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Exact spherical-mode reflection under the local impedance truncation

Statement

Fix $k, R > 0$ and $ℓ \geq 0$ . On $r = R$ , write one radial mode as $u (r) = A h_{ℓ}^{(1)} (k r) + B h_{ℓ}^{(2)} (k r)$ , where $h_{ℓ}^{(2)} = \overline{h_{ℓ}^{(1)}}$ . If $u$ satisfies $\partial_{r} u - i k u = 0$ at $R$ , then $B = - \frac{(h_{ℓ}^{(1)})^{'} (k R) - i h_{ℓ}^{(1)} (k R)}{(h_{ℓ}^{(2)})^{'} (k R) - i h_{ℓ}^{(2)} (k R)} A .$ The denominator is nonzero. For $A \neq 0$ , the exact reflected modal-amplitude ratio is therefore $\frac{B}{A} = - \frac{(h_{ℓ}^{(1)})^{'} (k R) - i h_{ℓ}^{(1)} (k R)}{(h_{ℓ}^{(2)})^{'} (k R) - i h_{ℓ}^{(2)} (k R)} .$ If $A = 0$ , the coefficient equality forces $B = 0$ and no ratio is defined. No uniform-in- $ℓ$ smallness is asserted.

Facts & Assumptions

Given: The outgoing/incoming radial pair at positive real argument.

[F1]

First-order local impedance condition at an artificial plane: the spherical truncation condition uses the outgoing convention $\partial_{r} u - i k u = 0$ .

[F2]

Outgoing spherical Hankel modes have positive flux and no positive-real zeros: $Im (\overline{h_{ℓ}^{(1)}} (h_{ℓ}^{(1)})^{'}) = z^{- 2}$ and the incoming conjugate has the negative of this flux.

Proof

Given: $k, R > 0$ , $ℓ \geq 0$ , and [F1]–[F2].

1.1F1F2algebra

Let $z = k R$ . The boundary condition, divided by $k$ , is $A ((h_{ℓ}^{(1)})^{'} (z) - i h_{ℓ}^{(1)} (z)) + B ((h_{ℓ}^{(2)})^{'} (z) - i h_{ℓ}^{(2)} (z)) = 0.$ If the incoming factor in parentheses were zero, then $(h_{ℓ}^{(2)})^{'} = i h_{ℓ}^{(2)}$ , so $Im (\overline{h_{ℓ}^{(2)}} (h_{ℓ}^{(2)})^{'}) = ∣ h_{ℓ}^{(2)} ∣^{2} > 0$ by nonvanishing, contradicting its negative flux from [F2]. Hence that factor is nonzero.

2.1step 1.1algebra∎

Divide the equation in step 1.1 by its nonzero incoming factor to obtain the displayed coefficient equality for every $A, B$ . For $A \neq 0$ , division by $A$ gives the displayed ratio. If $A = 0$ , the coefficient equality forces $B = 0$ , so the boundary condition has no nonzero purely incoming mode and the zero mode carries no ratio.

Depends on

Used by

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Sources

NIST Digital Library of Mathematical Functions, Chapter 10: Bessel Functions (standard reference, not scraped)
NIST Digital Library of Mathematical Functions, §§10.49–10.50 (standard reference, not scraped)