| { |
| "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": "" |
| } |
| } |
| } |
|
|
