Datasets:
File size: 3,640 Bytes
92f32a3 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 | {
"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": ""
}
}
}
|