The second-order law every non-dispersive wave obeys — and it is solved not by a special function but by any shape that simply travels. Poke a taut string and a hump slides off at speed c without spreading. d'Alembert saw why: split the motion into a right-mover and a left-mover, and the wave equation is satisfied identically. Rendered, not quoted.
source d'Alembert, Recherches sur la courbe que forme une corde tenduëe mise en vibration (1747), Hist. Acad. Roy. Sci. et Belles-Lettres de Berlin, vol. 3, pp. 214–219 — facsimile (Xavier / Oresme)
One law, one line. The transverse displacement u(x,t) of a taut string obeys
d'Alembert's insight (1747): the general solution is a wave going right plus a wave going left,
Any twice-differentiable f, g works — the shape is set by the initial pluck,
not by the equation. c = 1 in the panel's units
This is the law of a travelling wave. The same uₜₜ = c²uₓₓ,
confined between two fixed ends, forces the right- and left-movers to interfere into
the-standing-wave; freed into three dimensions and carried by fields rather than
string, its energy flux is the-poynting-vector. One second-order line;
a whole family downstream.
Re-runs the full selfcheck() live and reads the disclosed tamper flag. Green while the
operator's speed matches the wave's speed; flips red the instant window 6 corrupts it.
A smooth Gaussian pulse launched rightward, f(s)=exp(−s²/2w²), plus a second
pulse launched leftward. Speed c = 1, width w = 0.8. No random inputs —
deterministic seeds only.
uₜₜ − c₀ₙ²uₓₓ (max on grid) =
0 non-dispersive — shape preserved
Proven, live: a true right-mover f(x−ct) gives a discrete residual of
exactly 0 (the t- and x-second-differences are the same stencil); the peak advances by exactly
c·Δt with the profile unchanged; right-mover + left-mover and any linear
combination are also solutions.
"Waves must spread — everything diffuses." No: the 1-D wave equation is
non-dispersive. Every Fourier mode travels at the same c, so the packet keeps its shape
forever. Dispersion appears only when the medium makes c depend on frequency (deep-water waves, a stiff
beam, waveguides) — a different equation. The bare uₜₜ=c²uₓₓ
does not spread, and the panel's residual proves it.
"The solution is a sine wave."
→ A sine is one solution; d'Alembert's f(x−ct)+g(x+ct) covers all of
them, including sharp fronts.
"Doubling c doubles the residual of a fixed pulse."
→ The residual is zero only when the operator's c matches the pulse's c;
mismatch (window 6) makes it non-zero — that is the whole tamper.
"You need calculus of f and g to check it."
→ A second difference on a grid is enough; here it lands on 0 to machine precision.
Disclosed planted void: force the differential operator to use the wrong speed
c₀ₙ = c/2 while the pulse still travels at c. A genuine
f(x−ct) then leaves a non-zero residual — the WITNESS (7) catches it at once.