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