The ladder network of the baffled-piston radiation impedance: an elementary proof of existence to all orders and of the limits k²e_k → 1/4, k²d_k → 3/4
Ronald M. Aarts
Abstract
The radiation impedance of a rigid circular piston in an infinite baffle, Z = 1 − J₁(2ka)/(ka) + j H₁(2ka)/(ka), has a continued-fraction expansion in s = jka whose truncations are ladder networks of masses, compliances and resistances. We prove, without any computer-assisted step, that this expansion exists to every order, that all its coefficients e_k are positive, that the resistive coefficients d_k are positive for k ≥ 10, and that k²e_k → 1/4 and k²d_k → 3/4. A nine-entry table of explicit finite sums (reproduced by a short ancillary script) settles d₁, …, d₉ > 0, so all coefficients are positive and every truncation is a passive ladder.
The proof compares the piston with an explicitly solvable Kummer-function model that has the same high-frequency ripple. Three simple facts make the comparison work. Multiplying the model's delay weight by (2 − u) corresponds to multiplying by x in the dual variable, so every correction is a power of the model's own Jacobi matrix. The piston weight differs from the model by a series of such corrections whose coefficients all have one sign, so their total size is fixed by the ratio of the two impedances at infinite frequency. Finally, the first correction is tridiagonal with closed-form entries, and the remaining ones are bounded by products of Pochhammer symbols. Existence, e_k > 0, d_k > 0 for k ≥ 10 and the limits reduce to inequalities that can be checked by hand. We also explain the limits physically: they are fixed by the delay (2a/c) and the decay exponent (3/2) of the edge-diffraction ripple of Z at high frequency.
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