10-EFT.WP.Core.Tension v1.0 | Chapter 4 — Dynamic Tension and Waves

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Chapter 4 — Dynamic Tension and Waves

I. Scope and Model Tiers

• Objects of interest. Small‐deflection transverse motion u(ell,t) on a one-dimensional filament path gamma(ell), axial tension field T_fil(ell,t), line density rho_l(ell), and distributed forcing q(ell,t).
• Model tiers.
  Uniform string (T_fil = T0, rho_l = const); slowly varying non-uniform strings (either T_fil(ell) or rho_l(ell) vary smoothly); weakly dissipative systems with viscous/structural damping.
• Measure and span.
  Arc-length measure d ell; path length L_gamma = ( ∫ 1 d ell ). Boundary points are indexed by ell=0 and ell=L_gamma.

II. Governing Dynamics and Boundary Data

1. Transverse force balance (minimal form).
   S72-5 : rho_l(ell) * ∂_tt u = ∂_ell( T_fil(ell,t) * ∂_ell u ) + q(ell,t)
2. Quasi-static tension reduction.
   When T_fil(ell,t) ≈ T0, the balance reduces to the standard wave equation
   rho_l * ∂_tt u = T0 * ∂_ell ell u.
3. Boundary types (units must be explicit).
   • Dirichlet: u |_{∂} = u_bar (displacement constraint).
   • Neumann: ( T_fil * ∂_ell u ) |_{∂} = T_bar (end traction).
   • Robin: alpha * u + beta * ( T_fil * ∂_ell u ) = g (mixed).

III. Uniform String: Traveling Waves, Modes, and Wave Speed

• D’Alembert traveling waves.
  u(ell,t) = f( ell - c t ) + g( ell + c t )
• Wave speed–tension–density relation (uniform).
  S72-6 : c = sqrt( T0 / rho_l )
• Clamped–clamped modes and identification.
  f_n = ( n / ( 2 * L_gamma ) ) * c and
  T0 = rho_l * ( 2 * L_gamma * f_n / n )^2
• Clamped–free approximation.
  f_n = ( (2n - 1) / ( 4 * L_gamma ) ) * c

IV. Non-uniform Tension: Slowly Varying (WKB) Approximation

• Local phase speed.
  c(ell) = sqrt( T_fil(ell) / rho_l(ell) )
• WKB phase and envelope (slowly varying medium).
  Phase phi(ell) = ( ∫ ( d ell / c(ell) ) ) and approximate field
  u(ell,t) ≈ A(ell) * cos( omega * t - omega * phi(ell) )
• Amplitude conservation (lossless).
  A(ell) * sqrt( Z(ell) ) = const, with mechanical impedance Z(ell) = rho_l(ell) * c(ell).
• Time-of-arrival along a path.
  T_toa(path) = ( ∫ ( d ell / c(ell) ) ), consistent with the T_arr conventions in §VIII.

V. Damping and External Forcing (Weak Dissipation)

• Kelvin–Voigt viscous augmentation (generic form).
  rho_l * ∂_tt u = ∂_ell( T0 * ∂_ell u + eta * ∂_t ∂_ell u ) + q
  The effective viscosity eta has units N·s; frequency response exhibits first-order low-pass roll-off versus omega.
• Harmonic steady forcing q = Q0 * cos( omega t ).
  Frequency response H(omega) = U(omega) / Q0 depends on damping and boundary data. Spectral reporting must follow the S_xx(f) conventions from Core.Sea.

VI. Energy, Power Flow, and Mechanical Impedance

• Energy density and flux.
  e = ( 1 / 2 ) * [ rho_l * ( ∂_t u )^2 + T0 * ( ∂_ell u )^2 ]
  P_flow = - T0 * ( ∂_ell u ) * ( ∂_t u ) (negative sign indicates net flow toward +ell when the product is positive).
• Traveling-wave impedance.
  Z = rho_l * c
• Lossless interface power balance.
  Use the reflection/transmission conventions of Chapter 6 (S72-9 … S72-11). Here, Z is used for matching at uniform–uniform interfaces in the traveling-wave limit.

VII. Dispersion via Effective Bending Stiffness (Optional)

• Small-deflection dispersion with bending stiffness B (units N·m²).
  omega^2 = c^2 * k^2 + ( B / rho_l ) * k^4
• Group velocity v_g = ∂ omega / ∂ k is frequency-dependent.
  Time-of-arrival estimates must be band-limited; report ENBW_Hz and window power U_w per Core.Sea.

VIII. Time-of-Arrival → Speed → Tension Calibration (Mx-73)

1. Acquire waveform and path. Provide gamma(ell), L_gamma = ( ∫ 1 d ell ), and perform timebase alignment.
2. Two TOA conventions (cross-volume uniformity):
   • Constant-pulled: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell )
   • Path-wise: T_arr = ( ∫ ( n_eff / c_ref ) d ell )
   • Record delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |
3. Phase-speed estimate. c_hat = L_gamma / T_arr (with band limits and group-speed correction if dispersion is non-negligible).
4. Back-out tension. T_hat = rho_l * c_hat^2
5. Audit and publish. Units check, TOA convention disparity, measure declarations, and boundary description—consistent with Chapter 8 normalization.

IX. Spectral Identification and Uncertainty

• Modal method. Infer T0 from f_n; propagate cov(T0) using the Fisher/CRLB framework of Chapter 9.
• Spectral density. One-sided S_xx(f) must be reported with ENBW_Hz and U_w to avoid energy bias.
• Time–frequency. From STFT, estimate instantaneous frequency f_inst(t); then c(t) = ( 2 * L_gamma / n ) * f_n(t) yields slow tension drift.

X. Boundaries, Reflections, and Pulse Timing

• Reflection types. Clamped end: phase inversion; free end: in-phase reflection. For an impedance termination Z_L, the amplitude reflection is
  r_amp = ( Z_L - Z ) / ( Z_L + Z ) (see S72-9).
• Multi-bounce arrival sequence.
  t_k ≈ ( 2k - 1 ) * ( L_gamma / c ) or t_k ≈ ( 2k ) * ( L_gamma / c ) depending on end types. Always document end conditions and measures.

XI. Discretization and Steady-State Verification (aligned with Chapter 7)

• Explicit central differences (CFL). Δt ≤ CFL * ( Δell / c_max ) (see S72-12).
• Energy checks. Discrete energy E^n decays monotonically with damping; is approximately conserved without damping.
• Meshing strategy. Refine Δell where c(ell) varies sharply to control phase error; always spell out d ell.

XII. Engineering Interfaces (I70-3 and I70-6)

• string_wave_solve(path,T,rho_l,bc,damping) -> WaveRef
  Solves S72-5, returns traveling-wave/modal views, TOA estimates, and energy checks.
• estimate_tension_from_modes(freqs,L,rho_l) -> float
  Recovers T_fil from modal frequencies.
• calibrate_tension_by_toa(trace,L_gamma,rho_l,n_eff,c_ref) -> float
  Executes Mx-73, enforcing TOA dual-form recording and alignment via delta_form.

XIII. Publication Checklist (Mx-73-CHK)

• Declare gamma(ell), measure d ell, L_gamma, rho_l, boundary types, and q(ell,t).
• State which approximations are used in S72-5 (uniform / slowly varying / damped) and pass the unit audit (N, kg/m, m/s, Pa).
• When using TOA, report both T_arr forms and delta_form; source provenance for n_eff and c_ref.
• Spectral outputs must include ENBW_Hz, U_w, window family, and sampling rate—symbols aligned with Core.Sea.
• Numerical runs must include energy checks, CFL constraint, mesh/time-step settings, and reproducibility hashes.

Terminology and Anchors (quick recap)

• Minimal equations.
  S72-5 : rho_l * ∂_tt u = ∂_ell( T_fil * ∂_ell u ) + q
  S72-6 : c = sqrt( T0 / rho_l )
• Geometry. gamma(ell), L_gamma = ( ∫ 1 d ell )
• Impedance & power. Z = rho_l * c, P_flow = - T0 * ( ∂_ell u ) * ( ∂_t u )
• TOA dual-form & disparity. the two T_arr expressions and delta_form (consistent with Core.Density, Chapter 9)
• Spectral dialect. S_xx(f), ENBW_Hz, U_w (per Core.Sea)