10-EFT.WP.Core.Tension v1.0 | Appendix A Symbols and Units Reference

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Appendix A Symbols and Units Reference

I. Unit Baselines and Dimensional Conventions

1. SI base and common derived units
   • Length [m], Mass [kg], Time [s].
   • Force N = kg * m / s^2, Pressure Pa = N / m^2, Surface tension N / m.
   • Angular frequency rad / s (dimensionless angle labeled with rad).
2. Dimension notation
   Let M, L, T denote mass, length, and time. Example: the dimension of T_fil is M L T^{-2}.
3. Name-collision constraints
   • T_fil (tension, N) and T_trans (power transmission coefficient, dimensionless) must not be mixed.
   • n (number density) and n_eff (effective refractive index, dimensionless) must be strictly distinguished.

II. Geometry and Measures

• gamma(ell) — path parametrization; [SI] ell in m; [dim] L.
• ell — arc-length parameter; [SI] m; [dim] L.
• L_gamma = ( ∫ 1 d ell ) — total path length; [SI] m; [dim] L.
• d ell — arc-length measure element; [SI] m; [dim] L.
• dA — surface-element measure; [SI] m^2; [dim] L^2.
• dV — volume-element measure; [SI] m^3; [dim] L^3.
• t_hat — unit tangent; [SI] 1; [dim] 1.
• n_hat — unit normal; [SI] 1; [dim] 1.
• | d ell / ds | — Jacobian from parameter s to arc length; [SI] m; [dim] L.
• kappa1, kappa2 — principal curvatures; [SI] 1 / m; [dim] L^{-1}.

III. Tension, Stress, Strain, and Material Constants

• T_fil(ell,t) — line tension (along gamma(ell)); [SI] N; [dim] M L T^{-2}.
• sigma_s(x,t) — interfacial/membrane tension; [SI] N / m; [dim] M T^{-2}.
• sigma_ij(x,t) — Cauchy stress; [SI] Pa; [dim] M L^{-1} T^{-2}.
• epsilon_ij — small-strain tensor; [SI] 1; [dim] 1; definition epsilon_ij = (1/2) * ( ∂_i u_j + ∂_j u_i ).
• E — Young’s modulus; [SI] Pa; [dim] M L^{-1} T^{-2}.
• A — cross-sectional area; [SI] m^2; [dim] L^2.
• sigma_sv, sigma_sl, sigma_lv — solid–vapor, solid–liquid, liquid–vapor surface energies/surface tensions; [SI] N / m.
• theta_c — contact angle; [SI] rad; [dim] 1.
• Linear constitutive law (small deformation)
  S72-2 : T_fil(ell) = E * A * epsilon(ell) with epsilon = ∂_ell u.

IV. Kinematics, Loads, and Mass Parameters

• u(x,t) — displacement; [SI] m; [dim] L.
• ∂_t u, ∂_tt u — velocity and acceleration; [SI] m / s, m / s^2.
• rho_m — volumetric density; [SI] kg / m^3; [dim] M L^{-3}.
• rho_l = rho_m * A — line density; [SI] kg / m; [dim] M L^{-1}.
• b(x,t) — body-force density; [SI] N / m^3; [dim] M L^{-2} T^{-2}.
• q(ell,t) — distributed line load (along gamma); [SI] N / m; [dim] M T^{-2}.
• f_ext — nodal/end external force; [SI] N.
• Dynamic equilibrium
  S72-5 : rho_l ∂_tt u = ∂_ell( T_fil ∂_ell u ) + q.

V. Waves, Frequency Domain, and Impedance/Transmission

1. c — wave speed; [SI] m / s; [dim] L T^{-1}; S72-6 : c = sqrt( T0 / rho_l ).
2. omega — angular frequency; [SI] rad / s; [dim] T^{-1}.
3. k — wavenumber; [SI] rad / m; [dim] L^{-1}.
4. Z — mechanical impedance (string): Z = rho_l * c; [SI] kg / s; [dim] M T^{-1}.
5. r_amp, t_amp — amplitude reflection/transmission coefficients; [SI] 1.
6. R_ref, T_trans — power reflection/transmission coefficients; [SI] 1; lossless junctions satisfy R_ref + T_trans = 1.
7. Two-medium interface (lossless)
   • S72-9 : r_amp = ( Z2 - Z1 ) / ( Z2 + Z1 ).
   • S72-10 : t_amp = 2 * Z2 / ( Z1 + Z2 ).
   • S72-11 : R_ref = |r_amp|^2 , T_trans = 1 - R_ref.
8. Modal frequencies (uniform string, fixed–fixed)
   f_n = ( n / ( 2 * L_gamma ) ) * sqrt( T0 / rho_l ); [SI] Hz.

VI. Membrane/Interface Tension and Curvature

• Young–Laplace
  S72-3 : Δp = sigma_s * ( kappa1 + kappa2 ); Δp in Pa.
• Young–Dupré (contact angle)
  S72-4 : sigma_sv = sigma_sl + sigma_lv * cos(theta_c).
• Boundary line integral
  ∮ sigma_s d s (with d s the boundary line element; [SI] m).

VII. Junctions, Branches, and Networks

• Node equilibrium
  S72-7 : node_equilibrium (notation: ∑ T_i * t_hat_i + f_ext = 0); all terms in N.
• Path energy
  S72-8 : path_energy = ( ∫ ( T_fil * (∂_ell u)^2 / 2 ) d ell ); [SI] J.
• Interface impedance
  Z1, Z2 — impedances on both sides of a junction; [SI] kg / s.

VIII. Arrival Time, Normalization, and Calibration

1. T_arr — time of arrival; [SI] s; two conventions
   • Constant factored out: T_arr = ( 1 / c_ref ) * ( ∫ n_eff d ell ).
   • General form: T_arr = ( ∫ ( n_eff / c_ref ) d ell ).
2. n_eff(x,t) — effective refractive index; [SI] 1; consistent with channel definition.
3. c_ref — reference speed; [SI] m / s.
4. Non-dimensionalization
   T_star = T_fil / T_ref (with T_ref the reference tension); c_star = c / c_ref; rho_star = rho_l / rho_ref.
5. Consistency metrics
   • ratio_c = ( L_gamma / T_arr ) / sqrt( T_fil / rho_l ) (dimensionless).
   • delta_form = | ( 1 / c_ref ) * ( ∫ n_eff d ell ) - ( ∫ ( n_eff / c_ref ) d ell ) |; [SI] s.

IX. Identification, Uncertainty, and Information Bounds

• theta — parameter vector (e.g., T0, rho_l, ...); [SI] depends on components.
• L(theta) — likelihood; [SI] 1.
• I_F(theta) — Fisher information matrix; [SI] 1; S72-15 : I_F(theta) = E[ (∂_theta log L)^T (∂_theta log L) ].
• CRLB — Cramér–Rao lower bound; [SI] same as theta; S72-16 : cov(theta) ≥ I_F^{-1}(theta).
• y(t) — observed time series; [SI] per sensor; sampling rate fs in Hz.
• ENBW_Hz — equivalent noise bandwidth; [SI] Hz.

X. Boundary Conditions and Tractions/Operators

• Dirichlet (displacement boundary)
  u |_{∂Omega_u} = u_bar; u_bar in m.
• Neumann (traction boundary)
  t |_{∂Omega_t} = sigma • n_hat; t in N / m^2 (continuum) or N (line-end).
• Robin (mixed)
  alpha * u + beta * t = g; alpha, beta scaled per model.
• Derivatives/operators
  ∂_i (spatial partial), ∂_t (time partial), ∂_ell (arc-length partial); dot •, norm | • |.

XI. Numerical Discretization and Stability (Chapter 7 Gauge)

1. Δell — mesh step; [SI] m.
2. Δt — time step; [SI] s.
3. c_max — maximum phase/group speed; [SI] m / s.
4. CFL — Courant number; [SI] 1; stability condition
   S72-12 : Δt ≤ CFL * ( Δell / c_max ).
5. Rayleigh damping parameters
   C = a * M + b * K; a, b ≥ 0; a in s^{-1}, b in s.
6. Energy-conservation check
   • E_total(t) = ( ∫ ( T_fil * (∂_ell u)^2 / 2 ) d ell ) + ( ∫ ( rho_l * (∂_t u)^2 / 2 ) d ell ); [SI] J.
   • ΔE_rel — relative energy drift; [SI] 1.

XII. Other Common Symbols and Constants

• g — gravitational acceleration; [SI] m / s^2.
• H_z — Hertz (frequency unit); 1 / s.
• R, r — radius or characteristic length; [SI] m; example (sphere): Δp = 2 * sigma_s / R.
• Z1, Z2 — impedances at both sides of a junction; [SI] kg / s.
• f_n — n-th modal frequency; [SI] Hz.

XIII. Dimensionless Summary (for Comparison and Checks)

• T_star = T_fil / T_ref, c_star = c / c_ref, rho_star = rho_l / rho_ref, ratio_c, R_ref, T_trans, r_amp, t_amp, CFL.
• All starred quantities are dimensionless [1].

XIV. Notation and Subscript Rules

• Indices i, j denote Cartesian components; repeated indices are not summed unless an explicit ∑ is written.
• Along-line quantities must use the explicit measure d ell; membranes/interfaces use dA; volume integrals use dV.
• Path-dependent quantities explicitly carry an index/argument like (..., ell, ...), e.g., T_fil(ell,t).
• Frequency-domain quantities are denoted by omega or f, e.g., S_xx(f) (aligned with Core.Sea).

XV. Equation Numbering and Cross-Volume Anchors

• Minimal equations in this volume: S72-*; Postulates: P71-*; Processes: Mx-7*; Implementation bindings: I70-*.
• Cross-volume anchors: n_eff, c_ref, T_arr, S_xx(f) as defined in Core.Density, Core.Sea, and Core.Threads.

XVI. Quick Audit Checklist (Units and Gauges)

• T_fil must be in N, sigma_s in N / m, sigma_ij in Pa.
• Use d ell / dA / dV for line/surface/volume integrals respectively, and state the domain.
• Lossless junctions must satisfy R_ref + T_trans = 1.
• Wave-speed consistency: c = L_gamma / T_arr = sqrt( T_fil / rho_l ) (for uniform strings under steady conditions).
• Record the two-convention discrepancy delta_form in seconds and store it for traceability.