Datasets:
File size: 4,107 Bytes
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"simulation": "driven_string_fixed",
"physical_observation": "A taut string of length L=4.175 m, density mu=0.1331 kg/m, tension T=5.308 N is fixed at x=L and driven sinusoidally at x=0 with amplitude A=0.107 m and frequency f=2.402 Hz. Phase speed c=\\sqrt(T/mu)=6.315 m/s; wavelength lambda=c/f=2.629 m; ~1.59 wavelengths fit in L. Linear viscous damping gamma_w=1.451 1/s gives decay length c/gamma_w=4.351 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(5.308495280754792/0.13310537692450025) = 6.3152 m/s.",
"Driving angular frequency omega = 2 pi f = 15.0949 rad/s; wavelength lambda = c/f = 2.6287 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 = 4.351 m, ensuring the response remains bounded."
],
"physical_constants_derived": {
"c_m_s": 6.31521176329535,
"omega_rad_s": 15.094919259712645,
"lambda_m": 2.628675588134308,
"n_wavelengths_in_L": 1.5882839840657452,
"decay_length_m": 4.351157833177315
},
"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=6.315 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 = 1.314 m.",
"After a few transit times (~L/c = 0.661 s), the response is nearly stationary in shape, modulated at omega."
],
"parameters": {
"L": {
"value": 4.175083335938324,
"unit": "",
"description": ""
},
"mu": {
"value": 0.13310537692450025,
"unit": "",
"description": ""
},
"T": {
"value": 5.308495280754792,
"unit": "",
"description": ""
},
"A": {
"value": 0.10652940639011876,
"unit": "",
"description": ""
},
"f_drive": {
"value": 2.402431015756321,
"unit": "",
"description": ""
},
"gamma_w": {
"value": 1.451386505711708,
"unit": "",
"description": ""
},
"N_grid": {
"value": 200,
"unit": "",
"description": ""
},
"n_per_frame": {
"value": 16,
"unit": "",
"description": ""
}
}
}
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