Shock Math 2.0 — redesigned, with charts, the standard atmosphere, a real Apple Watch app and iPad support. A NEW LOOK • Every calculator now shares one clean layout: inputs, Calculate, results and a chart, all in one scrolling page that works at every text size. • Your inputs stay put when you switch between calculators. • The calculator picker lives in the navigation bar, with Theory & Equations one tap away. CHARTS • See your answer on the curve: isentropic, Fanno, Rayleigh and normal-shock ratios across the Mach range. • The full θ-β-M diagram for oblique shocks, with your Mach number and case highlighted, the maximum-deflection line and the sonic line. • Shock angle against cone angle for conical shocks, and temperature, pressure, density and speed-of-sound profiles for the atmosphere. NEW: STANDARD ATMOSPHERE • The U.S. Standard Atmosphere 1976 from −5 km to 86 km. Enter an altitude in metres or feet — or a pressure or density to find the matching altitude — and get temperature, pressure, density, speed of sound, viscosity and the θ, δ, σ ratios. IDEAL GAS, REBUILT • Solve p = ρRT or pV = nRT for any quantity, with a unit menu on every value: Pa, kPa, MPa, bar, atm, psi; K, °C, °F, °R; m³, L, cm³, ft³; mol, kmol. • Pick a gas and its gas constant R is used automatically. • Ideal gas results can now be saved to History and exported. THEORY & EQUATIONS • Every calculator now has a reference page with its assumptions and key equations — the θ-β-M relation, Taylor–Maccoll, Fanno and Rayleigh relations and more. APPLE WATCH • A real watch app: isentropic flow, normal shock and ideal gas, set with the Digital Crown, with results that update as you turn it — plus the scratch pad. IPAD • A full iPad layout with a sidebar of every calculator, and support for Split View and Stage Manager. SIRI & SHORTCUTS • "Isentropic ratios in Shock Math", "Normal shock in Shock Math" and "Standard atmosphere in Shock Math" — or use the actions in your own shortcuts. FIXES • Numbers typed with a decimal comma were read as zero. They now work. • The scratch pad no longer rounds to one decimal place mid-calculation (1 ÷ 3 × 3 now gives 1), handles divide-by-zero, and is easier to read in dark mode. • Very small results show in scientific notation instead of a row of zeros. • History now names every input for two-input calculators, e.g. "γ = 1.4 · M₁ = 2 · Angle (weak) = 10". • In the ideal gas law, "mPa" is gone (it was silently treated as pascals) and the volume button no longer says "Find Temperature".

Export your results — plus easier reading and smoother navigation. EXPORT • Share any result as plain text, a spreadsheet (CSV) or a PDF report. Tap Export on the results card, or on any calculation in History. • CSV files keep full precision — the screen rounds to five significant figures, your spreadsheet doesn't — and open cleanly in Excel and Numbers. • Exporting unlocks for 24 hours after one short video. Copy All stays free, as always. EASIER TO READ • Stronger contrast: links, menus and buttons use a deeper green that meets accessibility guidelines for readability. • Better with larger text: buttons and labels grow with your text size instead of cutting off. • VoiceOver now reads results naturally — "p sub 2 over p sub 1, 4.5" rather than a string of symbols — and confirms when a value is copied. SMOOTHER • Reference pages ("What are Isentropic Flow Relations?") now open as full pages. • Under the hood: modernized navigation and settings, ready for what's next.

Save and reload your cases — plus important fixes to Rayleigh and Fanno flow. SAVED CASES • Name a calculation to keep it. Open any calculation from history and tap "Save as Case" to give it a name (like "Nozzle throat"). Named cases are pinned in their own starred section. • Reload with one tap. Tap "Load into Calculator" on a saved case and the app jumps to the right calculator with gamma and all your inputs already filled in — and recomputes automatically. FIXES • Rayleigh flow input list. The picker offered two inputs that belong to Fanno flow (4fL*/D), and two more that silently solved for the wrong thing. It now lists exactly the eleven quantities Rayleigh flow can be solved from, including entropy, (s*-s)/R. • Rayleigh flow reference notes. The link at the top of the Rayleigh screen opened the Fanno flow explanation. It now opens the Rayleigh one. • Fanno flow from T/T*. Only supersonic values were accepted. Subsonic values — anything above 1, up to the limit the app already told you was valid — now solve correctly. • Rayleigh flow from U/U*. The upper limit was fixed at air's value and applied to every gas, so results could be wrong for gases such as argon and helium. The limit now follows the gamma you enter. The calculation engine behind all of this was also rebuilt and re-verified against the NACA Report 1135 tables, for every calculator and every input, across a range of specific-heat ratios. Builds on the calculation history added in the last update.

Results just got a lot easier to read. • New results layout. Every calculator now presents its answers as a clean, labeled table — proper symbols like T/T₀, p/p₀, ρ/ρ*, and A/A*, each with its value neatly aligned — instead of a wall of raw text. • Sensible precision. Values are rounded to a tidy 5 significant figures, so you'll see 0.12780 instead of 0.12780452546295096. • Consistent across the board — isentropic, Fanno, Rayleigh, and normal, oblique & conical shocks all share the same crisp format. All of the proven-accurate math from version 1.6 is unchanged. Thanks for the feedback — keep it coming from the in-app form.

Version 1.6 is a major accuracy and reliability release. We rebuilt and independently verified the entire calculation engine. PROVEN-ACCURATE MATH • Every relation is now cross-checked against NACA Report 1135 reference tables and independent reference implementations, across a wide range of Mach numbers and specific-heat ratios (γ = 1.3, 1.4, 1.667). Hundreds of checks, zero discrepancies. • Forward and inverse solutions now round-trip exactly — enter a Mach number to get a ratio, or a ratio to get the Mach number, and land back on the same value. BUGS FIXED • Corrected the isentropic inverse solvers so entering a pressure, temperature, density, or area ratio returns the right Mach number. • Fixed Fanno temperature ratios and Rayleigh pressure, stagnation-pressure, and velocity ratios, which previously used the wrong formulas. • Corrected the Fanno and Rayleigh supersonic inverse branches and the Rayleigh stagnation-temperature inverse. • Fixed the conical shock cone-angle calculation. • Replaced a slow normal-shock inverse with an exact closed-form solution. NO MORE FREEZES • Several input combinations could previously cause the app to hang while solving. Every solver now has guaranteed termination, so valid inputs always return a result. UNDER THE HOOD • Now requires iOS 17. • Added an automated test suite so accuracy stays locked in for future updates. • Removed dead code and hardened the feedback form. As always, thank you for the feedback — keep it coming from the in-app form.

Added descriptions for: • Conical Relations • Isentropic Relations • Fanno Relations • Rayleigh Relations Fixed some bugs.

added Fanno Flow relations added Rayleigh Flow relations killed some bugs

Added isentropic flow relations Removed some unnecessary code Killed some bugs

Added Conical Shock Relation Calculator

Oblique shock relations calculator Added Shock Descriptions Flattened some bugs

Added traditional Ideal Gas Law calculations (PV=nRT) Added description for Ideal Gas Law Killed some bugs

Killed some bugs Removed developer only outputs