{
  "simulation": "driven_string_ring",
  "physical_observation": "A taut string of length L=4.979 m, linear density mu=0.116 kg/m, tension T=4.497 N runs from a driven oscillator at x=0 to a frictionless ring on a vertical rod at x=L. Driver: u(0,t) = A \\sin(omega t) with A=0.116 m, f_drive=1.636 Hz.",
  "physical_law": {
    "name": "1-D Linear Wave Equation with Linear Viscous Damping; Driven Dirichlet (LEFT), Free-end Neumann (RIGHT)",
    "statement": "u_tt = c^2 u_xx - 2 gamma_w u_t with c = \\sqrt(T/mu); u(0,t) = A \\sin(omega t) (driver) and u_x(L,t) = 0 (free end via massless ring on frictionless rod).",
    "formula": "\\ddot{u}= c^2 d^2u/dx^2 - 2 gamma_w \\dot{u}"
  },
  "ode_system": {
    "state_variables": [
      "u(x,t) -- transverse displacement, m"
    ],
    "equations_latex": [
      "u_tt = c^2 u_xx - 2 gamma_w u_t",
      "Driven BC: u(0,t) = A \\sin(2 pi f_drive t)",
      "Free BC: u_x(L,t) = 0 (Neumann via ghost-cell mirror)"
    ],
    "equations_python": [
      "u_tt = c**2 * u_xx - 2 * gamma_w * u_t",
      "Driven * BC: u(0,t) = A * np.sin (2 * pi * f_drive * t)",
      "Free BC: u_x(L,t) = 0 (Neumann via ghost-cell mirror)"
    ]
  },
  "derivation_steps": [
    "Newton II on a small string element: T u_xx dx = mu dx u_tt -> u_tt = c^2 u_xx with c=\\sqrt(T/mu).",
    "Linear viscous-damping term -2 gamma_w u_t bounds long-time amplitude under continuous driving.",
    "Massless ring on frictionless rod transmits no transverse force -> u_x(L,t) = 0 (free end).",
    "Discretise with explicit leapfrog FDM: (1+gamma_w dt) u_{i,n+1} = 2 u_{i,n} - (1-gamma_w dt) u_{i,n-1} + (c dt/dx)^2 lap.",
    "Free-end implementation: ghost cell u_{N+1} = u_{N-1}, so the laplacian at i=N becomes 2 (u_{N-1,n} - u_{N,n}).",
    "CFL: (c dt/dx)^2 < 1 must hold for stability; sub-step n_per_frame times per video frame."
  ],
  "physical_constants_derived": {
    "c_phase_speed_mps": 6.2198,
    "lambda_wavelength_m": 3.8016,
    "n_wavelengths_in_L": 1.3097,
    "T_drive_period_s": 0.6112,
    "decay_length_m": 7.1784
  },
  "numerical_method": {
    "solver": "Explicit leapfrog FDM, second-order centred in time and space, with sub-stepping.",
    "tolerance": "Bit-exact within IEEE float; CFL must satisfy (c dt/dx)^2 < 1."
  },
  "code_validation": {
    "ode_source": "Standard 1-D wave equation with viscous damping; Dirichlet+Neumann BCs.",
    "energy_check": "Driver pumps energy in at x=0; gamma_w dissipation balances input in steady state so envelope saturates.",
    "physical_plausibility": true
  },
  "expected_behavior": [
    "Wavetrain forms and propagates rightward at c~6.220 m/s.",
    "Leading edge reaches the free end at t = L/c ~ 0.801 s and reflects WITHOUT phase inversion.",
    "After ~1-2 round trips a near-standing-wave pattern emerges; the ring is a displacement antinode."
  ],
  "parameters": {
    "L": {
      "value": 4.979190582885917,
      "unit": "",
      "description": ""
    },
    "mu": {
      "value": 0.11623857735239022,
      "unit": "",
      "description": ""
    },
    "T": {
      "value": 4.496838431134701,
      "unit": "",
      "description": ""
    },
    "A": {
      "value": 0.11586384072724647,
      "unit": "",
      "description": ""
    },
    "f_drive": {
      "value": 1.6360911434504202,
      "unit": "",
      "description": ""
    },
    "gamma_w": {
      "value": 0.8664687210282511,
      "unit": "",
      "description": ""
    },
    "N_grid": {
      "value": 200,
      "unit": "",
      "description": ""
    },
    "n_per_frame": {
      "value": 16,
      "unit": "",
      "description": ""
    }
  }
}