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Wave interfaces and conductors — Examples

1 · Prerequisites

2 · Summary

The three worked cases are deterministic computations of the interface and conducting-medium model with declared, non-measured parameters. The first checks the good-conductor skin-depth bounds at the declared ratio $ε ω / σ = 3 / 25$ with exact identities and bounded remainders, and records the exclusion of $σ = 0$ from the finite-depth approximation. The second evaluates the equal-permeability Brewster configuration $n_{1} = 1$ , $n_{2} = 3 / 2$ : the unique Brewster angle with $\tan θ_{B} = 3 / 2$ , the vanishing TM amplitude with $t_{p} = 1$ , and the surviving TE reflection $r_{s} = - 5 / 13$ with $R_{s} + T_{s} = 1$ exactly. The third treats total internal reflection with $n_{1} = 3 / 2$ , $n_{2} = 1$ at $\sin θ_{i} = 4 / 5$ : the unit-modulus reflection factor with its explicit phase, the evanescent transmitted field with decay factor $e^{- \sqrt{11} / 5}$ per distance $c / ω$ , the vanishing normal averaged flux and the positive tangential averaged flux. Each example states its own hypotheses, reports no measurement and asserts no empirical claim.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

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Numerical check of the good-conductor skin-depth bounds and the $σ = 0$ exclusion

Physical scope

Numerical illustration for declared model parameters in the isotropic scalar Ohmic branch: the good-conductor bounds |q-sqrt(b/2)|<=2tsqrt(b/2) and |delta-sqrt(2/b)|<=4tsqrt(2/b) at eps*omega/sigma=3/25 and the exclusion of sigma=0 from the finite-depth approximation. Excluded: any measured conductivity, any laboratory material, and any experimental or statistical claim.

Example

Adopt the isotropic scalar Ohmic branch of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch at rest and the exact quantities of Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth and Conducting-medium propagation: the exact attenuation constant, the skin depth and its good- and poor-conductor regimes. Let the declared parameters be $ε = 4 F m^{- 1}, μ = 1 H m^{- 1}, σ = 100 S m^{- 1}, ω = 3 s^{- 1};$ these are declared model parameters of the illustration, not measured values of any material. The numerical evaluations below use these SI units: $a, b, r$ have unit $m^{- 2}$ , $p, q$ have unit $m^{- 1}$ , and $δ$ has unit m; $t = a / b$ is dimensionless. Then: (i) $a = μ ε ω^{2} = 36$ , $b = μ σ ω = 300$ , $r = \sqrt{a^{2} + b^{2}} = \sqrt{91296}$ and $t = a / b = 3 / 25 = 0.12 \leq 1 / 4$ , so the good-conductor regime of Conducting-medium propagation: the exact attenuation constant, the skin depth and its good- and poor-conductor regimes(c) applies; (ii) the exact propagation and attenuation constants are $p = \sqrt{(r + 36) / 2}$ and $q = \sqrt{(r - 36) / 2}$ , with $p q = b / 2 = 150$ exactly; the decimal evaluations $r \approx 302.15228$ , $p \approx 13.00293$ , $q \approx 11.53586$ satisfy $q < \sqrt{b / 2} = \sqrt{150} \approx 12.24745$ with $∣ q - \sqrt{150} ∣ \approx 0.71159 \leq 2 t \sqrt{150} = 0.24 \sqrt{150} \approx 2.93939,$ and for the skin depth $δ = 1 / q \approx 0.08669$ against $\sqrt{2 / b} = 1 / \sqrt{150} \approx 0.08165$ one has $∣ δ - \sqrt{2 / b} ∣ \approx 0.00504 \leq 4 t \sqrt{2 / b} = 0.48 \sqrt{2 / b} \approx 0.03919.$ Thus the exact skin depth satisfies the stated good-conductor bound with the margin predicted by the theorem. (iii) for $σ = 0$ (same $ε, μ, ω$ ) one has $b = 0$ , $q = 0$ , no attenuation and $δ = + \infty$ by the convention of Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth(c); the inequalities of (ii) do not apply, consistently with the exclusion recorded in Conducting-medium propagation: the exact attenuation constant, the skin depth and its good- and poor-conductor regimes(f). (iv) [Scope] the computation is an exact check of the stated inequalities for the declared model parameters (with dimensionless ratio $t$ ); it reports no measurement, no material identification and no empirical claim, and the decimal evaluations are displayed only as approximate illustrations of the exact expressions.

Verification

technique · direct substitution in the exact identities and bounds of the skin-depth theorem, with the decimal displays read as roundings of the exact radicals

Given: The declared parameters $ε = 4$ , $μ = 1$ , $σ = 100$ , $ω = 3$ as the stated dimensionless representatives of the SI quantities of the Example, the branch assumption of Declared material-closure branches: linear nondispersive Ohmic response and the causal memory branch, and the exact quantities $a, b, r, p, q, δ, t$ of Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth and Conducting-medium propagation: the exact attenuation constant, the skin depth and its good- and poor-conductor regimes.

[A1] The conditional plane-wave attenuation theorem Conducting-medium propagation: the exact attenuation constant, the skin depth and its good- and poor-conductor regimes: in the declared scalar Ohmic branch with $ε, μ, ω, σ > 0$ , its exact identities (b) and good-conductor bounds (c) for $t \leq 1 / 4$ apply to a nonzero transverse plane-wave field with $k = λ n$ , $∣ n ∣ = 1$ , satisfying the four amplitude conditions of Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch(b). A field meeting these hypotheses is constructed in step 1.1.

[F1] The medium parameters (Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth): $a = μ ε ω^{2}$ , $b = μ σ ω$ , $r = \sqrt{a^{2} + b^{2}}$ , $p = \sqrt{(r + a) / 2}$ , $q = \sqrt{(r - a) / 2}$ , $δ = 1 / q$ for $σ > 0$ and $δ = + \infty$ for $σ = 0$ ; $t : = a / b$ .

[F2] The plane-wave solution criterion of Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch(b)-(c): for $ω > 0$ , complex amplitudes $e, h$ and wavevector $k$ , the smooth fields $E (x, τ) = Re (e \exp (i (k \cdot x - ω τ)))$ , $H (x, τ) = Re (h \exp (i (k \cdot x - ω τ)))$ , $D = ε E$ , $B = μ H$ , with $ρ_{f} = 0$ , $J_{e x t} = 0$ and $J_{f} = σ E$ , satisfy the source-free macroscopic equations exactly when $k \cdot e = k \cdot h = 0$ , $k \times e = ω μ h$ and $k \times h = - (ω ε + i σ) e$ . The products are the componentwise complex bilinear products used there.

1.1givenA1F1F2constructalgebra

[Substitution and solution witness, clause (i)] By [F1], $a = μ ε ω^{2} = 1 \cdot 4 \cdot 9 = 36$ and $b = μ σ ω = 1 \cdot 100 \cdot 3 = 300$ ; hence $r = \sqrt{a^{2} + b^{2}} = \sqrt{36^{2} + 300^{2}} = \sqrt{91296}$ and $t = a / b = 36 / 300 = 3 / 25$ . To meet [A1]'s field hypotheses, write $u_{1} = (1, 0, 0)$ , $u_{2} = (0, 1, 0)$ , $u_{3} = (0, 0, 1)$ for the Cartesian unit directions, choose $E_{*} = 1 V m^{- 1}$ , and set $λ = p + i q$ , $n = u_{3}$ , $k = λ u_{3}$ , $e = E_{*} u_{1} \neq 0$ and $h = λ E_{*} u_{2} / (μ ω)$ , which has units $A m^{- 1}$ . By [F1], $λ^{2} = μ ε ω^{2} + i μ σ ω$ , so $k \cdot e = k \cdot h = 0$ , $k \times e = λ E_{*} u_{2} = μ ω h$ and $k \times h = - λ^{2} E_{*} u_{1} / (μ ω) = - (ε ω + i σ) e$ . Thus the $C^{\infty}$ maps $E, H : R^{3} \times R \to R^{3}$ of [F2], with these amplitudes and the stated $D, B, J_{f}$ , satisfy the equations and are transverse; restrict them to $Ω \times I$ with $Ω = {x_{3} > 0}$ and $I = R$ , where the decaying branch is bounded. This is an auxiliary ideal solution, with no interface or empirical assertion. Since $∣ n ∣ = 1$ and $3 / 25 \leq 1 / 4$ (equivalently $12 \leq 25$ ), all hypotheses of A1,(c) now hold and its good-conductor bounds apply.

1.2A1F1algebra

[Exact identities, clause (ii)] By [F1] and A1, $p = \sqrt{(r + a) / 2} = \sqrt{(\sqrt{91296} + 36) / 2}$ , $q = \sqrt{(r - a) / 2} = \sqrt{(\sqrt{91296} - 36) / 2}$ and $p q = b / 2 = 150$ exactly. Since $91296 < 336^{2} = 112896$ , positivity and the square-root order give $r = \sqrt{91296} < 336$ , hence $q^{2} = (r - 36) / 2 < 300 / 2 = b / 2$ , so $q < \sqrt{150} = \sqrt{b / 2}$ ; and the exact bounds of A1 read $∣ q - \sqrt{150} ∣ \leq (6 / 25) \sqrt{150} = \frac{2 ε ω}{σ} \sqrt{\frac{μ σ ω}{2}}$ and $∣ δ - \sqrt{2 / b} ∣ \leq (12 / 25) \sqrt{2 / b} = \frac{4 ε ω}{σ} \sqrt{\frac{2}{μ σ ω}}$ .

1.3givenA1F1algebra

[The $σ = 0$ exclusion, clause (iii)] For the same $ε, μ, ω$ and $σ = 0$ , [F1] gives $b = 0$ , $r = a$ , $q = \sqrt{(r - a) / 2} = 0$ and $δ = + \infty$ by the convention of [F1]; the identities and bounds of A1,(c) carry the hypothesis $b > 0$ , equivalently $σ > 0$ , and are not evaluated at $b = 0$ , which is the exclusion recorded in A1.

2.1step 1.2A1algebra

[Decimal displays and the bound comparisons] The decimal numbers displayed in clause (ii) of the Example are the stated roundings of the exact quantities $r = \sqrt{91296}$ , $p = \sqrt{(r + 36) / 2}$ , $q = \sqrt{(r - 36) / 2}$ , $δ = 1 / q$ , $\sqrt{150}$ and $\sqrt{2 / 300}$ ; reading them as roundings, the comparisons $∣ q - \sqrt{150} ∣ \leq (6 / 25) \sqrt{150}$ and $∣ δ - \sqrt{2 / b} ∣ \leq (12 / 25) \sqrt{2 / b}$ displayed there are instances of the exact inequalities of step 1.2, and the numerical magnitudes quoted in the Example are not independent assertions of exact decimal arithmetic. In particular the inequalities hold with the margin predicted by A1 for the declared dimensionless ratio $t = 3 / 25$ .

3.1givenstep 2.1step 1.3A1F1∎

[Scope, clause (iv)] The computation used only the declared parameters, the branch assumption [A1] and the exact definitions [F1]; it reports no measurement, no material identification and no empirical or statistical claim, and the decimal displays of the Example are approximate illustrations of the exact expressions, as stated. No choice principle is used.

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The Brewster angle at an equal-permeability interface and the surviving $s$ -reflection

Physical scope

Deterministic example of the oblique-incidence interface model for two declared lossless media with equal permeabilities and indices n_1=1, n_2=3/2: the unique Brewster angle tan(theta_B)=n_2/n_1, r_p=0, t_p=1, the surviving s-reflection r_s=-5/13 with R_s+T_s=1 exactly. Excluded: measured refractive indices, absorption, interfaces between unequal permeabilities, unpolarised-beam statistics and every empirical claim.

Example

Adopt the setting of Oblique-incidence TE and TM Fresnel amplitudes, the flux budget, Brewster's angle and total internal reflection with two lossless media at rest, equal permeabilities $μ_{1} = μ_{2} = : μ > 0$ and declared refractive indices $n_{1} = 1$ , $n_{2} = 3 / 2$ (declared model parameters, not optical measurements), so that $ε_{j} = n_{j}^{2} / (c^{2} μ)$ and $K_{j} = n_{j} ω / c$ in the notation of Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth. Then: (i) [Brewster angle] by Oblique-incidence TE and TM Fresnel amplitudes, the flux budget, Brewster's angle and total internal reflection(c) the TM amplitude vanishes exactly at the unique angle $θ_{B} \in (0, π / 2)$ with $\tan θ_{B} = n_{2} / n_{1} = 3 / 2$ , i.e. $\sin θ_{B} = 3 / \sqrt{13}$ , $\cos θ_{B} = 2 / \sqrt{13}$ ; the transmitted angle satisfies $n_{1} \sin θ_{B} = n_{2} \sin θ_{t}$ , hence $\sin θ_{t} = 2 / \sqrt{13}$ , and $θ_{B} + θ_{t} = π / 2$ ; (ii) [Amplitudes] at $θ_{i} = θ_{B}$ one has $Y_{1}^{p} = Y_{2}^{p} = \sqrt{13} / (2 c μ)$ , hence $r_{p} = 0$ and $t_{p} = 1$ ; the remaining $s$ -amplitudes are $Y_{1}^{s} = 2 / (\sqrt{13} c μ)$ , $Y_{2}^{s} = 9 / (2 \sqrt{13} c μ)$ and $r_{s} = \frac{2 / \sqrt{13} - 9 / (2 \sqrt{13})}{2 / \sqrt{13} + 9 / (2 \sqrt{13})} = - \frac{5}{13}, t_{s} = \frac{8}{13};$ (iii) [Flux budget] with $R_{s} = ∣ r_{s} ∣^{2} = 25 / 169$ and $T_{s} = (Y_{2}^{s} / Y_{1}^{s}) ∣ t_{s} ∣^{2} = (9 / 4) (64 / 169) = 144 / 169$ one has $R_{s} + T_{s} = 169 / 169 = 1$ exactly, while the TM channel has $R_{p} = 0$ , $T_{p} = 1$ ; (iv) [Scope] the numbers are declared model parameters; the example is a deterministic computation of the classical interface model, reports no measurement, and a pure TM incident wave has zero reflected field at the Brewster angle. For incident data with a nonzero TE component, the surviving reflected field is TE-polarised because $r_{p} = 0$ and $r_{s} \neq 0$ ; no unpolarised-beam statistics are modelled here.

Verification

technique · direct substitution in the oblique-incidence theorem and the medium parameters, with all numerical values kept exact

Given: The lossless two-media setting of Oblique-incidence TE and TM Fresnel amplitudes, the flux budget, Brewster's angle and total internal reflection with $μ_{1} = μ_{2} = μ > 0$ , $n_{1} = 1$ , $n_{2} = 3 / 2$ , $ε_{j} = n_{j}^{2} / (c^{2} μ)$ , and the Brewster angle $θ_{B}$ of A1.

[A1] The oblique-incidence theorem Oblique-incidence TE and TM Fresnel amplitudes, the flux budget, Brewster's angle and total internal reflection: its clause (a) gives the two admittance systems with $Y_{1}^{s} = k_{13} / (μ_{1} ω)$ , $Y_{1}^{p} = ε_{1} ω / k_{13}$ , $Y_{2}^{s} = k_{t 3} / (μ_{2} ω)$ , $Y_{2}^{p} = ε_{2} ω / k_{t 3}$ and the amplitude formulas; clause (b) gives the angle law $n_{1} \sin θ_{i} = n_{2} \sin θ_{t}$ , the normal fluxes $Y_{1}^{χ} ∣ E_{i}^{χ} ∣^{2} / 2$ , $- Y_{1}^{χ} ∣ E_{r}^{χ} ∣^{2} / 2$ , $Y_{2}^{χ} ∣ E_{t}^{χ} ∣^{2} / 2$ and $R_{χ} + T_{χ} = 1$ with $R_{χ} = ∣ r_{χ} ∣^{2}$ , $T_{χ} = (Y_{2}^{χ} / Y_{1}^{χ}) ∣ t_{χ} ∣^{2}$ ; clause (c) gives the unique Brewster angle with $\tan θ_{B} = n_{2} / n_{1}$ for $n_{1} \neq n_{2}$ and the nonzero $s$ -reflection there.

[F1] Medium parameters (Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth): $K_{j} = ω (ε_{j} μ_{j})^{1 / 2} = ω n_{j} / c$ and $ε_{j} = n_{j}^{2} / (c^{2} μ)$ for $μ_{j} = μ$ . In the propagating interface setting of Oblique-incidence TE and TM Fresnel amplitudes, the flux budget, Brewster's angle and total internal reflection (Statement and clause (b)), $k_{13} = \sqrt{K_{1}^{2} - ∣ κ ∣^{2}} = K_{1} \cos θ_{i}$ and $k_{t 3} = K_{2} \cos θ_{t}$ for $∣ κ ∣ < K_{2}$ ; the cosine formulas follow from the stated positive normal roots and the principal angle convention.

[F2] Trigonometry: $\sin (\arcsin y) = y$ with sine strictly increasing on $[- π / 2, π / 2]$ (Principal inverse sine and inverse cosine); $\tan : (- π / 2, π / 2) \to R$ is a continuous strictly increasing bijection whose principal inverse is $\arctan : R \to (- π / 2, π / 2)$ , with $\tan (\arctan y) = y$ for every real $y$ (The principal inverse tangent $\arctan : R \to (- π / 2, π / 2)$ ); $\tan x = \sin x / \cos x$ (Tangent, cotangent, secant, and cosecant on their exact natural domains); $\sin^{2} x + \cos^{2} x = 1$ and $\cos (- x) = \cos x$ (Pythagorean and parity identities for all six trigonometric functions on their natural domains); sine strictly increasing on $[0, π / 2]$ with $\sin 0 = 0$ (Signs, monotonicity intervals, and ranges of sine and cosine); and $\sin (x + π / 2) = \cos x$ (Quarter-turn values and shifts by pi/2 and pi).

1.1A1F2algebra

[The Brewster angle and its sine and cosine, clause (i)] Since $n_{1} = 1 \neq 3 / 2 = n_{2}$ and $μ_{1} = μ_{2}$ , A1 gives a unique Brewster angle $θ_{B} \in (0, π / 2)$ with $\tan θ_{B} = n_{2} / n_{1} = 3 / 2$ . By [F2], $\sin^{2} θ_{B} = \tan^{2} θ_{B} / (1 + \tan^{2} θ_{B}) = (9 / 4) / (13 / 4) = 9 / 13$ ; since $θ_{B} \in (0, π / 2)$ and sine is strictly increasing on $[0, π / 2]$ with $\sin 0 = 0$ , $\sin θ_{B} > 0$ , hence $\sin θ_{B} = 3 / \sqrt{13}$ ; and $\cos θ_{B} = \sin θ_{B} / \tan θ_{B} = (3 / \sqrt{13}) / (3 / 2) = 2 / \sqrt{13}$ .

2.1step 1.1A1F2algebra

[Transmitted angle and the right-angle relation, clause (i)] Snell's law $n_{1} \sin θ_{B} = n_{2} \sin θ_{t}$ of A1 gives $\sin θ_{t} = (2 / 3) (3 / \sqrt{13}) = 2 / \sqrt{13}$ . Since also $\cos θ_{B} = 2 / \sqrt{13} = \sin θ_{t}$ and $\sin (π / 2 - θ_{B}) = \cos θ_{B}$ by [F2], the equality $\sin θ_{t} = \sin (π / 2 - θ_{B})$ holds with $θ_{t}, π / 2 - θ_{B} \in (0, π / 2)$ ; sine is strictly increasing on that interval, hence injective, so $θ_{t} = π / 2 - θ_{B}$ , i.e. $θ_{B} + θ_{t} = π / 2$ .

3.1step 1.1step 2.1A1F1algebra

[The admittances, clause (ii)] By [F1], $K_{1} = ω / c$ , $K_{2} = 3 ω / (2 c)$ , $k_{13} = K_{1} \cos θ_{B} = (ω / c) (2 / \sqrt{13})$ and $k_{t 3} = K_{2} \cos θ_{t} = K_{2} \sin θ_{B} = (3 ω / (2 c)) (3 / \sqrt{13}) = 9 ω / (2 \sqrt{13} c)$ . With $ε_{1} = 1 / (c^{2} μ)$ and $ε_{2} = 9 / (4 c^{2} μ)$ , the admittances of A1 are $Y_{1}^{s} = k_{13} / (μ ω) = 2 / (\sqrt{13} c μ)$ , $Y_{2}^{s} = k_{t 3} / (μ ω) = 9 / (2 \sqrt{13} c μ)$ , $Y_{1}^{p} = ε_{1} ω / k_{13} = \sqrt{13} / (2 c μ)$ and $Y_{2}^{p} = ε_{2} ω / k_{t 3} = (9 / (4 c^{2} μ)) ω \cdot 2 \sqrt{13} c / (9 ω) = \sqrt{13} / (2 c μ)$ , so $Y_{1}^{p} = Y_{2}^{p}$ exactly.

4.1step 3.1A1algebra

[The amplitudes, clause (ii)] By A1 and step 3.1, $r_{p} = (Y_{1}^{p} - Y_{2}^{p}) / (Y_{1}^{p} + Y_{2}^{p}) = 0$ and $t_{p} = 2 Y_{1}^{p} / (Y_{1}^{p} + Y_{2}^{p}) = 1$ . For the $s$ -channel, factoring the common positive scale $1 / (\sqrt{13} c μ)$ from numerator and denominator, $r_{s} = (2 - 9 / 2) / (2 + 9 / 2) = (- 5 / 2) / (13 / 2) = - 5 / 13$ and $t_{s} = 2 \cdot 2 / (2 + 9 / 2) = 4 / (13 / 2) = 8 / 13$ .

5.1step 4.1A1algebra

[Flux budget, clause (iii)] By A1, $R_{s} = ∣ r_{s} ∣^{2} = 25 / 169$ and $T_{s} = (Y_{2}^{s} / Y_{1}^{s}) ∣ t_{s} ∣^{2} = (9 / 4) (64 / 169) = 144 / 169$ , so $R_{s} + T_{s} = (25 + 144) / 169 = 169 / 169 = 1$ exactly; and in the TM channel $R_{p} = ∣ r_{p} ∣^{2} = 0$ , $T_{p} = (Y_{2}^{p} / Y_{1}^{p}) ∣ t_{p} ∣^{2} = 1$ , so $R_{p} + T_{p} = 1$ as well.

6.1givenstep 5.1A1∎

[Scope, clause (iv)] All quantities are declared model parameters and the computation is exact rational or quadratic-irrational algebra of the interface model; no measured refractive index, no absorption, no unequal-permeability interface, no unpolarised-beam statistics and no empirical claim is used. Since $r_{p} = 0$ and $r_{s} \neq 0$ at $θ_{B}$ , an incident wave with a nonzero TE component has a surviving reflected TE component; no statement about random or unpolarised beams is made. No choice principle is used.

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Total internal reflection: decaying transmitted field, vanishing normal flux and the reflection phase

Physical scope

Deterministic example of total internal reflection in the lossless interface model for declared indices n_1=3/2, n_2=1 at sin(theta_i)=4/5: the unit-modulus reflection factor with its phase, the evanescent transmitted field with decay factor e^{-beta x_3}, the vanishing normal and positive tangential averaged fluxes. Excluded: measured indices, absorption, information-speed or signal-propagation claims, and every empirical statement.

Example

Adopt the setting of Oblique-incidence TE and TM Fresnel amplitudes, the flux budget, Brewster's angle and total internal reflection with equal permeabilities $μ_{1} = μ_{2} = : μ > 0$ and declared indices $n_{1} = 3 / 2$ , $n_{2} = 1$ (declared model parameters, not measurements), choose the tangential wavevector $κ = (6 ω / (5 c)) e_{1}$ , so $\hat{κ} = e_{1}$ and $\hat{y} = n \times \hat{κ} = e_{2}$ , and take the transverse-electric polarisation $e_{j} = E_{j} \hat{y}$ . Let the incidence angle be given by $\sin θ_{i} = 4 / 5$ , which exceeds the critical angle $θ_{c}$ with $\sin θ_{c} = n_{2} / n_{1} = 2 / 3$ , so $∣ κ ∣ = K_{1} \sin θ_{i} = (6 / 5) (ω / c) > (ω / c) = K_{2}$ . Then with $β = \sqrt{∣ κ ∣^{2} - K_{2}^{2}} = (\sqrt{11} / 5) (ω / c)$ and $k_{13} = (ω / c) (3 / 2) (3 / 5) = (9 / 10) (ω / c)$ : (i) [Reflection] $Y_{1}^{s} = 9 / (10 μ c)$ , $Y_{2}^{s} = i \sqrt{11} / (5 μ c)$ and $r_{s} = \frac{9 / 10 - i \sqrt{11} / 5}{9 / 10 + i \sqrt{11} / 5}, ∣ r_{s} ∣ = 1, t_{s} = \frac{9 / 5}{9 / 10 + i \sqrt{11} / 5}, ∣ t_{s} ∣^{2} = \frac{324}{125};$ the reflection factor is the phase $r_{s} = e^{- 2 i φ}$ with $φ = \arctan ⁣ (2 \sqrt{11} / 9) \approx 0.63514703$ (about ${36.39124}^{\circ}$ ), i.e. the reflected wave acquires the phase $- 2 φ \approx - 1.27029406$ (about $- {72.78249}^{\circ}$ ); (ii) [Transmitted field] the transmitted field is $E_{t} (x, t) = e^{- β x_{3}} Re (E_{t} \hat{y} e^{i (∣ κ ∣ x_{1} - ω t)})$ with $β = (\sqrt{11} / 5) (ω / c)$ : it decays by the factor $e^{- \sqrt{11} / 5} \approx 0.51514$ when $x_{3}$ increases by $c / ω$ , its period-averaged normal flux vanishes identically, $⟨ S_{t} \cdot n ⟩ \equiv 0$ , while its tangential averaged flux is positive, $⟨ S_{t} \cdot \hat{κ} ⟩ = e^{- 2 β x_{3}} \frac{(6 / 5) (ω / c) ∣ E_{t} ∣^{2}}{2 μ ω} = e^{- 2 β x_{3}} \frac{3 ∣ E_{t} ∣^{2}}{5 μ c}$ ; (iii) [Flux budget] $∣ r_{s} ∣^{2} + (Re Y_{2}^{s} / Y_{1}^{s}) ∣ t_{s} ∣^{2} = 1 + 0 = 1$ , so the whole incident normal averaged flux is reflected, while the evanescent field carries a tangential averaged flux only; (iv) [Scope] the example is an exact deterministic computation of the lossless interface model for declared parameters; it reports no measurement, and the presence of a tangential evanescent flux is not a claim about information speed or signal propagation.

Verification

technique · direct substitution in the oblique-incidence and medium plane-wave theorems with exact rational and quadratic-irrational arithmetic, and an explicit phase computation via Euler's form

Given: The lossless two-media setting of Oblique-incidence TE and TM Fresnel amplitudes, the flux budget, Brewster's angle and total internal reflection with $μ_{1} = μ_{2} = μ > 0$ , $n_{1} = 3 / 2$ , $n_{2} = 1$ , the tangential wavevector $κ = (6 ω / (5 c)) e_{1}$ , the transverse-electric data $e_{j} = E_{j} \hat{y}$ with $\hat{y} = e_{2}$ , and the incidence data $\sin θ_{i} = 4 / 5$ .

[A1] The oblique-incidence theorem Oblique-incidence TE and TM Fresnel amplitudes, the flux budget, Brewster's angle and total internal reflection: clause (a) and (d) give $Y_{1}^{s} = k_{13} / (μ_{1} ω)$ , $Y_{2}^{s} = k_{t 3} / (μ_{2} ω)$ , the amplitude formulas, and in the evanescent branch $∣ r_{s} ∣ = 1$ , $∣ t_{s} ∣^{2} = 4 (Y_{1}^{s})^{2} / ((Y_{1}^{s})^{2} + ∣ Y_{2}^{s} ∣^{2}) > 0$ together with the flux budget; clause (d) also records that the transmitted field decays and its normal averaged flux vanishes.

[A2] The medium plane-wave structure Plane-wave fields in a homogeneous Ohmic medium: amplitude conditions, dispersion, averaged flux and the evanescent branch: the transmitted evanescent field of clause (e) with the substitution $n^{'} = n$ , $κ_{0} = κ$ : the exact factorization, the vanishing normal component $⟨ S \cdot n^{'} ⟩ \equiv 0$ and the positive tangential averaged flux $⟨ S \cdot {\hat{κ}}_{0} ⟩ = ∣ κ_{0} ∣ ∣ e ∣^{2} e^{- 2 β n^{'} \cdot x} / (2 μ ω)$ for the transverse-electric data $e = E_{0} (n^{'} \times {\hat{κ}}_{0})$ .

[F1] Medium parameters (Homogeneous isotropic medium parameters: phase speed, impedance, refractive index, complex propagation constant and skin depth): $K_{j} = ω (ε_{j} μ_{j})^{1 / 2} = ω n_{j} / c$ and $ε_{j} = n_{j}^{2} / (c^{2} μ)$ for the common permeability. The selected evanescent wavevectors in Oblique-incidence TE and TM Fresnel amplitudes, the flux budget, Brewster's angle and total internal reflection give $k_{13}^{2} = K_{1}^{2} - ∣ κ ∣^{2}$ , $β^{2} = ∣ κ ∣^{2} - K_{2}^{2}$ and $k_{t 3} = i β$ .

[F2] Complex arithmetic and modulus (Conjugation is an involutive real-field automorphism, $z \overline{z} = ∣ z ∣^{2}$ , and modulus is definite, multiplicative, and subadditive): $\overline{z + w} = \overline{z} + \overline{w}$ , $\overline{z w} = \overline{z} \overline{w}$ , $∣ z w ∣ = ∣ z ∣ ∣ w ∣$ , $∣ z ∣^{2} = z \overline{z}$ and $∣ z ∣ \geq 0$ with $∣ z ∣ = 0$ only for $z = 0$ ; in particular a complex number of modulus one is nonzero and the sum of a complex number and its conjugate is twice its real part.

[F3] Trigonometry and phases: $\arctan$ is the principal inverse tangent with $\tan (\arctan y) = y$ (The principal inverse tangent $\arctan : R \to (- π / 2, π / 2)$ ); $\tan x = \sin x / \cos x$ (Tangent, cotangent, secant, and cosecant on their exact natural domains); $\sin^{2} x + \cos^{2} x = 1$ and $1 + \tan^{2} x = \sec^{2} x$ (Pythagorean and parity identities for all six trigonometric functions on their natural domains); cosine is strictly decreasing on $[0, π]$ with $\cos (π / 2) = 0$ (Signs, monotonicity intervals, and ranges of sine and cosine, The zero sets of sine and cosine and the least positive common period 2 pi); and for real $y$ , $\exp (i y) = \cos y + i \sin y$ with $∣ \exp (i y) ∣ = 1$ ( $\exp (x + i y) = e^{x} (\cos y + i \sin y)$ , $∣ \exp (x + i y) ∣ = e^{x}$ , and $e^{i π} + 1 = 0$ ).

1.1givenA1F1algebra

[The geometry and the evanescent data] With $κ = (6 ω / (5 c)) e_{1}$ one has $\hat{κ} = e_{1}$ , $∣ κ ∣ = (6 / 5) (ω / c)$ and $\hat{y} = n \times \hat{κ} = e_{3} \times e_{1} = e_{2}$ ; $K_{1} = (3 / 2) (ω / c)$ , $K_{2} = ω / c$ by [F1], so $∣ κ ∣ = K_{1} \sin θ_{i}$ with $\sin θ_{i} = 4 / 5$ , and $∣ κ ∣ > K_{2}$ because $6 / 5 > 1$ . Hence the evanescent branch applies: $β = \sqrt{∣ κ ∣^{2} - K_{2}^{2}} = \sqrt{(36 / 25 - 1)} (ω / c) = \sqrt{11 / 25} (ω / c) = (\sqrt{11} / 5) (ω / c)$ and $k_{13} = \sqrt{K_{1}^{2} - ∣ κ ∣^{2}} = \sqrt{(9 / 4 - 36 / 25)} (ω / c) = \sqrt{81 / 100} (ω / c) = (9 / 10) (ω / c)$ ; the critical angle satisfies $\sin θ_{c} = n_{2} / n_{1} = 2 / 3 < 4 / 5 = \sin θ_{i}$ , both angles being in $(0, π / 2)$ .

2.1step 1.1A1F2algebra

[Admittances and reflection data, clause (i)] By A1 and [F1], $Y_{1}^{s} = k_{13} / (μ_{1} ω) = 9 / (10 μ c)$ , $Y_{2}^{s} = k_{t 3} / (μ_{2} ω) = i β / (μ ω) = i \sqrt{11} / (5 μ c)$ and $t_{s} = 2 Y_{1}^{s} / (Y_{1}^{s} + Y_{2}^{s}) = (9 / (5 μ c)) / ((9 / 10 + i \sqrt{11} / 5) / (μ c)) = (9 / 5) / (9 / 10 + i \sqrt{11} / 5)$ ; the numerator $9 / 10 - i \sqrt{11} / 5$ of $r_{s} = (Y_{1}^{s} - Y_{2}^{s}) / (Y_{1}^{s} + Y_{2}^{s})$ is the complex conjugate of its denominator, so $∣ r_{s} ∣ = 1$ by [F2] (and A1). For the transmitted modulus, $∣ t_{s} ∣^{2} = 4 (Y_{1}^{s})^{2} / ((Y_{1}^{s})^{2} + ∣ Y_{2}^{s} ∣^{2})$ by A1; computing $(Y_{1}^{s})^{2} = 81 / (100 μ^{2} c^{2})$ and $∣ Y_{2}^{s} ∣^{2} = 11 / (25 μ^{2} c^{2}) = 44 / (100 μ^{2} c^{2})$ gives $∣ t_{s} ∣^{2} = (324 / 100) / (125 / 100) = 324 / 125$ .

2.2step 1.1A1A2algebra

[The transmitted field and its fluxes, clause (ii)] By $∣ t_{s} ∣^{2}$ -normalised data we take the transmitted electric amplitude $E_{t} \hat{y}$ with $E_{t} = t_{s} E_{i}$ ; the evanescent structure of [A2] applied with $n^{'} = n = e_{3}$ , $κ_{0} = κ$ , $β$ as in step 1.1 and $e = E_{t} \hat{y} = E_{t} (n^{'} \times {\hat{κ}}_{0})$ gives exactly $E_{t} (x, t) = e^{- β x_{3}} Re (E_{t} \hat{y} e^{i (∣ κ ∣ x_{1} - ω t)})$ , the identity $⟨ S_{t} \cdot n ⟩ \equiv 0$ , and the tangential averaged flux $⟨ S_{t} \cdot \hat{κ} ⟩ = e^{- 2 β x_{3}} ∣ κ ∣ ∣ E_{t} ∣^{2} / (2 μ ω)$ ; substituting $∣ κ ∣ = (6 / 5) (ω / c)$ gives $⟨ S_{t} \cdot \hat{κ} ⟩ = e^{- 2 β x_{3}} (6 / 5) (ω / c) ∣ E_{t} ∣^{2} / (2 μ ω) = e^{- 2 β x_{3}} 3 ∣ E_{t} ∣^{2} / (5 μ c)$ . The decay over a distance $c / ω$ is the factor $e^{- β c / ω} = e^{- \sqrt{11} / 5}$ , whose decimal display in clause (ii) is the stated rounding.

3.1step 2.1F2F3algebra

[The reflection phase, clause (i)] Put $Y_{1} : = 9 / 10$ and $β^{'} : = \sqrt{11} / 5$ in the units $1 / (μ c)$ and $α : = β^{'} / Y_{1} = 2 \sqrt{11} / 9$ , so that $r_{s} = (1 - i α) / (1 + i α)$ ; let $φ : = \arctan α \in (0, π / 2)$ and $w : = 1 + i α$ , $∣ w ∣ = \sqrt{1 + α^{2}}$ . By [F3], $\tan φ = α$ and $1 + \tan^{2} φ = \sec^{2} φ$ , so $\cos^{2} φ = 1 / (1 + α^{2}) = 1 / ∣ w ∣^{2}$ ; since $\cos φ > 0$ (cosine is strictly decreasing on $[0, π]$ with $\cos (π / 2) = 0$ and $φ \in (0, π / 2)$ ), $\cos φ = 1 / ∣ w ∣$ and $\sin φ = α \cos φ = α / ∣ w ∣$ . Euler's form of [F3] then gives $w = ∣ w ∣ (\cos φ + i \sin φ) = ∣ w ∣ e^{i φ}$ and $\overline{w} = ∣ w ∣ e^{- i φ}$ by [F2], so $r_{s} = \overline{w} / w = e^{- 2 i φ}$ . The decimal values displayed in clause (i) are the stated roundings of $φ = \arctan (2 \sqrt{11} / 9)$ and of the reflected phase $- 2 φ$ .

4.1givenstep 2.1step 2.2A1F2∎

[Flux budget and scope, clauses (iii)-(iv)] By A1, $∣ r_{s} ∣^{2} + (Re Y_{2}^{s} / Y_{1}^{s}) ∣ t_{s} ∣^{2} = 1$ ; here $Re Y_{2}^{s} = 0$ by step 2.1, so the identity reads $1 + 0 = 1$ : the whole incident normal averaged flux is reflected, while the tangential averaged flux of step 2.2 is positive and transports no normal flux. All quantities are declared model parameters; the computation is deterministic classical interface algebra, reports no measurement, and neither the tangential evanescent flux nor the reflection phase is a claim about information speed, signal propagation or empirical data. No choice principle is used.

Wave interfaces and conductors — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

Numerical check of the good-conductor skin-depth bounds and the σ=0σ=0 exclusion

Example

Verification

The Brewster angle at an equal-permeability interface and the surviving ss-reflection

Example

Verification

Total internal reflection: decaying transmitted field, vanishing normal flux and the reflection phase

Example

Verification

Sources

Numerical check of the good-conductor skin-depth bounds and the $σ = 0$ exclusion

The Brewster angle at an equal-permeability interface and the surviving $s$ -reflection