For one mode of propagation the acoustic potentials Φ (air) and Θ (lining) share a common axial propagation constant Γ. Applying continuity of normal particle velocity and acoustic pressure at the lining–air interface, with a rigid outer wall, leads to Scott's transcendental equation, rewritten by Christie in the dimensionless form:
f(w) = w·tan w + M·√(w² + p²)·tan(b·√(w² + p²)) = 0 (2.2.9)where w = √((Γℓ)² + (kℓ)²), p = √((hℓ)² − (kℓ)²), b = d/ℓ, and M = ρ/ρ′ is the ratio of air density to complex lining density. The lining wave number h and characteristic impedance Za are obtained from the Delany–Bazley/Mechel formulae with normalised flow resistivity A = σ/(ρ f).
Once w is found, the duct propagation constant follows from Γ = √((w/ℓ)² − k²), and the attenuation is 20·log₁₀(e)·Re{Γ} ≈ 8.686·Re{Γ} dB per metre. Here it is normalised to the duct half-width ℓ.
The iteration is seeded from the Morse (local-reaction) starting points w(0,0) ≈ 0
and w(0,2) ≈ π, then continued across the frequency axis using linear extrapolation
of the previous solution as the next initial value — exactly as in the 1988 BASIC code of the
original thesis.