Physics
Wave Equation
A string is a row of little masses, each one pulled by its two neighbours. A bead that sags below the line joining them gets pulled up, and the harder it sags the harder the pull. Write that one sentence down and you have the wave equation, second difference and all.
coarse, dx largefine
Drag the string itself to pull it into a new shape. Click along the lower strip to move the bead you are watching.
What to observe
- Look at the card on the right. The only thing the middle bead can feel ishow far it sits below the straight line joining its neighbours. That sag is what the two tensions fail to cancel, and it is the whole force.
- Sag is exactly half the second difference, so the pull isu(i−1) − 2u(i) + u(i+1). Divide by dx² and it is the curvature; the wave equation is that sentence written for every bead at once.
- The strip under the string is that pull, bead by bead. It is theacceleration field: teal where the string is being dragged up, orange where it is being dragged down. Notice it is largest exactly at the sharp bends and zero on the straight runs.
- Press One step a few times with a bead selected. Its position does not follow the force, its velocity does, and that one extra derivative is why a wave travels instead of just slumping flat like heat.
- Drag beads on the string down to 24 and watch dx grow. The pulse starts to wobble and trail ripples behind it: a coarse grid cannot carry the short wavelengths, so they travel at the wrong speed.
- Push the Courant number past 1 and the whole thing detonates. With an explicit step, information may not be allowed to cross more than one dx per tick.
Shortcuts: space run/pause · s step · r reset · f fullscreen