{
 "title": "Elastic wave in a bar from a pressure step (explicit)",
 "kind": "theory",
 "source": "d'Alembert's solution of the one-dimensional wave equation: c = √(E/ρ), particle velocity v = p/(ρc); Graff, Wave Motion in Elastic Solids, Dover 1991, §2.1",
 "quantities": [
  {
   "id": "u_end",
   "quantity": "displacement of the loaded end at t = 0.06",
   "measure": {
    "command": "solve.probe",
    "params": {
     "step": "Wave",
     "at": [
      [
       10.0,
       0,
       0
      ]
     ]
    },
    "pointer": "/nodes/0/displacement/0"
   },
   "value": -0.0006,
   "tolerance": {
    "relative": 0.01
   },
   "reason": "the end moves at the particle velocity p/(ρc) from the start; the discrete wave's ringing behind the front changes the end's displacement by well under 1%"
  },
  {
   "id": "v_mid",
   "quantity": "particle velocity at x = 5, 20 elements behind the front, at t = 0.06",
   "measure": {
    "command": "solve.probe",
    "params": {
     "step": "Wave",
     "at": [
      [
       5.0,
       0,
       0
      ]
     ]
    },
    "pointer": "/nodes/0/velocity/0"
   },
   "value": -0.01,
   "tolerance": {
    "relative": 0.02
   },
   "reason": "central differences ring behind a step front (dispersion); bulk viscosity damps it within a few elements, leaving about 1% twenty elements behind"
  },
  {
   "id": "balance",
   "quantity": "energy balance error",
   "measure": {
    "from": "solve",
    "pointer": "/energy_balance"
   },
   "value": 0.0,
   "tolerance": {
    "absolute": 0.01
   },
   "reason": "external work = kinetic + internal + dissipated energy; the solver warns above 1%"
  }
 ]
}