Baselproblem vs Shock Math Calculator

The Basel problem is a problem in mathematical analysis with relevance to number theory, first posed by Pietro Mengoli in 1650 and solved by Leonhard Euler in 1734.

Shock Math is a fast, accurate compressible-aerodynamics calculator for students, engineers, and anyone working in high-speed gas dynamics. Solve isentropic, shock, and duct-flow relations in seconds — from any input variable, for any gas. COMPRESSIBLE FLOW, SOLVED • Isentropic flow — pressure, temperature, density, and area ratios, Mach angle, and Prandtl–Meyer angle • Normal shock relations — downstream Mach, static and stagnation pressure, temperature, and density ratios • Oblique shock relations — weak and strong solutions, wave angle, deflection angle, and normal Mach components • Conical shock relations — Taylor–Maccoll solution for cone angle, surface Mach number, and surface property ratios • Fanno flow — adiabatic flow with friction in a constant-area duct • Rayleigh flow — frictionless flow with heat addition • Standard atmosphere — U.S. Standard Atmosphere 1976 from sea level to 86 km: temperature, pressure, density, speed of sound and viscosity, from altitude, pressure or density • Ideal gas law — p = ρRT and pV = nRT, solve for any quantity in the units you work in • Scratch Pad — a quick built-in calculator for the numbers in between SEE THE PHYSICS Every result comes with a chart: property ratios across the Mach range with your answer marked, the full θ-β-M oblique shock diagram with your case on it, the cone-angle curve for conical shocks, and atmosphere profiles by altitude. SOLVE FROM ANY VARIABLE Don't just go forward from Mach number. Enter a pressure ratio, temperature ratio, area ratio, wave angle, or Prandtl–Meyer angle and Shock Math finds the Mach number for you — subsonic or supersonic branch, your choice. ANY GAS, ANY GAMMA Set the ratio of specific heats (γ) to model air, combustion products, monatomic gases, or anything in between. Every relation respects your γ. PROVEN-ACCURATE MATH Version 1.6 rebuilt and independently verified the calculation engine. Every relation is cross-checked against the classic NACA Report 1135 tables and against independent reference implementations across a wide range of Mach numbers and specific-heat ratios — hundreds of checks, zero discrepancies. Forward and inverse solutions round-trip exactly. BUILT FOR REAL WORK • Clean, consistent calculators that remember your inputs • Theory & Equations: assumptions and key equations for every calculator, one tap away • History and saved cases — reload any past calculation • Export results as text, CSV or PDF • iPad layout with a sidebar, plus Split View and Stage Manager • Apple Watch app: isentropic flow, normal shock, ideal gas and a scratch pad, set with the Digital Crown • Siri and Shortcuts actions for isentropic ratios, normal shocks and the standard atmosphere • Light and dark mode, VoiceOver, and every text size • Show or hide the calculators you use Whether you're checking homework in a gas dynamics course, sizing a supersonic nozzle, or sanity-checking a CFD result, Shock Math gives you trustworthy numbers fast. Feedback and feature requests are always welcome from the in-app form.