{
  "simulation": "driven_string_fixed",
  "physical_observation": "A taut string of length L=4.451 m, density mu=0.0887 kg/m, tension T=5.524 N is fixed at x=L and driven sinusoidally at x=0 with amplitude A=0.076 m and frequency f=1.809 Hz. Phase speed c=\\sqrt(T/mu)=7.889 m/s; wavelength lambda=c/f=4.360 m; ~1.02 wavelengths fit in L. Linear viscous damping gamma_w=0.582 1/s gives decay length c/gamma_w=13.559 m.",
  "physical_law": {
    "name": "Damped wave equation on a string with Dirichlet BCs",
    "statement": "Small-amplitude transverse motion of a taut string obeys u_tt = c^2 u_xx - 2 gamma_w u_t with prescribed driver at the left end and zero displacement at the right end.",
    "formula": "u(0,t) = A \\sin(omega t);  u(L,t) = 0;  u_tt = c^2 u_xx - 2 gamma_w u_t,  c = \\sqrt(T/mu)."
  },
  "ode_system": {
    "state_variables": [
      "u(x_i, t) (transverse displacement at grid points)"
    ],
    "equations_latex": [
      "u^{n+1}_{j} = (1/(1+g dt)) \\cdot (2 u^n_j - (1-g dt) u^{n-1}_j + (c dt/dx)^2 (u^n_{j+1} - 2 u^n_j + u^n_{j-1}))",
      "u^{n+1}_0 = A \\sin(\\omega t_{n+1}); u^{n+1}_N = 0"
    ],
    "equations_python": [
      "u**(n+1)_j = (1/(1+g * dt)) * (2 * u**n_j - (1-g * dt) * u**(n-1)_j + (c * dt/dx)**2 * (u**n_j_1 - 2 * u**n_j + u**n_j_1))",
      "u**(n+1)_0 = A * np.sin ( omega * t_n_1); u**(n+1)_N = 0"
    ]
  },
  "derivation_steps": [
    "Phase speed c = \\sqrt(T/mu) = \\sqrt(5.523921460218052/0.08874660772689846) = 7.8895 m/s.",
    "Driving angular frequency omega = 2 pi f = 11.3685 rad/s; wavelength lambda = c/f = 4.3604 m.",
    "Discretise u(x,t) on a uniform grid; use centred second differences in x and centred-time damping for stability.",
    "Apply Dirichlet BCs at every step: prescribed driver at j=0, zero at j=N.",
    "CFL constraint: (c dt/dx)^2 < 1; using dt = 1/(fps \\cdot n_per_frame) so the simulation auto-tightens n_per_frame if needed.",
    "Damping reduces wave amplitudes over a length scale c/gamma_w = 13.559 m, ensuring the response remains bounded."
  ],
  "physical_constants_derived": {
    "c_m_s": 7.889470685405782,
    "omega_rad_s": 11.368480451818296,
    "lambda_m": 4.3603898077721714,
    "n_wavelengths_in_L": 1.0206725135798091,
    "decay_length_m": 13.558730194477583
  },
  "numerical_method": {
    "solver": "Explicit leapfrog FDM on a uniform grid with centred-time damping; CFL-stable when (c dt/dx)^2 < 1.",
    "tolerance": "Spatial second order; temporal second order; CFL margin enforced by adjusting n_per_frame.",
    "justification": "Hyperbolic PDE with smooth BCs; standard scheme provides good fidelity for the wavetrain and reflections."
  },
  "code_validation": {
    "ode_source": "Damped wave equation in 1D with Dirichlet BCs.",
    "energy_check": "With damping, total energy decays smoothly; without damping the standing-wave envelope is steady.",
    "physical_plausibility": true
  },
  "expected_behavior": [
    "Wavetrain emitted from x=0 propagates rightward at c=7.889 m/s; reflects from the fixed end with a 180-degree phase flip.",
    "Outgoing + reflected waves superpose into an approximate standing wave with a node at x=L and antinodes spaced by lambda/2 = 2.180 m.",
    "After a few transit times (~L/c = 0.564 s), the response is nearly stationary in shape, modulated at omega."
  ],
  "parameters": {
    "L": {
      "value": 4.450530025286603,
      "unit": "",
      "description": ""
    },
    "mu": {
      "value": 0.08874660772689846,
      "unit": "",
      "description": ""
    },
    "T": {
      "value": 5.523921460218052,
      "unit": "",
      "description": ""
    },
    "A": {
      "value": 0.07584862170850458,
      "unit": "",
      "description": ""
    },
    "f_drive": {
      "value": 1.8093498593504656,
      "unit": "",
      "description": ""
    },
    "gamma_w": {
      "value": 0.5818738607704674,
      "unit": "",
      "description": ""
    },
    "N_grid": {
      "value": 200,
      "unit": "",
      "description": ""
    },
    "n_per_frame": {
      "value": 16,
      "unit": "",
      "description": ""
    }
  }
}