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Two Films: the maths

Two exact solutions of a wave equation with two time directions, u_tt + u_ss − u_xx = 0, chosen to agree completely on the starting slice. Exact.

Equations

u_A = cos 2x · cos 2t

Film A.

u_B = u_A + ε cos 3x · (cos 3t − cos √5 t · cos 2s)

Film B. The extra term and all its first rates of change vanish at t = 0, s = 0 — yet it grows as t increases.

u_tt + u_ss − u_xx = 0

Both films satisfy this exactly (check: −9 + 0 + 9 = 0 for the first part, −5 − 4 + 9 = 0 for the second).

Constants

ε

0.10–0.60 (slider)

the size of the difference

Worked examples ✓ checked on every change

Largest difference on the starting slice	0.000	identical start
Largest difference at t = 2 (s = 0)	0.419	ε |cos 6 − cos 2√5|
Film B obeys the equation (largest error, finite differences)	0.0000	zero, up to rounding

The app's test suite puts the lab in each setup, reads the lab's own result, and fails if it strays from these values.

What's simplified

• One space dimension and two time dimensions.
• The point is mathematical: complete data on one slice doesn't fix the solution.

Where it breaks

It doesn't — both are exact solutions. It's a statement about equations with two times, the reason physics with two times struggles to predict.

Sources: I. Bars, 'Survey of two-time physics' (2001); W. Craig & S. Weinstein, Proc. R. Soc. A 465, 3023 (2009).