The Kinked String Was Fine, the Cut String Overshot by 9% — How the 1747 Wave-Equation Fight Ended Numerically
·6 min read
d'Alembert's traveling waves and Bernoulli's sines agree until the series is cut. A jump keeps a 9% overshoot; a smooth bell is already exact by 32 modes.
The Outlet Is Open, Yet the Wave Comes Back — NSCBC and the Price of σ
·9 min read
At σ = 0 a duct reflects 0% and keeps a +0.30 pressure offset; at σ = 0.25 the reflection is 1.9% and the offset drops to 0.0007.
Supersonic Flow Can't See Upstream — Characteristics and Sound Waves in the Euler Equations
·6 min read
At 220 m/s in air the speeds are -118.06, 220, and 558.06 m/s. Past Mach 1 the left-going sound wave no longer reaches upstream.
When Characteristics Collide, a Shock Is Born — Burgers and the (x,t) Plane
·6 min read
A sine of amplitude 0.35 and wavenumber 1 breaks at t* ≈ 0.455 when characteristics cross, and the weak solution cuts that fold with one line.