin's rated capacity per unit volume.

Soil Test ppm to Mass per Hectare

M=c d ρbM = c \, d \, \rho_b

Crop ProductionConverts a soil test in parts per million to the nutrient actually present per unit of land, using the sampling depth and the soil's bulk density.

Solar Declination Angle (Cooper's Equation)

δ=23.45∘sin⁡ ⁣(360∘ (284+n)365)\delta = 23.45^\circ \sin\!\left( \frac{360^\circ \, (284 + n)}{365} \right)

Solar & Wind PowerThe angle between the sun's rays and the plane of the equator on a given day of the year. Cooper's 1969 approximation, the one line of solar geometry every collector calculation begins from.

Solar Panel Power Output

P=GAηP = G A \eta

Solar & Wind PowerThe electrical power a photovoltaic module delivers: the irradiance falling on it, times its area, times its conversion efficiency. The definition that lies behind every nameplate wattage on the back of a panel.

Solids Capture in Dewatering

R=Cc(Cf−Cfil)Cf(Cc−Cfil)×100R = \frac{C_c (C_f - C_{fil})}{C_f (C_c - C_{fil})} \times 100

Water & WastewaterWater TreatmentPercent solids capture of a belt press, centrifuge or filter from the feed, cake and filtrate solids concentrations by mass balance.

Solow Steady-State Capital per Effective Worker

k∗=(s An+g+δ)11−αk^{*} = \left(\frac{s\,A}{n + g + \delta}\right)^{\frac{1}{1 - \alpha}}

Prices & MarketsCapital per effective worker at which investment exactly replaces what depreciation and growth dilute away, for a Cobb–Douglas economy. Robert Solow published the model in 1956; this is the level the capital stock converges toward and never quite reaches.

Solubility Product of a 1:1 Salt

Ksp=s2K_{sp} = s^{2}

ChemistryLinks the solubility product of an AB salt such as AgCl or BaSO4 to its molar solubility, since each formula unit releases one cation and one anion.

Solubility Product of an AB₂ Salt

Ksp=4s3K_{sp} = 4s^{3}

ChemistryLinks the solubility product of an AB2 or A2B salt such as CaF2 or Mg(OH)2 to its molar solubility, with the factor 4 from the doubled ion.

Solve an Exponential Equation for the Exponent

x=ln⁡(y/a)ln⁡bx = \frac{\ln (y / a)}{\ln b}

AlgebraSolves a·bˣ = y for the unknown exponent x using logarithms, the standard move behind every growth and decay problem.

Sommerfeld Number (Bearing Characteristic Number)

S=(rc)2μNPS = \left( \frac{r}{c} \right)^{2} \frac{\mu N}{P}

Machine DesignThe one dimensionless group that characterises a journal bearing: clearance ratio squared, times viscosity and speed, divided by the unit load. Two bearings with the same Sommerfeld number behave alike — same eccentricity, same friction variable, same minimum film — no matter their size, and that is what makes journal-bearing design charts possible.

Sound Intensity (I = P/A)

I=PAI = \frac{P}{A}

Waves & OscillationsPhysicsAcoustics & NoiseSound intensity is the acoustic power passing through each square meter of surface.

Sound Power Level to Sound Pressure Level

Lp=LW+10log⁡10 ⁣(Q4πr2)L_p = L_W + 10\log_{10}\!\left(\frac{Q}{4\pi r^{2}}\right)

Acoustics & NoiseA machine's sound POWER is a property of the machine; its sound PRESSURE is what a meter reads at a given place. This converts one to the other in a free field, using the distance and the directivity of wherever the machine is sitting.

Sound Transmission Loss

TL=10log⁡10 ⁣(IiIt)TL = 10\log_{10}\!\left(\frac{I_i}{I_t}\right)

Acoustics & NoiseHow much of the sound striking a partition never gets out the other side, in decibels. A 40 dB wall lets one ten-thousandth of the incident energy through — which still sounds like something, because hearing is logarithmic too.

Space Time and Space Velocity

τ=Vv0\tau = \frac{V}{v_0}

ChemistryWater & WastewaterHow long the feed nominally spends in the reactor: vessel volume divided by volumetric feed rate. Space velocity is simply its reciprocal, 1/τ, and is the number catalyst vendors quote as LHSV or GHSV in reciprocal hours.

Space Velocity

SV=1τ\mathrm{SV} = \frac{1}{\tau}

Chemical EngineeringHow many reactor volumes of feed pass through per unit time — the reciprocal of space time, and the number catalyst vendors quote on a datasheet. Reported in reciprocal hours: an LHSV of 2 h⁻¹ means two vessel volumes of liquid feed every hour.

Species-Area Relationship (S = cA^z)

S=c AzS = c\,A^{z}

Ecology & BiodiversityStatisticsExpected number of species in an area, as a power law fitted to survey data. The exponent z is typically 0.20 to 0.35 for nested samples within a region and 0.25 to 0.45 for true islands; c is a fitted constant whose units depend on z and on the area unit it was fitted in.

Specific Capacity of a Well

Sc=QsS_c = \frac{Q}{s}

Water & WastewaterCivil & SurveyingSpecific capacity of a water well: the pumping rate divided by the drawdown it produces, reported in cubic metres per day per metre of drawdown.

Specific Energy in an Open Channel

E=y+V22gE = y + \frac{V^2}{2g}

Fluid MechanicsCivil & SurveyingWater & WastewaterSpecific energy: the total head measured from the channel bed, being the flow depth plus the velocity head. It is the quantity that stays constant across a smooth transition.

Specific Gravity

SG=ρρwaterSG = \frac{\rho}{\rho_{water}}

Fluid MechanicsWater TreatmentChemistryDensity expressed as a multiple of water's 1000 kg/m³.

Specific Gravity of Sludge

1Ssl=P/100Ss+(1−P100)\frac{1}{S_{sl}} = \frac{P/100}{S_s} + \left(1 - \frac{P}{100}\right)

Water & WastewaterWater TreatmentSpecific gravity of a wet sludge from its percent solids and the specific gravity of the dry solids, by a reciprocal mass balance.

Specific Gravity to Degrees Plato

P=259−259SGP = 259 - \frac{259}{SG}

Brewing & FermentationConverts a hydrometer's specific gravity to degrees Plato, the percent of dissolved extract by weight that a refractometer and every commercial brewery report.

Specific Impulse and Exhaust Velocity

Isp=veg0I_{sp} = \frac{v_e}{g_0}

Aerospace & FlightSpecific impulse in seconds is the effective exhaust velocity divided by standard gravity — exactly 9.80665 m/s², a defined constant and not the local gravity wherever the engine happens to be firing.

Specific Stiffness (E / ρ)

Es=EρE_s = \frac{E}{\rho}

Strength of MaterialsModulus divided by density — the number that explains why an aircraft is worth building out of carbon fibre and a bridge usually is not. Steel, aluminium, magnesium and titanium all sit within a few percent of the same value, which is one of the quiet surprises of materials engineering; a unidirectional carbon laminate sits three or four times higher along its fibres and no better than the resin across them.

Specific Strength (σ / ρ)

σs=σρ\sigma_s = \frac{\sigma}{\rho}

Strength of MaterialsStrength divided by density: how much load a material carries for each kilogram of itself. Unlike specific stiffness, where all the structural metals tie, this one separates them sharply — and it is where fibre composites win by a margin big enough to change what an aeroplane is made of.

Specificity (True Negative Rate)

Sp=TNTN+FP\mathrm{Sp} = \frac{\mathrm{TN}}{\mathrm{TN} + \mathrm{FP}}

Epidemiology & Diagnostic AccuracyThe share of people who do not have the condition that the test correctly calls negative — read straight down the disease-absent column of a 2×2 table.

Speed at Natural Roll

vf=57v0+27Rω0v_f = \tfrac{5}{7} v_0 + \tfrac{2}{7} R\omega_0

Billiards & Cue SportsMechanicsThe speed a ball settles to once cloth friction has turned its slide into a roll. Friction acts at the contact point, so angular momentum about that point is conserved through the whole slide — and the answer comes out as a fixed weighted average of the starting speed and the starting spin. A ball struck with no spin loses exactly two-sevenths of its speed, always.

Speed in Circular Motion (v = 2πr/T)

v=2πrTv = \frac{2\pi r}{T}

MechanicsPhysicsSpeed of an object in uniform circular motion: one circumference (2πr, with π ≈ 3.14159265) per period.

Speed of Sound in Air

v=331.3+0.606 TCv = 331.3 + 0.606\, T_C

Waves & OscillationsPhysicsAcoustics & NoiseThe speed of sound in dry air grows about 0.6 m/s for every degree Celsius above freezing.

Speed, Distance & Time

v=dtv = \tfrac{d}{t}

PhysicsAverage speed from distance travelled and elapsed time.

Sphere Surface Area

S=4πr2S = 4 \pi r^{2}

GeometrySurface area of a sphere of radius r, using π ≈ 3.14159265.

Sphere Volume

V=43πr3V = \frac{4}{3} \pi r^{3}

GeometryVolume enclosed by a sphere of radius r, using π ≈ 3.14159265.

Spherical Cap Volume

V=πh23(3R−h)V = \frac{\pi h^2}{3}(3R - h)

GeometryVolume of the piece a flat plane slices off a sphere, from the sphere radius R and the cap height h.

Split Cylinder Tensile Strength

fct=2PπLDf_{ct} = \dfrac{2P}{\pi L D}

ConcreteA cylinder laid on its side and squeezed across a diameter splits along that diameter, because the diametral compression sets up a nearly uniform TENSILE stress across the loaded plane. This is the elasticity solution for that stress, it is pure test geometry, and it is how the tensile strength of concrete, rock and asphalt is actually measured.

Sprayer Application Rate

V=Qw vV = \frac{Q}{w \, v}

Crop ProductionSpray volume applied per unit of land from one nozzle's output, the nozzle spacing and the ground speed — the dimensional form of the 600 and 5940 calibration constants, so any mix of metric and imperial units may be entered.

SPT Overburden Correction (Liao–Whitman)

(N1)60=N60paσv′(N_1)_{60} = N_{60}\sqrt{\frac{p_a}{\sigma'_v}}

Soil MechanicsMechanicsCorrects a field SPT blow count to a reference overburden of one atmosphere using the Liao and Whitman square-root factor CN.

Square Area

A=s2A = s^2

GeometryArea of a square from its side length.

Square Diagonal

d=s2d = s\sqrt{2}

GeometryDiagonal of a square from its side length.

Square Perimeter

P=4sP = 4s

GeometryPerimeter of a square as four times its side length.

Square Pyramid Slant Height

l=h2+(s2)2l = \sqrt{h^2 + \left(\tfrac{s}{2}\right)^2}

GeometrySlant height of a square pyramid — apex down the middle of a face — from the vertical height and the base side.

Square Pyramid Surface Area

A=s2+2slA = s^2 + 2sl

GeometryTotal surface area of a square pyramid: the square base plus four identical triangular faces, from the base side and the slant height.

Squeezing Competence Factor

Nc=σcmσvN_c = \frac{\sigma_{cm}}{\sigma_v}

Rock MechanicsRock mass strength divided by in-situ stress: the first screening question in any deep tunnel. Below about one the ground cannot carry its own overburden elastically and will squeeze into the opening; well above one it behaves, and the problem becomes structurally controlled blocks instead. It is a ratio, not a prediction, and the boundary is a smear rather than a line.

Stack Draft Pressure (Chimney Effect)

Δp=h g (ρa−ρs)\Delta p = h \, g \, (\rho_a - \rho_s)

Air Quality & DispersionHVAC & HydronicsThe pressure a chimney generates on its own, from the height of the column and the density difference between cold outside air and hot flue gas. The same equation explains why a tall building's lobby doors are hard to open in January.

Stack Exit Velocity

vs=4Qvπd2v_s = \frac{4 Q_v}{\pi d^2}

Air Quality & DispersionHVAC & HydronicsThe speed exhaust leaves a round stack, from the volumetric flow and the inside diameter. Every plume-rise calculation starts here, and so does the check against stack-tip downwash.

Stack Load on the Bottom Case

F=(n−1)mgF = \left( n - 1 \right) m g

Packaging & Box StrengthWhat the bottom case in a stack actually carries: every case above it, and not its own weight. The minus one is the whole content of the equation and it is the term most often got wrong — a six-high stack puts five cases on the bottom one, not six.

Stacking Safety Factor

SF=BCTsvcFSF = \frac{BCT_{svc}}{F}

Packaging & Box StrengthThe ratio of what the box can carry to what the stack puts on it. Simple arithmetic with a hard warning attached: a safety factor applied on top of a full chain of derate multipliers double-counts, because the multipliers already contain most of the margin the trade rule was invented to supply.

Stadia Distance from Rod Intercept

D=K s+CD = K\,s + C

Civil & SurveyingHorizontal distance from a stadia rod intercept, with the instrument's stadia interval factor and additive constant.

Stair Comfort Rule (2R + T)

2R+T=C2R + T = C

Trades & ConstructionThe stride rule that keeps a stair comfortable: twice the riser plus one tread should land near 25 inches, or 630 mm in metric practice.

Stair Riser Count from Total Rise

N=HrN = \frac{H}{r}

Trades & ConstructionDivides the total rise, finished floor to finished floor, by the target riser height to give the number of risers a stair needs.

Stair Stringer Length

L=H2+Rt2L = \sqrt{H^2 + R_t^2}

Trades & ConstructionGeometryThe straight-line length of the stringer board, the hypotenuse of the total rise and the total run.

Stair Total Run from Risers and Tread Depth

Rt=(N−1) TR_t = (N - 1)\,T

Trades & ConstructionThe floor space a straight flight consumes: one fewer tread than there are risers, because the upper floor serves as the last tread.

Staircase Surface Roughness

Ra≈t4 ∣cos⁡θ∣R_a \approx \frac{t}{4} \, \lvert \cos\theta \rvert

Additive ManufacturingA sloped surface built out of flat layers is a staircase, and this is the arithmetic-mean roughness of those steps: a quarter of the layer thickness, scaled by the cosine of the angle the surface makes with the build plate. Pure geometry — no melting, no powder, no material. It is a floor under the real roughness and never a prediction of it.

Stall Speed

Vs=2Wρ S CL,maxV_s = \sqrt{\frac{2W}{\rho \, S \, C_{L,max}}}

Aerospace & FlightThe slowest speed at which a wing can still carry the weight, reached when the lift coefficient is at its maximum. Everything an aircraft does slowly — approach, touchdown, minimum turn radius, runway length — is a multiple of this number.

Stand Basal Area per Hectare

G=∑BAAG = \frac{\sum BA}{A}

Forestry & Forest EcologyAdd up the cross-sections of every stem on a plot, divide by the plot's area, and you have the standard measure of how heavily a site is stocked — the number every thinning prescription, stocking guide and growth model is written against. Reported here in m²/ha, with the ft²/ac equivalent alongside.

Standard Atmosphere Density (Troposphere)

ρ=ρ0(1−LhT0)g0RL−1\rho = \rho_0 \left(1 - \frac{L h}{T_0}\right)^{\frac{g_0}{R L} - 1}

Aerospace & FlightAir density at altitude on a standard day, from the troposphere layer of the ICAO and U.S. Standard Atmosphere: a lapse rate of 6.5 K per kilometre from 15 °C at sea level, in hydrostatic balance. Valid to the tropopause at 11 km and no further.

Standard Cell Potential from Half-Cells

Ecell∘=Ecathode∘−Eanode∘E^{\circ}_{\text{cell}} = E^{\circ}_{\text{cathode}} - E^{\circ}_{\text{anode}}

ChemistryBuilds a galvanic cell's standard voltage by subtracting the anode's standard reduction potential from the cathode's, using tabulated half-cell values.

Standard Enthalpy of Reaction from Formation Enthalpies

ΔHrxn∘=∑ΔHf,prod∘−∑ΔHf,react∘\Delta H^{\circ}_{\text{rxn}} = \sum \Delta H^{\circ}_{f,\text{prod}} - \sum \Delta H^{\circ}_{f,\text{react}}

ChemistryThermodynamicsThe tabulated form of Hess's law: standard enthalpy of reaction equals the summed formation enthalpies of the products minus those of the reactants.

Standard Error of a Proportion

SE=p(1−p)nSE = \sqrt{\frac{p (1 - p)}{n}}

StatisticsAlgebraHow much a sample percentage typically wanders from the true population proportion, given the proportion and the sample size.

Standard Error of the Mean

SE=σnSE = \frac{\sigma}{\sqrt{n}}

StatisticsAlgebraHow much a sample mean typically wanders from the true mean, shrinking with the square root of the sample size.

Standard Time from a Time Study

Ts=To R (1+A)T_s = T_o\,R\,(1 + A)

Industrial EngineeringTurns a stopwatch reading into an allowed time: observed time, adjusted for how fast the operator was working, then padded for rest, personal needs and unavoidable delay.

Stanton Number for Heat Transfer

St=hρ v cp\mathrm{St} = \frac{h}{\rho \, v \, c_p}

Heat TransferThe convection coefficient measured against the heat the stream is carrying past: what fraction of the oncoming flow's thermal capacity is actually delivered to the wall. The thermal twin of the mass-transfer Stanton number, and the group the Colburn j-factor is written in.

Stanton Number for Mass Transfer

StD=kcu\mathrm{St}_D = \frac{k_c}{u}

Chemical EngineeringThe mass transfer coefficient measured against the velocity that produced it: what fraction of the oncoming stream is actually delivered to the surface. The group the Chilton–Colburn j-factor is built from.

Steam Coil Condensate Load

m˙=Q˙hfg\dot m = \frac{\dot Q}{h_{fg}}

HVAC & HydronicsThermodynamicsWater TreatmentSteam a coil condenses per unit time from its heat duty and the latent heat at the coil pressure — the same figure that sizes the trap and the condensate return.

Steam Quality from Enthalpy

x=h−hfhfgx = \frac{h - h_f}{h_{fg}}

ThermodynamicsWater TreatmentDryness fraction of wet steam from its measured enthalpy, the saturated liquid enthalpy and the latent heat at the same pressure.

Steam Turbine Power Output

P=m˙ wP = \dot{m} \, w

ThermodynamicsShaft power a steam turbine develops from its steam rate and the specific work taken out of each kilogram, before generator and gearbox losses.

Steam Turbine Specific Work

w=h1−h2w = h_1 - h_2

ThermodynamicsWork a steam turbine takes out of every kilogram of steam: the enthalpy at the stop valve less the enthalpy at the exhaust, from the steady-flow energy equation.

Stefan Frost Penetration Depth

X=2kf n ΔT tLX = \sqrt{\frac{2 k_f \, n \, \Delta T \, t}{L}}

Heat TransferJosef Stefan's 1891 result for how deep a freezing front reaches into wet ground. All the heat removed goes into freezing the pore water, so the depth grows as the square root of the accumulated freezing index — twice the cold season for only 1.4 times the depth. The classic first estimate for a frost line, a footing depth, or the cover a buried water line needs.

Stefan-Boltzmann Law

P=εσAT4P = \varepsilon \sigma A T^4

ThermodynamicsPhysicsPower radiated by a hot surface, using the Stefan–Boltzmann constant σ = 5.670374419×10⁻⁸ W/(m²·K⁴).

Stellar Luminosity

L=4πR2σT4L = 4\pi R^{2} \sigma T^{4}

Astronomy & GravitationPhysicsTotal power a star radiates — the Stefan–Boltzmann law wrapped over a sphere of radius R glowing at effective temperature T.

Stellar Parallax Distance

d=1 AUpd = \frac{1\,\text{AU}}{p}

Astronomy & GravitationPhysicsDistance to a star from its annual parallax angle p, using the small-angle form of d = AU/tan p.

Stemming, Subdrilling and Spacing from the Burden

X=fB⋅BX = f_B \cdot B

Rock MechanicsOnce the burden is fixed, the rest of a bench pattern follows from it as simple multiples: stemming about 0.7 B, subdrilling about 0.3 B, spacing between about 1.15 and 1.4 B for delayed holes. Enter the burden and the ratio you are working to and this returns the dimension. The ratios themselves come from FHWA-HI-92-001 and are starting points, not laws.

Step-Drawdown Test: Aquifer Loss and Well Loss

s=BQ+CQ2s = BQ + CQ^{2}

Water & WastewaterJacob's (1947) separation of the drawdown inside a pumping well into two parts: BQ, the laminar loss the aquifer imposes, and CQ², the turbulent loss the well's own screen, gravel pack and borehole impose. Q is taken in cubic metres per day, so B is in d/m² and C in d²/m⁵.

Stern-Geary Corrosion Current

Icorr=βaβc2.303 (βa+βc) RpI_{corr} = \frac{\beta_a \beta_c}{2.303 \, (\beta_a + \beta_c) \, R_p}

Corrosion & ProtectionStern and Geary's 1957 result: the corrosion current is the Stern-Geary constant divided by the polarization resistance. The equation that made an instantaneous corrosion rate a field measurement rather than a laboratory exercise.

Stock Solution for an Injector

cs=r ctc_s = r \, c_t

Indoor & GreenhouseHow strong to mix the concentrate for a fertiliser injector: the dilution ratio times the concentration wanted at the emitter.

Stocking Rate in Animal Units

SR=AUASR = \frac{AU}{A}

Livestock & FeedAnimal units carried per unit of grazing land. A stocking rate is a decision the manager makes — how many head went out on the field — and it is not the same quantity as the carrying capacity that decides whether the decision was right.

Stockpile Volume (Truncated Pyramid)

V=h3(A1+A2+A1A2)V = \frac{h}{3}\left(A_1 + A_2 + \sqrt{A_1 A_2}\right)

Civil & SurveyingGeometryVolume of a flat-topped stockpile or borrow pit from its base area, top area and height; the answer is reported in cubic yards.

Stokes Number (particle inertia)

Stk=ρp d2 v18 μ L\mathrm{Stk} = \frac{\rho_p \, d^{2} \, v}{18 \, \mu \, L}

Air Quality & DispersionThe ratio of a particle's stopping time to the time the flow takes to go round an obstacle. It is the number that decides whether a particle follows the air or hits the thing the air is flowing past — the whole of cyclone capture, impactor sampling, filtration and whether a droplet lands on your windscreen or slides around it.

Stokes Settling Velocity

vs=g(ρs−ρ)d218μv_s = \frac{g (\rho_s - \rho) d^2}{18 \mu}

Water & WastewaterFluid MechanicsTerminal settling velocity of a small sphere in laminar flow by Stokes' law — the grit chamber and clarifier design relation.

Stokes' Drag (F = 6πμrv)

F=6πμrvF = 6\pi \mu r v

Fluid MechanicsPhysicsViscous drag on a small sphere creeping through a fluid at low Reynolds number.

Stopping Sight Distance

d=v tr+v22ad = v\,t_r + \frac{v^{2}}{2a}

Civil & SurveyingDistance to stop from a design speed, adding the reaction distance travelled before braking to the braking distance itself.

Storativity from Specific Storage

S=Ss bS = S_s \, b

Water & WastewaterStorativity of a confined aquifer as the specific storage multiplied by the saturated thickness. Specific storage is the volume of water a unit volume of aquifer releases per unit drop in head, and it comes from the compressibility of the skeleton and of the water itself.

Straight-Line Depreciation

D=C−SnD = \frac{C - S}{n}

Money & BusinessEqual annual write-down of an asset: cost minus salvage value, spread evenly over its useful life.

Strain Energy Release Rate

G=K2(1−ν2)EG = \dfrac{K^{2} \left( 1 - \nu^{2} \right)}{E}

Strength of MaterialsThe bridge Irwin built between Griffith's energy picture and his own stress picture: the energy a crack releases per unit of new area is the square of the stress intensity divided by an effective modulus. Written here in the PLANE STRAIN form. For plane stress — a thin sheet — enter Poisson's ratio as 0, which makes E prime equal E exactly.

Stream Duty from Mass Flow (Q = ṁcΔT)

Q˙=m˙ cp ΔT\dot{Q} = \dot{m} \, c_p \, \Delta T

Heat TransferThermodynamicsHVAC & HydronicsHeat picked up or given off by one exchanger stream, from its mass flow, specific heat and the temperature change across the unit.

Stress Concentration (σmax = Kt σnom)

σmax=Kt σnom\sigma_{max} = K_t \, \sigma_{nom}

Strength of MaterialsMechanicsPeak elastic stress at a hole, notch, groove or fillet: the nominal stress on the net section multiplied by a geometric factor read from a Peterson chart.

Stress Intensity Factor

K=Y σ πaK = Y \, \sigma \, \sqrt{\pi a}

Strength of MaterialsIrwin's 1957 result, and the single most useful equation in fracture mechanics: the entire elastic stress field around a crack tip is set by one number. Compare K against the material's fracture toughness K_IC and you have the answer to whether a cracked part will break — or, turned around, the crack size that part can tolerate.

Stripping Operating Line (McCabe–Thiele)

y=VB+1VB x−xBVBy = \frac{V_B + 1}{V_B}\,x - \frac{x_B}{V_B}

Chemical EngineeringThe straight line below the feed, written in terms of the boilup ratio. Slope greater than 1, and it passes through (x_B, x_B) on the 45° line — the mirror of the rectifying line, anchored on the bottoms instead of the distillate.

Strouhal Number (Vortex Shedding Frequency)

St=fLvSt = \frac{f L}{v}

Fluid MechanicsHow often something in a flow repeats itself, made dimensionless: shedding frequency times a length, over the speed. A cylinder in a steady stream sheds alternate vortices at a Strouhal number close to 0.2 over an enormous range of Reynolds numbers, which is why one number predicts the hum of a wire, the sing of a chimney stack and the frequency a heat-exchanger tube will be shaken at.

Structural Number of a Flexible Pavement

SN=a1D1+a2D2m2+a3D3m3SN = a_1 D_1 + a_2 D_2 m_2 + a_3 D_3 m_3

Axle Loads & PavementThe single number that stands for the whole strength of a flexible pavement: each layer's thickness multiplied by a coefficient for what it is made of, and by a drainage coefficient for the unbound layers. Layer coefficients are INPUTS — they are regionally calibrated, they live in copyrighted design guides, and no honest tool bakes them in.

Studs on a Wall at a Given Spacing

N=Ls+1N = \frac{L}{s} + 1

Trades & ConstructionCounts the studs, joists or fence posts along a run at a fixed on-centre spacing, including the one that closes the far end.

Stun Shot — Cue Ball Speed

vCB=usin⁡φv_{CB} = u \sin\varphi

Billiards & Cue SportsMechanicsWhat the cue ball keeps after a stun shot: the component of its velocity ALONG THE TANGENT LINE, which the collision cannot touch. Together with the object ball's u cos φ this is the 90° rule — two perpendicular velocities whose squares add back to u², so no energy is lost and no momentum goes missing.

Stun Shot — Object Ball Speed

vOB=ucos⁡φv_{OB} = u \cos\varphi

Billiards & Cue SportsMechanicsHow much of the cue ball's speed the object ball actually receives on a cut. The balls are smooth, so the impulse runs along the line of centres and only the component of the cue ball's velocity along that line is transferred — cos φ of it. A thin cut sends the object ball nowhere, which is why thin cuts have to be hit hard.

Submerged (Buoyant) Unit Weight

γ′=γsat−γw\gamma' = \gamma_{sat} - \gamma_w

Soil MechanicsFluid MechanicsEffective or buoyant unit weight of soil below the water table, the saturated unit weight less the uplift of the water it displaces.

Submerged Culvert Inlet (Orifice) Headwater

Q=Cd A2g(HW−D2)Q = C_d \, A \sqrt{2g\left(HW - \tfrac{D}{2}\right)}

Civil & SurveyingFluid MechanicsWater & WastewaterCapacity of a culvert in inlet control with the inlet submerged, treated as an orifice discharging under the head between the headwater surface and the centroid of the barrel opening.

Sum of Divisors of a Prime Power

σ(pk)=pk+1−1p−1\sigma(p^k) = \frac{p^{k+1} - 1}{p - 1}

AlgebraEvery divisor of a prime power added together, which is a geometric series in disguise: 1 + p + p² + … + p^k.

Sum of the Roots of a Quadratic

S=−baS = -\frac{b}{a}

AlgebraVieta's relation giving the sum of a quadratic's two roots directly from its coefficients, without solving the equation.

Superelevation Rate for a Horizontal Curve

e100+f=v2g R\frac{e}{100} + f = \frac{v^{2}}{g\,R}

Civil & SurveyingBalances banking and side friction against centripetal demand on a horizontal curve, using standard gravity of 9.80665 m/s².

Support Reaction — Simple Beam, Off-Centre Point Load

RA=P(L−a)LR_A = \frac{P (L - a)}{L}

Strength of MaterialsCivil & SurveyingMechanicsReaction at the near support of a simply supported beam with one point load at distance a from it. The far reaction is the remainder, P − R_A = Pa/L.

Surface Overflow Rate

vo=QAv_o = \frac{Q}{A}

Water & WastewaterWater TreatmentFluid MechanicsSurface overflow (surface loading) rate of a settling basin — flow divided by plan area, reported here in metres per day.

Surface Temperature Depression Under a Clear Sky

Ts=Tair−qnethcT_s = T_{air} - \frac{q_{net}}{h_c}

Heat TransferAir Quality & DispersionHow far a surface settles below air temperature when it loses net radiation to a clear sky and gains heat back only by convection. This is why frost appears on a windscreen while the porch thermometer reads a few degrees above freezing.

Surface Twist Angle of a Yarn

tan⁡θ=π d T\tan\theta = \pi \, d \, T

Textiles & FabricUnroll one turn of the outermost fibre and you get a right triangle: the circumference πd along one side, the length of one turn 1/T along the other. The angle between them is what twist actually does to a yarn, and it is the quantity the twist multiplier exists to hold constant across counts.

Swamee–Jain Friction Factor

f=0.25[log⁡10 ⁣(ε3.7D+5.74Re0.9)]2f = \frac{0.25}{\left[\log_{10}\!\left(\frac{\varepsilon}{3.7D} + \frac{5.74}{Re^{0.9}}\right)\right]^{2}}

HVAC & HydronicsFluid MechanicsPhysicsAn explicit turbulent friction factor within about 1% of the implicit Colebrook–White equation, valid for Re from 5000 to 10⁸.

Swell: Loose Volume from Bank Volume

VL=VB(1+S100)V_L = V_B\left(1 + \frac{S}{100}\right)

Civil & SurveyingSoil MechanicsConverts undisturbed bank volume into the loose volume the same soil occupies once excavated, using its percent swell.

Synchronous Speed from Frequency and Poles

Ns=2fpN_{s} = \frac{2f}{p}

Electrical TradeElectricity & MagnetismSpeed of an AC machine's rotating field from supply frequency and pole count — the familiar 120f/p when read out in rpm.

System Volume from Turnover Time

V=R tV = R \, t

Water TreatmentFluid MechanicsSystem water volume estimated from the recirculation rate and the measured turnover time — the field method when no drawings exist.

Systemic Vascular Resistance (SVR)

SVR=80 (MAP−CVP)CO\mathrm{SVR} = \frac{80\,(\mathrm{MAP} - \mathrm{CVP})}{\mathrm{CO}}

Biomedical & ClinicalSystemic vascular resistance in dyn·s·cm⁻⁵: the pressure drop across the systemic circuit divided by the flow through it, with the 80 that converts the units.