# Fluctuations of Power versus Energy for Random Fields Near a Perfectly
Conducting Boundary

Luk R. Arnaut

###### Abstract

The standard deviations of the energy and Poynting power densities for an
isotropic random field near a perfectly conducting planar boundary are
characterized, based on quartic plane-wave expansions. For normal and
transverse components, different rates of decay exist as a function of
electrical distance from the boundary. At large distances, the envelopes for
the power are more strongly damped than for the energy, both showing inverse
power law decay. The decay for the standard deviation is generally one order
faster than for the corresponding mean. For the normally directed power flux,
its standard deviation near the boundary increases linearly with distance. The
relative uncertainty of the scalar power is much smaller than for the Poynting
power. Poynting’s theorem for standard deviations is obtained and demonstrates
larger standard deviations of the energy imbalance and power flux than their
mean values.

###### Index Terms:

Boundary zone fields, complex cavity, energy, measurement, noise, power,
standard deviation.

##  I Introduction

The characterization of the fluctuations of complex random fields, energy, and
power near an electromagnetic (EM) boundary is an aspect of fundamental
importance. Some recent studies include the evaluation of linear and orbital
angular momentum near a ground plane [1], the probing of fields near a metal
wall in a mode-stirred reverberation chamber (MSRC) [2], and the propagation
of uncertainty and correlation in computational EM field solvers [3].

When measuring deterministic fields, practitioners have a choice between
electric (or magnetic) field sensors of wire dipole (or loop) type versus
aperture antennas or waveguides that measure EM power flow. The choice is
largely governed by the sensitivity at the frequency of interest, the size and
practicality. For plane waves, the conversion from voltage, field intensity,
electric (or magnetic) energy to EM power is straightforward.

Random fields near an EM boundary prompt a consideration of the effects of the
boundary conditions on the root-mean-square (RMS) fluctuations (standard
deviation) of power vs. energy. Any such effects complement those that have
been found to exist for their expected values (averages) [1], which governs
the optimal sensor placement for eliminating the effect of the boundary. As
will be shown here, the choice of sensor type and measurand leads to
supplementary differences in the signal-to-noise ratio (SNR) between random
energy vs. random power that affect their relative uncertainties.

In [4], the issue of defining power in an ideal reverberant space was raised.
While the average local power is null across all angles of incidence ($4\pi$
sr), this does not preclude significant nonzero fluctuations of power that may
still produce physical effects and require characterization. Near a perfect
electric conducting (PEC) boundary, the average power [1] and energy [5]–[8]
of random fields are both nonzero and exhibit different spatial dependencies,
through decaying interference at-a-distance, unlike for deterministic fields.
One consequence is the recommendation to avoid using a boundary layer zone of
quarter-wavelength thickness near any metallic surface inside a MSRC [9].
However, this guard distance creates practical problems in testing heavy
floor-standing equipment or inside small cabinets. More generally, it is
wasteful of the available interior space, particularly at relatively low
frequencies. Some studies have started to address this issue [1, sec. III-C],
[10], [11, sec. 5.3.5], [12], with a view to increase the working volume of a
MSRC up to its physical boundaries. To this end, the full statistical
characterization of the EM energy and power in the boundary zone is required.

In this article, the work in [1], [5], [6], [10] on average energy and
Poynting power densities for random fields near a PEC boundary is extended to
cover corresponding standard deviations. A time-harmonic dependence $\exp({\rm
j}\omega t)$ is assumed and suppressed. To simplify notation, the argument of
all spherical Bessel functions will be dropped, i.e., $j_{\ell}\equiv
j_{\ell}(2kz)$ for $\ell=0,1,2$, which are hence always evaluated at $2kz$.

##  II Spectral Plane Wave Expansions for Quartic Mixed Products of Circular
Gaussian Fields

The angular spectral plane-wave expansion is defined by

$\displaystyle\underline{F}(\underline{r})$
$\displaystyle=\frac{1}{\Omega}\iint_{\Omega}\underline{{\cal
F}}(\Omega)\exp(-{\rm j}\underline{k}\cdot\underline{r}){\rm d}\Omega$ (1)

where $\underline{F}=\underline{E}$ or $\underline{H}$ and $\underline{{\cal
F}}=\underline{{\cal E}}$ or $\underline{{\cal H}}$, which can be traced to
Cramér’s spectral representation of scalar stationary random functions [5],
[13]. For the time-averaged local electric and magnetic energy densities,
$U_{{\rm
e}}(\underline{r})=\epsilon_{0}\underline{E}(\underline{r})\cdot\underline{E}^{*}(\underline{r})/4$
and $U_{{\rm
m}}(\underline{r})=\mu_{0}\underline{H}(\underline{r})\cdot\underline{H}^{*}(\underline{r})/4$,
and for the Poynting power vector
$\underline{S}(\underline{r})=[\underline{E}(\underline{r})\times\underline{H}^{*}(\underline{r})]/2$,
the expansions involve two-factor pure or mixed products of plane wave fields,
viz., [1]

$\displaystyle F_{\alpha}(\underline{r})G^{*}_{\beta}(\underline{r})$
$\displaystyle=\frac{1}{\Omega}\left(\iint_{\Omega}\right)^{2}{{\cal
F}}_{1,\alpha}{{\cal G}}^{*}_{2,\beta}\,\delta(\Omega_{12})$
$\displaystyle~{}\times\exp\left[-{\rm
j}(\underline{k}_{1}-\underline{k}^{*}_{2})\cdot\underline{r}\right]{\rm
d}\Omega_{1}{\rm d}\Omega_{2}$ (2)

for ${F},{G}\in\\{{E},{H}\\}$; ${{\cal F}},{{\cal G}}\in\\{{{\cal E}},{{\cal
H}}\\}$; $\alpha,\beta\in\\{x,y,z\\}$, and where ${\cal
F}_{i,\alpha}\stackrel{{\scriptstyle\Delta}}{{=}}\underline{{\cal
F}}(\Omega_{i})\cdot\underline{1}_{\alpha}$, etc., are Cartesian field
components along the direction of $\underline{1}_{\alpha}$ for the angular
spectral plane-wave components $(\underline{{\cal
E}}(\Omega_{i}),\underline{{\cal H}}(\Omega_{i}),\underline{k}_{i})$ of the
observable random EM field $(\underline{E},\underline{H})$ at location of
incidence $\underline{r}$. The wave vector
$\underline{k}_{i}\stackrel{{\scriptstyle\Delta}}{{=}}\underline{k}(\Omega_{i})$
is quasi-monochromatic
($|\underline{k}_{i}|=\omega_{i}\sqrt{\mu_{0}\epsilon_{0}}$). The analysis
assumes a lossless medium of incidence (${\cal E}_{i}/{\cal H}_{i}={\cal
E}_{0}/{\cal H}_{0}=\eta_{0}=\sqrt{\mu_{0}/\epsilon_{0}}$;
$\underline{k}_{i}=\underline{k}^{*}_{i}$). The symmetric set difference
$\Omega_{ij}\stackrel{{\scriptstyle\Delta}}{{=}}\Omega_{i}\Delta\Omega_{j}=(\Omega_{i}\cup\Omega_{j})\setminus(\Omega_{i}\cap\Omega_{j})$
for overlap between solid angles $\Omega_{i}$ and $\Omega_{j}$ satisfies
$\delta(\Omega_{ij}=\emptyset)=1$ and $\delta(\Omega_{ij}\not=\emptyset)=0$.
The differentials ${\rm d}\Omega_{i}=\sin\theta_{i}{\rm d}\theta_{i}{\rm
d}\phi_{i}$ refer to standard spherical angles of azimuth $\phi_{i}$ and
elevation $\theta_{i}$, measured from zenith, for integration across the upper
hemisphere $\Omega=2\pi$ sr ($z\geq 0$).

The second-order expansions (2) suffice for the calculation of ensemble
averages (expected values) of energy and power, denoted by
$\langle\cdot\rangle$. Using a TE/TM decomposition and enforcing the EM
boundary conditions at the PEC plane $oxy$ [1], [5]

$\displaystyle\langle{S}_{\alpha}(kz)\rangle$
$\displaystyle=\langle{S}_{t}(kz)\rangle=0,~{}~{}\alpha=x,y$ (3)
$\displaystyle\langle{S}_{z}(kz)\rangle$
$\displaystyle=\langle{S}(kz)\rangle=-{\rm j}\frac{\langle|{\cal
E}_{0}|^{2}\rangle}{\eta_{0}}j_{1}$ (4) $\displaystyle\langle
U_{\alpha}(kz)\rangle$ $\displaystyle=\frac{\varphi_{0}\langle|{\cal
F}_{0}|^{2}\rangle}{3}\left(1\mp
j_{0}\pm\frac{j_{2}}{2}\right),~{}~{}\alpha=x,y$ (5) $\displaystyle\langle
U_{z}(kz)\rangle$ $\displaystyle=\frac{\varphi_{0}\langle|{\cal
F}_{0}|^{2}\rangle}{3}\left(1\pm j_{0}\pm j_{2}\right)$ (6)
$\displaystyle\langle U_{t}(kz)\rangle$
$\displaystyle=\frac{2\,\varphi_{0}\langle|{\cal
F}_{0}|^{2}\rangle}{3}\left(1\mp j_{0}\pm\frac{j_{2}}{2}\right)$ (7)
$\displaystyle\langle U(kz)\rangle$ $\displaystyle=\varphi_{0}\langle|{\cal
F}_{0}|^{2}\rangle\left(1\mp\frac{j_{0}}{3}\pm\frac{2j_{2}}{3}\right)$ (8)

where upper and lower signs correspond to electric $(U=U_{{\rm e}})$ and
magnetic $(U=U_{{\rm m}})$ energy densities, respectively, and
$\varphi_{0}\langle|{\cal
F}_{0}|^{2}\rangle\stackrel{{\scriptstyle\Delta}}{{=}}\epsilon_{0}\langle|{\cal
E}_{0}|^{2}\rangle=\mu_{0}\langle|{\cal H}_{0}|^{2}\rangle$. The subscripts
$\alpha$, $z$, and $t$ represent 1-D tangential, 1-D normal, and 2-D
tangential ($xy$-) components, respectively. Absence of a subscript refers to
the 3-D full (vector) field $\underline{E}$ or $\underline{H}$. The total
(combined, summed) electric and magnetic energy will be denoted as $U_{{\rm
em}}$.

The expectations (3)–(8) presume a set of orthogonality conditions to hold for
the second-order moments of the local plane-wave complex fields
$\underline{{\cal F}}_{i}\equiv\underline{{\cal F}}^{\prime}_{i}-{\rm
j}\underline{{\cal F}}^{\prime\prime}_{i}={{\cal
F}}_{i,\phi}\underline{1}_{\phi_{i}}+{{\cal
F}}_{i,\theta}\underline{1}_{\theta_{i}}$ in the transverse plane
$o\phi_{i}\theta_{i}$, viz., [4], [13]

$\displaystyle C_{{\cal F}^{\prime(\prime)}_{i,\phi}{\cal
F}^{\prime(\prime)}_{j,\phi}}=C_{{\cal F}^{\prime(\prime)}_{i,\theta}{\cal
F}^{\prime(\prime)}_{j,\theta}}=(\langle|{{\cal
E}_{0}|^{2}\rangle}/{4})\,\delta(\Omega_{ij})$ (9)

for the in-phase/in-phase (I/I) and quadrature/quadrature (Q/Q) covariances
$C_{{\cal F}^{\prime(\prime)}_{i,\cdot}{\cal
F}^{\prime(\prime)}_{j,\cdot}}=\langle{\cal F}^{\prime(\prime)}_{i,\cdot}{\cal
F}^{\prime(\prime)}_{j,\cdot}\rangle$. The delta correlation in (9) is a
consequence of the assumed stationarity (i.e., statistical homogeneity and
isotropy) of incident $\underline{F}$ with respect to $\underline{r}$. For the
I/Q and cross-component covariances, the conditions are

$\displaystyle C_{{\cal F}^{\prime}_{i,\phi}{\cal
F}^{\prime\prime}_{j,\phi}}=C_{{\cal F}^{\prime}_{i,\theta}{\cal
F}^{\prime\prime}_{j,\theta}}=C_{{\cal F}^{\prime(\prime)}_{i,\phi}{\cal
F}^{\prime(\prime)}_{j,\theta}}=0.$ (10)

The expansion (2) can be extended to quartic forms, as required in the
calculation of second-order moments of $U_{{\rm e}}$, $U_{{\rm m}}$, $U_{{\rm
em}}$ and $S$ involving pure or pairwise mixed fourth-order moments of the
fields, in the general form

$\displaystyle\/F_{\alpha}(\underline{r})F^{*}_{\beta}(\underline{r})G_{\gamma}(\underline{r})G^{*}_{\delta}(\underline{r})=\frac{1}{\Omega}\left(\iint_{\Omega}\right)^{4}{\cal
F}_{1,\alpha}{\cal F}^{*}_{2,\beta}{\cal G}_{3,\gamma}{\cal G}^{*}_{4,\delta}$
$\displaystyle\/\times\exp\left[-{\rm
j}(\underline{k}_{1}-\underline{k}^{*}_{2}+\underline{k}_{3}-\underline{k}^{*}_{4})\cdot\underline{r}\right]\delta(\Omega_{1234}){\rm
d}\Omega_{1}{\rm d}\Omega_{2}{\rm d}\Omega_{3}{\rm d}\Omega_{4}$

for any selection of pairwise conjugate orthogonal field components along
$\alpha,\beta,\gamma,\delta\in\\{x,y,z\\}$, with
$\delta(\Omega_{1234})\equiv\delta(\Omega_{1}\Delta\Omega_{2}\Delta\Omega_{3}\Delta\Omega_{4})=\delta(\Omega_{12})\delta(\Omega_{34})$.
Now, if all $\underline{{\cal F}}_{i}$ and $\underline{{\cal G}}_{j}$ are
taken to be centered circular Gaussian fields, then $\langle{\cal
F}_{1,\alpha}{\cal F}^{*}_{2,\beta}{\cal G}_{3,\gamma}{\cal
G}^{*}_{4,\delta}\rangle$ can be expressed in terms of second-order moments
only, without imposing additional fourth-order moment conditions, by applying
Isserlis’s theorem for Gaussian moments to the individual real and imaginary
parts of ${\cal F}_{i,\alpha}$ and ${\cal G}_{j,\gamma}$. Expressing the
kernel of (LABEL:eq:FalphaFbetaGgammaGdelta) as a sum of products of pairwise
covariances for real variates, followed by a recombination to the sought
covariances of the complex fields, Isserlis’s theorem for the complex plane-
wave field components follows as

$\displaystyle\/\langle{\cal F}_{1,\alpha}{\cal F}^{*}_{2,\beta}{\cal
G}_{3,\gamma}{\cal G}^{*}_{4,\delta}\rangle=\langle{\cal F}_{1,\alpha}{\cal
F}^{*}_{2,\beta}\rangle\langle{\cal G}_{3,\gamma}{\cal
G}^{*}_{4,\delta}\rangle$ $\displaystyle\/+\langle{\cal F}_{1,\alpha}{\cal
G}_{3,\gamma}\rangle\langle{\cal F}^{*}_{2,\beta}{\cal
G}^{*}_{4,\delta}\rangle+\langle{\cal F}_{1,\alpha}{\cal
G}^{*}_{4,\delta}\rangle\langle{\cal F}^{*}_{2,\beta}{\cal
G}_{3,\gamma}\rangle.$ (12)

For the mixed fourth moment $\langle|{\cal F}_{i,\alpha}|^{2}|{\cal
G}_{j,\gamma}|^{2}\rangle$, its terms in (12) contain $\langle{\cal
F}_{i,\alpha}{\cal G}_{j,\gamma}\rangle$ and $\langle{\cal F}_{i,\alpha}{\cal
G}^{*}_{j,\gamma}\rangle$, which may vanish or not [1], as will become
apparent in the further analysis.

In summary, the conditions (9) and (10) on the second-order moments are
necessary and sufficient to fully characterize all even-order moments of
circular complex Gaussian ${\cal F}_{i,\alpha}$ and ${\cal G}_{j,\gamma}$ and
hence all (even and odd) moments of energy and power, following (12). If some
fields are elliptic and/or non-Gaussian, then additional conditions on their
pseudovariances and/or fourth-order moments apply, respectively.

Note that Isserlis’s theorem can equally be applied to $\langle
F_{1,\alpha}F^{*}_{2,\beta}G_{3,\gamma}G^{*}_{4,\delta}\rangle$, i.e., just as
well to the actual (i.e., physical) fields $F$ and $G$ as to their spectral
plane-wave source fields ${\cal F}$ and ${\cal G}$, because of linearity for
the superposition of Gaussian fields. For non-Gaussian ${\cal F}$ and ${\cal
G}$, however, (12) does not necessarily extend to moments of $F$ and/or $G$.

##  III Power

###  III-A 1-D Cartesian Tangential or Normal Power Flow

The complex Poynting vector $\underline{S}$ has Cartesian components
$S_{\alpha}\underline{1}_{\alpha}=[(E_{\beta}H^{*}_{\gamma}-E_{\gamma}H^{*}_{\beta})/2]\underline{1}_{\alpha}$
where $(\alpha,\beta,\gamma)$ is a cyclic permutation of $x$, $y$ and $z$.
Their second-order moments are

$\displaystyle\langle|S_{\alpha}|^{2}\rangle$
$\displaystyle=\frac{1}{4}\left(\langle|E_{\beta}|^{2}|H_{\gamma}|^{2}\rangle+\langle|E_{\gamma}|^{2}|H_{\beta}|^{2}\rangle\right.$
$\displaystyle~{}~{}~{}\left.-\langle
E_{\beta}H^{*}_{\gamma}E^{*}_{\gamma}H_{\beta}\rangle-\langle
E^{*}_{\beta}H_{\gamma}E_{\gamma}H^{*}_{\beta}\rangle\right)$ (13)

and are evaluated using (12) for ${\cal F}={\cal E}$ and ${\cal G}={\cal H}$,
as follows.

First, for a 1-D component of the power flowing parallel to the boundary,
i.e., tangential $S_{\alpha}$ with $\alpha=x$ or $y$, Isserlis’s theorem for
complex $E_{\beta}$, $E_{z}$, $H_{\beta}$ and $H_{z}$ leads to

$\displaystyle\langle E_{\beta}H^{*}_{z}E^{*}_{z}H_{\beta}\rangle=\langle
E^{*}_{\beta}H_{z}E_{z}H^{*}_{\beta}\rangle=0$ (14)

with $\beta=y$ or $x$, respectively ($\beta\not=\alpha$), because $\langle
E_{\beta}H^{*}_{z}\rangle=\langle E_{\beta}E^{*}_{z}\rangle=\langle
E_{\beta}H_{\beta}\rangle=0$, as follows from (2) [1]. The first two terms in
(13) with $\langle E_{\beta}H^{(*)}_{z}\rangle=\langle
E_{z}H^{(*)}_{\beta}\rangle=0$ are

$\displaystyle\left\\{\begin{array}[]{l}\langle|E_{\beta}(kz)|^{2}|H_{z}(kz)|^{2}\rangle\\\
\langle|E_{z}(kz)|^{2}|H_{\beta}(kz)|^{2}\rangle\end{array}\right\\}$
$\displaystyle=\frac{4\,\langle|{\cal
E}_{0}|^{2}\rangle^{2}}{9\,\eta^{2}_{0}}\left(1\mp
2j_{0}\mp\frac{j_{2}}{2}\right.$ (17)
$\displaystyle~{}~{}~{}\left.+j^{2}_{0}-\frac{j^{2}_{2}}{2}+\frac{j_{0}j_{2}}{2}\right)$
(18)

where upper and lower signs refer to
$\langle|E_{\beta}|^{2}|H_{z}|^{2}\rangle$ and
$\langle|E_{z}|^{2}|H_{\beta}|^{2}\rangle$. Substituting (14) and (18) into
(13) yields

$\displaystyle\langle|S_{\alpha}(kz)|^{2}\rangle$
$\displaystyle=\frac{2\,\langle|{\cal
E}_{0}|^{2}\rangle^{2}}{9\,\eta^{2}_{0}}\left(1+j^{2}_{0}-\frac{j^{2}_{2}}{2}+\frac{j_{0}j_{2}}{2}\right).$
(19)

Combining (19) with $\langle S_{\alpha}\rangle=0$ results in the standard
deviation $\sigma_{S_{\alpha}}=\sqrt{\langle|S_{\alpha}|^{2}\rangle-|\langle
S_{\alpha}\rangle|^{2}}$ as

$\displaystyle\sigma_{S_{\alpha}}(kz)=\sigma_{S_{\alpha,\infty}}\sqrt{1+j^{2}_{0}-\frac{j^{2}_{2}}{2}+\frac{j_{0}j_{2}}{2}},~{}~{}\alpha=x,y$
(20)

where its asymptotic value for $kz\rightarrow+\infty$ is

$\displaystyle\sigma_{S_{\alpha,\infty}}=\frac{\sqrt{2}\,\langle|{\cal
E}_{0}|^{2}\rangle}{3\,\eta_{0}}.$ (21)

Second, for the normally directed power flow ${S}_{z}$, we have

$\displaystyle\langle E_{x}H^{*}_{y}E^{*}_{y}H_{x}\rangle$
$\displaystyle=\langle
E_{x}^{*}H_{y}E_{y}H^{*}_{x}\rangle=-\frac{\langle|{\cal
E}_{0}|^{2}\rangle^{2}}{\eta^{2}_{0}}j^{2}_{1}$ (22)

on account of $\langle E_{x}H^{*}_{y}\rangle=-\langle
E^{*}_{y}H_{x}\rangle=-{\rm j}\langle|{\cal E}_{0}|^{2}\rangle j_{1}/\eta_{0}$
and $\langle E_{x}E^{*}_{y}\rangle=0=\langle E_{x}H_{x}\rangle$, the other
terms in (13) being

$\displaystyle\/\langle|E_{x}(kz)|^{2}|H_{y}(kz)|^{2}\rangle=\langle|E_{y}(kz)|^{2}|H_{x}(kz)|^{2}\rangle=$
$\displaystyle\/\frac{4\,\langle|{\cal
E}_{0}|^{2}\rangle^{2}}{9\,\eta^{2}_{0}}\left(1-j^{2}_{0}+\frac{9j^{2}_{1}}{4}-\frac{j^{2}_{2}}{4}+j_{0}j_{2}\right)+\frac{|\langle{\cal
E}^{2}_{0}\rangle|^{2}}{\eta^{2}_{0}}j^{2}_{1}.$ (23)

In order to arrive at (23), the pseudocovariances $\langle
E_{x}H_{y}\rangle=-\langle E_{y}H_{x}\rangle=-{\rm j}\langle{{\cal
E}^{2}_{0}}\rangle j_{1}/\eta_{0}$ were employed. However, $\langle{\cal
E}^{2}_{0}\rangle=0$ owing to the assumed complex circularity of the plane-
wave fields, whence the second term in (23) vanishes.111In the evaluation of
$\langle|E_{x}(kz)|^{2}|H_{y}(kz)|^{2}\rangle$, particularly for relatively
small data sets and in case of (residual) noncircularity, a small value of the
pseudovariance $\langle{\cal E}^{2}_{0}\rangle$ may still have a significant
effect when $\langle|E_{x}(kz)|^{2}|H_{y}(kz)|^{2}\rangle$ is also small,
particularly when $kz\not\ll 1$. Hence, substituting (22) and (23) into (13)
for $\alpha=z$ yields

$\displaystyle\langle|S_{z}(kz)|^{2}\rangle$
$\displaystyle=\frac{2\,\langle|{\cal
E}_{0}|^{2}\rangle^{2}}{9\,\eta^{2}_{0}}\left(1-j^{2}_{0}+\frac{9j^{2}_{1}}{2}-\frac{j^{2}_{2}}{4}+j_{0}j_{2}\right)$
(24)

which, combined with the mean (4), results in

$\displaystyle\sigma_{S_{z}}(kz)=\sigma_{S_{z,\infty}}\sqrt{1-j^{2}_{0}-\frac{j^{2}_{2}}{4}+{j_{0}j_{2}}}$
(25)

where

$\displaystyle\sigma_{S_{z,\infty}}=\frac{\sqrt{2}\,\langle|{\cal
E}_{0}|^{2}\rangle}{3\,\eta_{0}}.$ (26)

In Fig. 1(a), the standard deviations (20) and (25) are compared with
numerical results obtained via Monte Carlo (MC) simulation of random plane
waves, based on 30 random values of $|{\cal E}_{0}|$, 32 elevation angles, 16
azimuthal angles, and $16$ polarization angles, evaluated at 362
logarithmically spaced distances $kz$ ranging from 0.05 to 50. The value of
$\sigma_{S_{x}}$ reaches its maximum on the PEC surface, where
$\sigma_{S_{z}}$ vanishes. Local maxima and minima of $\sigma_{S_{x}}(kz)$ are
found at distances that are marginally shorter than those for
$\sigma_{S_{z}}(kz)$. The oscillations of $\sigma_{S_{z}}(kz)$ are more
strongly damped than those for $\langle S_{z}(kz)\rangle$ [1, Fig. 1], as will
be confirmed in sec. V by comparing their envelopes. For
$kz\rightarrow+\infty$, it is verified that (20) and (25) merge, as expected
for ideal isotropic fields in the absence of an EM boundary.

---
(a)
(b)
Figure 1:  Standard deviations, in units W/m2, based on $\langle{{\cal
E}^{\prime}_{0}}^{2}\rangle^{1/2}=\langle{{\cal
E}^{\prime\prime}_{0}}^{2}\rangle^{1/2}=0.01$ V/m, for (a) Poynting power
$\sigma_{S_{x}}(kz)$, $\sigma_{S_{z}}(kz)$, $\sigma_{S_{t}}(kz)$, and
$\sigma_{S}(kz)$ with upper and lower asymptotic envelopes
${\mit\Upsilon}^{\pm}_{S_{(\alpha)}}(kz\gg 3/2)$. Solid: theory; dashed: MC
simulation; dotted: envelopes. Olive/green: $S_{x}$; maroon/red: $S_{z}$;
blue/cyan: $S_{t}$; black/gray: $S$; and for (b) scalar power $\sigma_{{\rm
c}U_{{\rm em},x}}(kz)$, $\sigma_{{\rm c}U_{{\rm em},z}}(kz)$, $\sigma_{{\rm
c}U_{{\rm em},t}}(kz)$, and $\sigma_{{\rm c}U_{{\rm em}}}(kz)$ with upper
asymptotic envelopes ${\mit\Upsilon}^{+}_{{\rm c}U_{{\rm em},\alpha}}(kz\gg
3/2)$, shown using the same color scheme as in (a).

###  III-B 2-D Tangential Power Flow

For the 2-D tangential Poynting vector
$\underline{S}_{t}=S_{x}\underline{1}_{x}+S_{y}\underline{1}_{y}$, it follows
that
$|S_{t}|^{2}\equiv\underline{S}^{\dagger}_{t}\cdot\underline{S}_{t}=|S_{x}|^{2}+|S_{y}|^{2}$.
With (12) and (3), this yields $\sigma_{S_{t}}=\sqrt{2}\,\sigma_{S_{x}}$ as

$\displaystyle\sigma_{S_{t}}(kz)=\sigma_{S_{t,\infty}}\sqrt{1+j^{2}_{0}-\frac{j^{2}_{2}}{2}+\frac{j_{0}j_{2}}{2}}$
(27)

where

$\displaystyle\sigma_{S_{t,\infty}}=\frac{2\,\langle|{\cal
E}_{0}|^{2}\rangle}{3\,\eta_{0}}.$ (28)

###  III-C 3-D Spatial Power Flow

For the full Poynting vector
$\underline{S}=\sum_{\alpha=x,y,z}S_{\alpha}\underline{1}_{\alpha}$, we have
$|S|^{2}\equiv\underline{S}^{\dagger}\cdot\underline{S}=|S_{t}|^{2}+|S_{z}|^{2}$.
With (12), (19) and (24),

$\displaystyle\langle|S(kz)|^{2}\rangle$ $\displaystyle=\frac{2\,\langle|{\cal
E}_{0}|^{2}\rangle^{2}}{3\,\eta^{2}_{0}}\left(1+\frac{j^{2}_{0}}{3}+\frac{3j^{2}_{1}}{2}-\frac{5j^{2}_{2}}{12}+\frac{2j_{0}j_{2}}{3}\right)$

which, combined with (4), yields

$\displaystyle\sigma_{S}(kz)=\sigma_{S_{\infty}}\sqrt{1+\frac{j^{2}_{0}}{3}-\frac{5j^{2}_{2}}{12}+\frac{2j_{0}j_{2}}{3}}$
(30)

where

$\displaystyle\sigma_{S_{\infty}}=\sqrt{\frac{2}{3}}\,\frac{\langle|{\cal
E}_{0}|^{2}\rangle}{\eta_{0}}.$ (31)

##  IV Energy

The standard deviations of $U_{{\rm e}(,\alpha)}$ and $U_{{\rm m}(,\alpha)}$
for circular Gaussian fields near an impedance boundary were derived in [8],
[10], enabling all moments to be obtained merely by a single variate
transformation.

For $|F_{\alpha}|^{2}$, setting ${\cal F}_{\alpha}={\cal F}_{\beta}={\cal
G}_{\gamma}={\cal G}_{\delta}$ in (12) and using $\langle{\cal
F}^{2}_{\alpha}\rangle=0$ owing to the complex circularity ($\langle{\cal
E}^{2}_{0}\rangle=0$), this yields $\langle|{\cal
F}_{\alpha}|^{4}\rangle=2\langle|{\cal
F}_{\alpha}|^{2}\rangle^{2}+|\langle{\cal
F}^{2}_{\alpha}\rangle|^{2}=2\langle|{\cal F}_{\alpha}|^{2}\rangle^{2}$,
leading to the familiar result
$\sigma_{|F_{\alpha}|^{2}}=\langle|F_{\alpha}|^{2}\rangle$ for $\alpha=x,y,z$
[10]. For the energies, this results in explicit functions of $kz$ as

$\displaystyle\sigma_{U_{\alpha}}(kz)$ $\displaystyle=\left(1\mp
j_{0}\pm\frac{j_{2}}{2}\right)\sigma_{U_{\alpha,\infty}},~{}~{}\alpha=x,y$
(32) $\displaystyle\sigma_{U_{z}}(kz)$ $\displaystyle=\left(1\pm j_{0}\pm
j_{2}\right)\sigma_{U_{\alpha,\infty}}$ (33)

in which upper and lower signs refer to electric ($U=U_{{\rm e}}$, ${\cal
F}_{0}={\cal E}_{0}$, $\varphi_{0}=\epsilon_{0}$) and magnetic ($U=U_{{\rm
m}}$, ${\cal F}_{0}={\cal H}_{0}={\cal E}_{0}/\eta_{0}$,
$\varphi_{0}=\mu_{0}$) energy densities, respectively, and with

$\displaystyle\sigma_{U_{\alpha,\infty}}=\langle
U_{\alpha,\infty}\rangle=\frac{{\varphi_{0}\langle|{\cal
F}_{0}|^{2}\rangle}}{3},~{}~{}\alpha=x,y,z.$ (34)

For the energy density of the 2-D tangential electric or magnetic field, it
follows from (32) and $U_{t}=U_{x}+U_{y}$ that

$\displaystyle\sigma_{U_{t}}(kz)$ $\displaystyle=\left(1\mp
j_{0}\pm\frac{j_{2}}{2}\right)\sigma_{U_{t,\infty}}$ (35)

where

$\displaystyle\sigma_{U_{t,\infty}}=\frac{\sqrt{2}\,\varphi_{0}\langle|{\cal
F}_{0}|^{2}\rangle}{3}.$ (36)

For the full (vectorial) electric or magnetic energy density, the standard
deviation follows from (32) and (33) as

$\displaystyle\sigma_{U}(kz)$
$\displaystyle=\sigma_{U_{\infty}}\sqrt{1\mp\frac{2j_{0}}{3}\pm\frac{4j_{2}}{3}+j^{2}_{0}+\frac{j^{2}_{2}}{2}}$
(37)

where upper and lower signs again refer to electric and magnetic densities,
respectively, and with

$\displaystyle\sigma_{U_{\infty}}=\frac{{\varphi_{0}\langle|{\cal
F}_{0}|^{2}\rangle}}{\sqrt{3}}.$ (38)

These standard deviations are plotted in Fig. 2, comparing favourably with
results in Figs. 6(b) and 7(b) of [10].222Note that in [10], energy densities
were denoted by the symbol $S$.

For the combined EM energy density $U_{{\rm em}(,\alpha)}\equiv U_{{\rm
e}(,\alpha)}+U_{{\rm m}(,\alpha)}$, the variances of its Cartesian
tangential/normal, planar tangential, and full vectorial densities are found
as

$\displaystyle\sigma^{2}_{U_{{\rm em},\alpha}}$
$\displaystyle=\sigma^{2}_{U_{{\rm e},\alpha}}+\sigma^{2}_{U_{{\rm
m},\alpha}},~{}~{}\alpha=x,y,z$ (39) $\displaystyle\sigma^{2}_{U_{{\rm
em}(,t)}}$ $\displaystyle=\sigma^{2}_{U_{{\rm e}(,t)}}+\sigma^{2}_{U_{{\rm
m}(,t)}}+\left|\langle E_{x}H^{*}_{y}\rangle\right|^{2}/(2{\rm c})^{2}.$ (40)

Explicit expressions for their standard deviations follow as

$\displaystyle\sigma_{U_{{\rm em},\alpha}}(kz)$ $\displaystyle=\sigma_{U_{{\rm
em},\alpha,\infty}}\sqrt{1+j^{2}_{0}+\frac{j^{2}_{2}}{4}-j_{0}j_{2}},~{}~{}\alpha=x,y$
(41) $\displaystyle\sigma_{U_{{\rm em},z}}(kz)$ $\displaystyle=\sigma_{U_{{\rm
em},z,\infty}}\sqrt{1+j^{2}_{0}+j^{2}_{2}+2j_{0}j_{2}}$ (42)
$\displaystyle\sigma_{U_{{\rm em},t}}(kz)$ $\displaystyle=\sigma_{U_{{\rm
em},t,\infty}}\sqrt{1+j^{2}_{0}+\frac{9j^{2}_{1}}{4}+\frac{j^{2}_{2}}{4}-j_{0}j_{2}}$
(43) $\displaystyle\sigma_{U_{{\rm em}}}(kz)$ $\displaystyle=\sigma_{U_{{\rm
em},\infty}}\sqrt{1+j^{2}_{0}+\frac{3j^{2}_{1}}{2}+\frac{j^{2}_{2}}{2}}$ (44)

where $\sigma_{U_{{\rm
em},(\alpha,)(t,)\infty}}=\sqrt{2}\,\sigma_{U_{(\alpha,)(t,)\infty}}$, i.e.,

$\displaystyle\sigma_{U_{{\rm em},\alpha,\infty}}$
$\displaystyle=\frac{\sqrt{2}\,\varphi_{0}\langle|{\cal
F}_{0}|^{2}\rangle}{3},~{}~{}\alpha=x,y,z$ (45) $\displaystyle\sigma_{U_{{\rm
em},t,\infty}}$ $\displaystyle=\frac{2\,\varphi_{0}\langle|{\cal
F}_{0}|^{2}\rangle}{3}$ (46) $\displaystyle\sigma_{U_{{\rm em},\infty}}$
$\displaystyle=\sqrt{\frac{2}{3}}\,\varphi_{0}\langle|{\cal
F}_{0}|^{2}\rangle.$ (47)

In (32), (33), (35) and (37), the terms linear in $j_{0}$ and $j_{2}$ cancel
in their contributions to (41)–(44), leaving only quadratic terms. As shown in
sec. V, this compensation between $U_{{\rm e}(,\alpha)}$ and $U_{{\rm
m}(,\alpha)}$ produces faster (higher-order) power-law decays with $kz$ in the
envelopes of their sums $U_{{\rm em}(,\alpha)}$.

In summary, a PEC boundary induces different RMS levels of fluctuation for the
Poynting vs. scalar power, i.e.,

$\displaystyle\sigma_{S_{(\alpha)}}(kz)\not=\sigma_{{\rm c}U_{{\rm
em}(,\alpha)}}(kz),~{}~{}\alpha=x,y,z,t$ (48)

complementing the result ${\rm j}\langle
S_{(\alpha)}(kz)\rangle\not=\langle{\rm c}U_{{\rm em}(,\alpha)}(kz)\rangle$
[1]. Fig. 1(b) shows (41)–(47), after scaling by ${\rm c}$.

---
(a)
(b)
Figure 2:  Standard deviations, in units J/m3, based on $\langle{{\cal
E}^{\prime}_{0}}^{2}\rangle^{1/2}=\langle{{\cal
E}^{\prime\prime}_{0}}^{2}\rangle^{1/2}=0.01$ V/m, for (a) $U_{{\rm e},x}$,
$U_{{\rm e},z}$, $U_{{\rm e},t}$, $U_{{\rm e}}$, and (b) $U_{{\rm m},x}$,
$U_{{\rm m},z}$, $U_{{\rm m},t}$, $U_{{\rm m}}$. Solid: theory; dashed: MC
simulation; dotted: asymptotic upper and lower envelopes. Olive/green:
$U_{({\rm e},)({\rm m},)x}$; maroon/red: $U_{({\rm e},)({\rm m},)z}$;
blue/cyan: $U_{({\rm e},)({\rm m},)t}$; black/gray: $U_{({\rm e})({\rm m})}$.

In applications to immunity, susceptibility or fading testing, the PEC
boundary thus induces a $kz$-dependent scaling of the probability density
functions (PDFs). For example, for the Cartesian field strength $|E_{\alpha}|$
with variance
$\sigma^{2}_{|E_{\alpha}|}(kz)=[(4-\pi)/\epsilon_{0}]\sigma_{U_{{\rm
e},\alpha}}(kz)$ for $\alpha=x,y,z$, the scaling of its PDF

$\displaystyle
f_{|E_{\alpha}|}(|e_{\alpha}|,kz)=\frac{\epsilon_{0}\,|e_{\alpha}|}{2\sigma_{U_{{\rm
e},\alpha}}(kz)}\exp\left[-\frac{\epsilon_{0}\,|e_{\alpha}|^{2}}{4\sigma_{U_{{\rm
e},\alpha}}(kz)}\right]$ (49)

is damped oscillatory, as governed by (32) or (33). In turn, this scaling
affects associated statistics and distributions, e.g., $F_{|E_{\alpha}|_{\rm
max}}(|e_{\alpha}|_{\rm max},kz)=[F_{|E_{\alpha}|}(|e_{\alpha}|_{\rm
max},kz)]^{N(kz)}$ for the maximum field strength $|E_{\alpha}|_{\rm max}$.
Fig. 3 demonstrates this scaling and bifurcation of (49) for actual (i.e.,
dimensioned, non-normalized) $|E_{x}|$ vs. $|E_{z}|$, at selected values of
$kz$.

---
Figure 3:  Rayleigh PDFs (49) of $|E_{x}|$ (green) and $|E_{z}|$ (red) for
$\langle{{\cal E}^{\prime}_{0}}^{2}\rangle^{1/2}=\langle{{\cal
E}^{\prime\prime}_{0}}^{2}\rangle^{1/2}=20$ V/m, exhibiting $kz$-dependent
scaling.

##  V Asymptotic Envelopes and Range of $\sigma$

In [1, sec. III-C], asymptotic approximations by envelopes for the mean power
and energy densities at $kz\gg 3/2$ were derived, resulting in simpler inverse
power law expressions for their damped oscillations. Such nonlocal
approximations are also useful when phase shifts arise that are caused by
perturbations, e.g., apertures, scatterers, other nearby walls, boundary
roughness, antenna-surface coupling, etc. Here, results for these envelopes of
the mean are extended to corresponding envelopes for the standard deviation.

###  V-A Linear Forms for Electric or Magnetic Energy Fluctuations

In (32), (33) and (35), the standard deviations have the form

$\displaystyle\sigma_{U_{\alpha}}(kz)=(1+a_{0}j_{0}+a_{2}j_{2})\sigma_{U_{\alpha,\infty}}.$
(50)

Their associated asymptotic envelopes ${\mit\Upsilon}^{\pm}_{U_{\alpha}}(kz\gg
3/2)$ can be obtained as the modulus of the corresponding analytic signal,
$|\tilde{\sigma}_{U_{\alpha}}(kz\gg 3/2)|$. To this end,
$\tilde{j}_{\ell}\stackrel{{\scriptstyle\Delta}}{{=}}j_{\ell}+{\rm j}{{\rm
H}}[j_{\ell}]$ for the individual terms in (50) are sought for $kz\gg 3/2$,
where ${\rm H}[\cdot]$ denotes the Hilbert transform. Recall the asymptotic
approximations

$\displaystyle j_{\ell}$
$\displaystyle\simeq\frac{\sin(2kz-\ell\pi/2)}{2kz}~{}~{}{\rm
for}~{}kz\gg\frac{\ell(\ell+1)}{4}$ (51)

which, for even $\ell\equiv 2m$, result in

$\displaystyle\,{\rm
H}[j_{2m}(2kz\gg(2m+1)m)]\simeq(-1)^{m}\,\frac{1-\cos(2kz)}{2kz}.$ (52)

On the other hand, $z^{-1}\sin(2kz-\ell\pi/2)$ in (51) represents an amplitude
modulated signal when $z\gg(2k)^{-1}$, for an arbitrary wavenumber $k>0$.
Therefore, on application of Bedrosian’s theorem [14], it follows that

$\displaystyle\tilde{j}_{\ell}=j_{\ell}+{\rm j}{{\rm
H}}[j_{\ell}]\simeq\frac{\exp[{\rm j}(2kz-(\ell+1)\pi/2)]}{2kz}$ (53)

whence $|\tilde{j}_{\ell}(2kz\gg 1)|\simeq(2kz)^{-1}$. Combining with ${\rm
H}[1]=0$ and applying the linearity of the Hilbert transform, upper (‘$+$’)
and lower (‘$-$’) envelopes of $\sigma_{U_{\alpha}}(kz\gg 3/2)$ follow as

$\displaystyle{\mit\Upsilon}^{\pm}_{U_{\alpha}}(kz\gg
3/2)\simeq\left\\{\begin{array}[]{ll}\left[1\pm\frac{|a_{0}-a_{2}|}{2(kz)}\right]\sigma_{U_{\alpha,\infty}},&a_{0}-a_{2}\not=0,\\\
\\\
\left[1\pm\frac{3|a_{2}|}{4(kz)^{2}}\right]\sigma_{U_{\alpha,\infty}},&a_{0}-a_{2}=0.\end{array}\right.$
(57)

Thus, the envelopes decay typically333 If $a_{0}-a_{2}=0$, e.g., for $U_{z}$,
the envelopes are found by first replacing $j_{2}$ by its exact expression
$[3/(2kz)]j_{1}-j_{0}$ instead of the approximation $-j_{0}$.  according to a
first-order power law, $(kz)^{-1}$. Application of (57) to (32), (33) and (35)
yields the asymptotic envelopes of $\sigma_{U_{(\alpha)(t)}}(kz\gg 3/2)$ as

$\displaystyle{\mit\Upsilon}^{\pm}_{U_{\alpha}}(kz\gg 3/2)$
$\displaystyle\simeq\left[1\pm\frac{3}{4kz}\right]\sigma_{U_{\alpha,\infty}},~{}~{}\alpha=x,y$
(58) $\displaystyle{\mit\Upsilon}^{\pm}_{U_{z}}(kz\gg 3/2)$
$\displaystyle\simeq\left[1\pm\frac{3}{4(kz)^{2}}\right]\sigma_{U_{z,\infty}}$
(59) $\displaystyle{\mit\Upsilon}^{\pm}_{U_{t}}(kz\gg 3/2)$
$\displaystyle\simeq\left[1\pm\frac{3}{4kz}\right]\sigma_{U_{t,\infty}}.$ (60)

By contrast, close to the PEC boundary, using

$\displaystyle
j_{\ell}\simeq\frac{(2kz)^{\ell}}{(2\ell+1)!!}\left[1-\frac{(2kz)^{2}}{2(2\ell+3)}\right]~{}~{}{\rm
for}~{}kz\ll\sqrt{2\ell+5}$ (61)

all $\sigma_{U_{(\alpha)(t)}}(kz)$ are quadratic in $kz$ to leading order,
viz.,

$\displaystyle\sigma_{U_{{\rm e}(,\alpha)(,t)}}(kz\ll\sqrt{5})$
$\displaystyle\simeq\frac{4(kz)^{2}}{5}\sigma_{U_{{\rm
e}(,\alpha)(,t),\infty}},~{}~{}\alpha=x,y$ (62) $\displaystyle\sigma_{U_{{\rm
e},z}}(kz\ll\sqrt{5})$ $\displaystyle\simeq
2\left[1-\frac{(kz)^{2}}{5}\right]\sigma_{U_{{\rm e},z,\infty}}$ (63)
$\displaystyle\sigma_{U_{{\rm m}(,\alpha)(,t)}}(kz\ll\sqrt{5})$
$\displaystyle\simeq 2\left[1-\frac{2(kz)^{2}}{15}\right]\sigma_{U_{{\rm
m}(,\alpha)(,t),\infty}}$ (64) $\displaystyle\sigma_{U_{{\rm
m},z}}(kz\ll\sqrt{5})$ $\displaystyle\simeq\frac{2(kz)^{2}}{5}\sigma_{U_{{\rm
m},z,\infty}}.$ (65)

###  V-B Quadratic Forms for EM Energy or Power Fluctuations

In (20), (25), (27) and (30), the variances have the form

$\displaystyle\sigma^{2}_{S_{(\alpha)}}(kz)=(1+a_{00}j^{2}_{0}+a_{22}j^{2}_{2}+a_{02}j_{0}j_{2})\sigma^{2}_{S_{(\alpha,)\infty}}.$
(66)

For their asymptotic envelopes, a derivation analogous to that in sec. V-A can
be performed. With

$\displaystyle j^{2}_{2m}\simeq\frac{2\left[1-\cos(4kz)\right]}{(4kz)^{2}}$
$\displaystyle,\,{\rm
H}[j^{2}_{2m}]\simeq\frac{2\left[4kz-\sin(4kz)\right]}{(4kz)^{2}}$
$\displaystyle j_{2m}j_{2m+2}\simeq\frac{2\sin(4kz)}{(4kz)^{2}}$
$\displaystyle,\,{\rm H}[j_{2m}j_{2m+2}]\simeq-{\rm H}[j^{2}_{2m}]$ (67)

for $kz\gg 3/2$, envelopes for $\sigma^{2}_{S_{(\alpha)}}(kz)$ can then be
converted to envelopes ${\mit\Upsilon}^{\pm}_{S_{(\alpha)}}(kz)$ for
$\sigma_{S_{(\alpha)}}(kz)$, on expanding $\sigma_{S_{(\alpha)}}(kz)$ in (66)
using $\sqrt{1+x}\simeq 1+{x}/{2}$ for $x\ll 1$. This results in444A similar
remark as in sec. V-A for $U_{z}$ holds if $a_{00}+a_{22}-a_{02}=0$.

$\displaystyle{\mit\Upsilon}^{\pm}_{S_{(\alpha)}}(kz\gg
3/2)\simeq\sigma_{S_{(\alpha,)\infty}}$
$\displaystyle\,\times\left\\{\begin{array}[]{ll}\left[1\pm\frac{|a_{00}+a_{22}-a_{02}|}{8(kz)^{2}}\right],&a_{00}+a_{22}-a_{02}\not=0,\\\
\\\
\left[1\pm\frac{3|2a_{22}-a_{02}|}{32(kz)^{3}}\right],&a_{00}+a_{22}-a_{02}=0.\end{array}\right.$
(71)

The fluctuations of the tangential power are decaying at a faster (cubic) rate
than the quadratic rate of the normal and full vector Poynting powers, viz.,

$\displaystyle{\mit\Upsilon}^{\pm}_{S_{(\alpha)(t)}}(kz\gg
3/2)\simeq\left[1\pm\frac{9}{64(kz)^{3}}\right]\sigma_{S_{(\alpha,)(t,)\infty}},~{}~{}\alpha=x,y$
(72)
$\displaystyle{\mit\Upsilon}^{+}_{S_{z}}=\sigma_{S_{z,\infty}},~{}{\mit\Upsilon}^{-}_{S_{z}}(kz\gg
3/2)\simeq\left[1-\frac{9}{32(kz)^{2}}\right]\sigma_{S_{z,\infty}}$ (73)
$\displaystyle{\mit\Upsilon}^{\pm}_{S}(kz\gg
3/2)\simeq\left[1\pm\frac{3}{32(kz)^{2}}\right]\sigma_{S_{\infty}}.$ (74)

Unlike ${\mit\Upsilon}^{\pm}_{U_{(\alpha)}}(kz)$, which exhibit the same
$kz$-dependence as the envelopes $\Xi^{\pm}_{U_{(\alpha)}}(kz)$ for $\langle
U_{(\alpha)}(kz)\rangle$ [1, eqs. (34)–(36)], the
${\mit\Upsilon}^{\pm}_{S_{z}}(kz)$ decay more rapidly than
${\mit\Xi}^{\pm}_{S_{z}}(kz)$ [1, eq. (33)]. Near the boundary,
$\sigma_{S_{z}}(kz)$ exhibits a linear dependence, all other Poynting powers
being quadratic in $kz$, i.e.,

$\displaystyle\sigma_{S_{(\alpha)(t)}}(kz\ll\sqrt{5})$
$\displaystyle\simeq\sqrt{2}\left[1-\frac{3(kz)^{2}}{10}\right]\sigma_{S_{(\alpha,)(t,)\infty}},~{}\alpha=x,y$
(75) $\displaystyle\sigma_{S_{z}}(kz\ll\sqrt{5})$
$\displaystyle\simeq\sqrt{\frac{8}{5}}\,kz\,\sigma_{S_{z,\infty}}$ (76)
$\displaystyle\sigma_{S}(kz\ll\sqrt{5})$
$\displaystyle\simeq\frac{2}{\sqrt{3}}\left[1-\frac{(kz)^{2}}{10}\right]\sigma_{S_{\infty}}.$
(77)

For the total EM energy densities (41)–(44), the upper envelopes
${\mit\Upsilon}^{+}_{U_{{\rm em}(,\alpha)(,t)}}(kz)$ follow in a similar way
as

$\displaystyle{\mit\Upsilon}^{+}_{U_{{\rm em},\alpha}}(kz\gg 3/2)$
$\displaystyle\simeq\left[1+\frac{9}{32(kz)^{2}}\right]\sigma_{U_{{\rm
em},\alpha,\infty}},~{}\alpha=x,y$ (78)
$\displaystyle{\mit\Upsilon}^{+}_{U_{{\rm em},z}}(kz\gg 3/2)$
$\displaystyle\simeq\left[1+\frac{9}{32(kz)^{4}}\right]\sigma_{U_{{\rm
em},z,\infty}}$ (79) $\displaystyle{\mit\Upsilon}^{+}_{U_{{\rm em},t}}(kz\gg
3/2)$ $\displaystyle\simeq\left[1+\frac{9}{32(kz)^{2}}\right]\sigma_{U_{{\rm
em},t,\infty}}$ (80) $\displaystyle{\mit\Upsilon}^{+}_{U_{{\rm em}}}(kz\gg
3/2)$ $\displaystyle\simeq\left[1+\frac{3}{16(kz)^{2}}\right]\sigma_{U_{{\rm
em},\infty}}.$ (81)

In particular, ${\mit\Upsilon}^{+}_{S}(kz\gg 3/2)$ and
${\mit\Upsilon}^{+}_{{\rm c}U_{{\rm em}}}(kz\gg 3/2)$ decay at identical
quadratic rates and magnitudes. The ${\mit\Upsilon}^{-}_{U_{{\rm
em},\alpha}}(kz\gg 3/2)$ coincide with $\sigma_{U_{{\rm em},\alpha,\infty}}$,
whereas ${\mit\Upsilon}^{-}_{{U_{{\rm em},t}}}(kz\gg 3/2)$ and
${\mit\Upsilon}^{-}_{{U_{{\rm em}}}}(kz\gg 3/2)$ exhibit no significant
oscillations, whence

$\displaystyle{\mit\Upsilon}^{-}_{U_{{\rm em},\alpha}}(kz\gg 3/2)$
$\displaystyle\simeq\sigma_{U_{{\rm em},\alpha,\infty}},~{}~{}\alpha=x,y,z$
(82) $\displaystyle{\mit\Upsilon}^{-}_{U_{{\rm em}(,t)}}(kz\gg 3/2)$
$\displaystyle\simeq{\mit\Upsilon}^{+}_{U_{{\rm em}(,t)}}(kz\gg 3/2).$ (83)

Near the boundary, $\sigma_{{\rm c}U_{{\rm
em}{(,x)(,t)}}}(kz)\simeq\sigma_{S_{(x)(t)}}(kz)$ while $\sigma_{{\rm
c}U_{{\rm em}{(,z)}}}(kz)>\sigma_{S_{(z)}}(kz)$, as seen in Fig. 1.
Explicitly,

$\displaystyle\sigma_{U_{{\rm em},\alpha}}(kz\ll\sqrt{5})$
$\displaystyle\simeq\sqrt{2}\left[1-\frac{2(kz)^{2}}{5}\right]\sigma_{U_{{\rm
em},\alpha,\infty}},~{}\alpha=x,y$ (84) $\displaystyle\sigma_{U_{{\rm
em},z}}(kz\ll\sqrt{5})$
$\displaystyle\simeq\sqrt{2}\left[1-\frac{(kz)^{2}}{5}\right]\sigma_{U_{{\rm
em},z,\infty}}$ (85) $\displaystyle\sigma_{U_{{\rm em},t}}(kz\ll\sqrt{5})$
$\displaystyle\simeq\sqrt{2}\left[1-\frac{3(kz)^{2}}{10}\right]\sigma_{U_{{\rm
em},t,\infty}}$ (86) $\displaystyle\sigma_{U_{\rm em}}(kz\ll\sqrt{5})$
$\displaystyle\simeq\sqrt{2}\left[1-\frac{(kz)^{2}}{2}\right]\sigma_{U_{{\rm
em},\infty}}.$ (87)

###  V-C Mixed Linear-Quadratic Forms for 3-D Vectorial Electric or Magnetic
Energy Fluctuations

If $\sigma(kz)$ contains both linear and quadratic terms in $j_{\ell}$, e.g.,
for $U$ in (37), then the weaker decay for the linear terms – typically a
$(kz)^{-1}$-law – prevails for $kz\gg 3/2$. Thus,

$\displaystyle{\mit\Upsilon}^{\pm}_{U}(kz\gg 3/2)$
$\displaystyle\simeq\left[1\pm\frac{1}{2kz}\right]\sigma_{U_{\infty}}$ (88)

for $U=U_{{\rm e}}$ or $U_{{\rm m}}$. Near the boundary,

$\displaystyle\sigma_{U_{{\rm e}}}(kz\ll\sqrt{5})$
$\displaystyle\simeq\frac{2}{\sqrt{3}}\left[1-\frac{2(kz)^{2}}{15}\right]\sigma_{U_{{\rm
e},\infty}}$ (89) $\displaystyle\sigma_{U_{{\rm m}}}(kz\ll\sqrt{5})$
$\displaystyle\simeq\sqrt{\frac{8}{3}}\left[1-\frac{2(kz)^{2}}{3}\right]\sigma_{U_{{\rm
m},\infty}}.$ (90)

##  VI Relative Levels of Fluctuation

###  VI-A Coefficient of Variation

The coefficient of variation (CV),
$\nu_{X}\stackrel{{\scriptstyle\Delta}}{{=}}\sigma_{X}/\langle X\rangle$,
represents the mean-normalized RMS level of fluctuation of $X$, as a simple
measure of its relative uncertainty. Its reciprocal, $1/\nu_{X}$, measures the
SNR of $X$. Close to the boundary, the CV differs considerably between the
various components of the powers $S$ or ${\rm c}U_{{\rm em}}$. On the boundary
itself,

$\displaystyle\,\nu_{S_{z}}(0)={\rm
j}{2}/{\sqrt{5}},~{}~{}\nu_{S}(0)=\lim_{kz\rightarrow 0+}{\rm
j}{\sqrt{2}}/(kz)=+{\rm j}\infty$ (91) $\displaystyle\,\nu_{{\rm c}U_{{\rm
em},\alpha}}(0)=\sqrt{2}\,\nu_{{\rm c}U_{{\rm em},t}}(0)=\sqrt{3}\,\nu_{{\rm
c}U_{{\rm em}}}(0)=1.$ (92)

Fig. 4(a) shows the SNRs ${\rm j}/\nu_{S_{z}}(kz)$ and ${\rm j}/\nu_{S}(kz)$,
confirming that $|\nu_{S}(kz)|\geq|\nu_{S_{z}}(kz)|$ for arbitrary $kz$ and
that both CVs are in phase for $kz\gg 3/2$. Fig. 4(b) shows selected CVs
$\nu_{U_{({\rm em})(\alpha)}}(kz)\equiv\nu_{{\rm c}U_{({\rm
em})(\alpha)}}(kz)$. Comparing with Fig. 4(a), a slightly smaller relative
uncertainty of $S_{z}$ over $U_{{\rm e},z}$ (up to $11.8\%$, on the boundary)
is achieved for $kz<1$, while for $kz>1$ the relative uncertainty for $S_{z}$
is much larger. The constancy of $\nu_{U_{{\rm e},z}}(kz)$ and $\nu_{U_{{\rm
m},t}}(kz)$ indicates that the $\chi^{2}_{2}$ and $\chi^{2}_{4}$ PDFs of the
respective deep energies are maintained at any distance, up to the boundary.
The variation of $\nu_{U_{\rm e}}(kz)$ from $1$ at $kz=0$ to $1/\sqrt{3}$ at
$kz\rightarrow+\infty$ is a testimony of the compound exponential PDF [15] for
the full $U_{\rm e}(kz)$, evolving from $\chi^{2}_{2}$ to $\chi^{2}_{6}$. The
$U_{{\rm em}}(kz)$ has a corresponding evolution, from $\chi^{2}_{6}$ for
$U_{{\rm em}}(0)=U_{{\rm e},z}(0)+U_{{\rm m},t}(0)$ with $\nu_{U_{{\rm
em}}}(0)=1/\sqrt{3}$, to $\chi^{2}_{12}$ for $U_{{\rm
e}}(kz\rightarrow\infty)+U_{{\rm m}}(kz\rightarrow\infty)$ and $\nu_{U_{{\rm
em}}}(\infty)=1/\sqrt{6}$. All $\nu_{{\rm c}U_{{\rm em}(,\alpha)}}(kz)$ are
rescalings of the $\sigma_{{\rm c}U_{{\rm em}(,\alpha)}}(kz)\equiv{\rm
c}\,\sigma_{U_{{\rm em}(,\alpha)}}(kz)$ shown in Fig. 1(b), because all
$\langle U_{{\rm em}(,\alpha)}\rangle$ are independent of $kz$. Fig. 4 also
indicates that

$\displaystyle\nu_{{\rm c}U_{{\rm em}}}(kz)\ll|\nu_{S}(kz)|$ (93)

at any $kz$, differing typically by an order of magnitude or more. As a
practical consequence, the conversion of the intensity or energy measured
using electric and magnetic field probes to the scalar EM power produces
considerably lower relative uncertainty than the direct measurement of
Poynting power using aperture antennas, at any distance from the boundary.

---
(a)
(b)
Figure 4:  Relative fluctuations as a function of $kz$: (a) SNRs ${\rm
j}/\nu_{S_{z}}$ (brown), ${\rm j}/\nu_{S}$ (gray), and $1/\nu_{U_{{\rm m}-{\rm
e}}}$ (purple); (b) CVs $\nu_{U_{{\rm m},x}}$ (green), $\nu_{U_{{\rm e},z}}$
(green), $\nu_{U_{{\rm m},t}}$ (cyan), $\nu_{U_{\rm e}}$ (red), $\nu_{U_{\rm
m}}$ (blue), and $\nu_{U_{\rm em}}$ (black).

###  VI-B Coefficient of Range Variation

The CV measures the actual standard deviation relative to the actual mean
value, both at the same location. Related to their respective envelopes, the
coefficient of range variation (CRV) of $X$, denoted as $\zeta_{X}$, is here
introduced as the ratio of the local range of variation between
${\mit\Upsilon}^{+}_{X}(kz)$ and ${\mit\Upsilon}^{-}_{X}(kz)$ for $\sigma_{X}$
at $kz$ to the corresponding range of ${\mit\Xi}^{\pm}_{X}(kz)$ for the real
or complex $\langle X\rangle$ at the same $kz$, i.e.,

$\displaystyle\zeta_{X}(kz)\stackrel{{\scriptstyle\Delta}}{{=}}\frac{{\mit\Upsilon}^{+}_{X}(kz)-{\mit\Upsilon}^{-}_{X}(kz)}{{\mit\Xi}^{+}_{X}(kz)-{\mit\Xi}^{-}_{X}(kz)}.$
(94)

This measure is particularly useful when nonlocal deviations are expected from
ideal $\sigma_{X}(kz)$ or $\langle X(kz)\rangle$ and hence $\nu_{X}(kz)$,
e.g., for causes of field perturbations listed in sec. V. For $kz\gg 3/2$, the
CRVs for the Poynting power components are

$\displaystyle\zeta_{S_{\alpha}}(kz\gg 3/2)$
$\displaystyle=\frac{9\,\sigma_{S_{\alpha,\infty}}/[32(kz)^{3}]}{\langle{S_{\alpha,\infty}}\rangle}\rightarrow\infty,~{}~{}\alpha=x,y$
(95) $\displaystyle\zeta_{S_{z}}(kz\gg 3/2)$
$\displaystyle=\frac{9\,\sigma_{S_{z,\infty}}/[32(kz)^{2}]}{-{\rm
j}\langle|{\cal E}_{0}|^{2}/\eta_{0}\rangle/(kz)}={\rm
j}\frac{3\sqrt{2}}{32\,kz}$ (96) $\displaystyle\zeta_{S_{t}}(kz\gg 3/2)$
$\displaystyle=\frac{9\,\sigma_{S_{t,\infty}}/[32(kz)^{3}]}{\langle{S_{t,\infty}}\rangle}\rightarrow\infty$
(97) $\displaystyle\zeta_{S}(kz\gg 3/2)$
$\displaystyle=\frac{3\,\sigma_{S_{\infty}}/[16(kz)^{2}]}{-{\rm
j}\langle|{\cal E}_{0}|^{2}/\eta_{0}\rangle/(kz)}={\rm
j}\frac{\sqrt{6}}{16\,kz}.$ (98)

The decrease of $\zeta_{S_{(z)}}(kz)$ according to $(kz)^{-1}$ signifies that
the nonlocal range of $\sigma_{S_{z}}$ decays more rapidly with $kz$ than that
of $\langle S_{z}\rangle$ for $kz\gg 3/2$. At these large distances, it
follows that $|\zeta_{S_{(z)}}(kz\gg 3/2)|\ll 1$.

Similarly, for the electric or magnetic energy densities

$\displaystyle\zeta_{U_{\alpha}}(kz\gg 3/2)$
$\displaystyle=\frac{3\,\sigma_{U_{\alpha,\infty}}/(2kz)}{3\,\langle
U_{\alpha,\infty}\rangle/(2kz)}=1,~{}~{}\alpha=x,y$ (99)
$\displaystyle\zeta_{U_{z}}(kz\gg 3/2)$
$\displaystyle=\frac{3\,\sigma_{U_{z,\infty}}/[2(kz)^{2}]}{3\,\langle
U_{z,\infty}\rangle/[2(kz)^{2}]}=1$ (100) $\displaystyle\zeta_{U_{t}}(kz\gg
3/2)$ $\displaystyle=\frac{3\,\sigma_{U_{t,\infty}}/(2kz)}{3\,\langle
U_{t,\infty}\rangle/(2kz)}=\frac{1}{\sqrt{2}}$ (101)
$\displaystyle\zeta_{U}(kz\gg 3/2)$
$\displaystyle=\frac{\sigma_{U_{\infty}}/(kz)}{\langle
U_{\infty}\rangle/(kz)}=\frac{1}{\sqrt{3}}.$ (102)

Unlike $\zeta_{S_{(\alpha)}}$, all $\zeta_{U_{(\alpha)}}$ are finite and
independent of $kz$. Note that $\zeta_{U_{(\alpha)}}(kz\gg
3/2)=\nu_{U_{(\alpha,)\infty}}$ even though all $\sigma_{U_{(\alpha)}}(kz)$
exhibit oscillations, as seen in Fig. 2. All $\zeta_{U_{{\rm
em}(,\alpha)}}(kz)$ are infinite because all $\langle U_{{\rm
em}(,\alpha)}(kz)\rangle$ are independent of $kz$.

##  VII Poynting’s Theorem for RMS Fluctuations

Poynting’s theorem of energy conservation links the local power flux for
deterministic fields to the energy imbalance as

$\displaystyle\underline{\nabla}\cdot\underline{S}(\underline{r})=-{\rm
j}\omega[U_{{\rm m}}(\underline{r})-U_{{\rm
e}}(\underline{r})]\stackrel{{\scriptstyle\Delta}}{{=}}-{\rm j}\omega U_{{\rm
m}-{\rm e}}(\underline{r}).$ (103)

For deterministic or random plane waves ($\underline{\nabla}=-{\rm
j}\underline{k}$) with fixed $\underline{k}$, (103) reduces to
$S_{k}(\underline{r})={\rm c}U_{{\rm m}}(\underline{r})-{\rm c}U_{{\rm
e}}(\underline{r})$, expressing the $\underline{k}$-directed Poynting
component $S_{k}\equiv\underline{S}\cdot\underline{1}_{k}$ as the imbalance
between the magnetic and electric scalar power densities. For stochastic
fields, it was shown in [1] that (103) also holds for the mean power and
energy densities , i.e.,

$\displaystyle\langle\underline{\nabla}\cdot\underline{S}(kz)\rangle=\underline{\nabla}\cdot\langle\underline{S}(kz)\rangle=-{\rm
j}\omega\langle U_{{\rm m}-{\rm e}}(kz)\rangle.$ (104)

Complex conjugation of (103) followed by side-by-side multiplication and
ensemble averaging for random fields shows that the second-order moments are
correspondingly related as

$\displaystyle\langle|\underline{\nabla}\cdot\underline{S}(kz)|^{2}\rangle=\omega^{2}\langle|U_{{\rm
m}-{\rm e}}(kz)|^{2}\rangle.$ (105)

With $\sigma_{U_{{\rm m}-{\rm e}}}=\sigma_{U_{{\rm em}}}$ and (47), Poynting’s
theorem for RMS fluctuations at any $kz$ follows from (104) and (105) as

$\displaystyle\sigma_{\underline{\nabla}\cdot\underline{S}}(kz)=\omega\sigma_{U_{{\rm
m}-{\rm e}}}(kz)=\omega\sigma_{U_{{\rm
em},\infty}}\sqrt{1+j^{2}_{0}+\frac{3j^{2}_{1}}{2}+\frac{j^{2}_{2}}{2}}.$
(106)

The average energy imbalance for arbitrary $kz$ [1, eq. (47)]

$\displaystyle\langle U_{{\rm m}-{\rm
e}}(kz)\rangle=\frac{2\,\varphi_{0}\langle|{\cal
F}_{0}|^{2}\rangle}{3}(j_{0}-2j_{2})$ (107)

is plotted in [1, Fig. 5]. Near the boundary,

$\displaystyle\langle U_{{\rm m}-{\rm e}}(kz\ll\sqrt{5})\rangle$
$\displaystyle\simeq\frac{2\,\varphi_{0}\langle|{\cal
F}_{0}|^{2}\rangle}{3}\left[1-\frac{6(kz)^{2}}{5}\right]$ (108)

i.e., quadratically approaching $\langle U_{{\rm m},x}(0)\rangle=\langle
U_{{\rm e},z}(0)\rangle=\langle U_{{\rm em},t}(0)\rangle=\langle U_{{\rm
em}}(0)\rangle/3$ for $kz\rightarrow 0$. Its asymptotic envelopes are

$\displaystyle{\mit\Xi}^{\pm}_{U_{{\rm m}-{\rm e}}}(kz\gg
3/2)\simeq\pm\frac{\varphi_{0}\langle|{\cal F}_{0}|^{2}\rangle}{kz}.$ (109)

The CV of $U_{{\rm m}-{\rm e}}$ (or ${\rm c}U_{{\rm m}-{\rm e}}$) at arbitrary
distances is

$\displaystyle\nu_{U_{{\rm m}-{\rm
e}}}(kz)=\sqrt{\frac{3\left(1+j^{2}_{0}+\frac{3}{2}j^{2}_{1}+\frac{1}{2}j^{2}_{2}\right)}{8\left(j_{0}-2j_{2}\right)^{2}}}.$
(110)

Its reciprocal is plotted in Fig. 4(a). From its range of values across $kz$,
it can be inferred that, even though Poynting’s law holds equally between the
average power flux and the average energy imbalance as well as between their
respective RMS fluctuation levels, the exchanges between the fluctuations are
stronger, i.e., $\sigma_{U_{{\rm m}-{\rm e}}}(kz)\gg\langle U_{{\rm m}-{\rm
e}}(kz)\rangle$ for general $kz$. This is a fortiori the case for $kz\gg 1$,
where $\langle\underline{S}\rangle$ and hence
$\underline{\nabla}\cdot\langle\underline{S}\rangle$ approach null.
Consequently, $\nu_{U_{{\rm m}-{\rm e}}}(kz)$ approaches infinity for
$kz\rightarrow+\infty$, also at quasi-periodic finite distances.

On the boundary, $\nu_{U_{{\rm m}-{\rm e}}}$ reaches its minimum value
$\sqrt{3}$ that exceeds $1$, i.e., the RMS fluctuation of the EM energy
imbalance always remains larger than the mean imbalance, further increasing
quadratically with distance as

$\displaystyle\nu_{U_{{\rm m}-{\rm e}}}(kz\ll\sqrt{5})$
$\displaystyle\simeq\sqrt{3}\left[1+\frac{7(kz)^{2}}{10}\right].$ (111)

From (81), (83) and (109), the CRV $\zeta_{U_{{\rm m}-{\rm e}}}(kz\gg 3/2)$
varies as $(kz)^{-1}$ and coincides with $|\zeta_{S}(kz\gg 3/2)|$, i.e.,

$\displaystyle\zeta_{U_{{\rm m}-{\rm e}}}(kz\gg 3/2)$
$\displaystyle=\frac{3\,\sigma_{U_{{\rm
em},\infty}}/[8(kz)^{2}]}{2\,\varphi_{0}\langle|{\cal
F}_{0}|^{2}\rangle/(kz)}$
$\displaystyle=\frac{\sqrt{6}}{16\,kz}=\left|\zeta_{S}(kz\gg 3/2)\right|.$
(112)

##  VIII Conclusion

In this article, the scalar EM power density ${\rm c}U_{{\rm em}}$ as a
converted and scaled EM energy density and the Poynting vector $\underline{S}$
for harmonic random fields were analyzed. The divergence of $\underline{S}$ is
proportional to $U_{\rm m}-U_{\rm e}$, whereas ${\rm c}U_{{\rm em}}$ is
proportional to $U_{\rm m}+U_{\rm e}$. For deterministic plane waves
($\underline{E}=\eta_{0}\underline{H}\times\underline{1}_{k}$,
$\underline{k}\cdot\underline{E}=0$), both power quantities yield the same
value. In random non-plane wave environments, only the latter produces a
nontrivial mean value in free space. The presence of an ideal PEC boundary
affects both their means and standard deviations in different ways; cf. (4)
vs. (8) and (30) vs. (44). For random fields near a PEC boundary, $U_{{\rm
e}}(kz)\not=U_{\rm m}(kz)$ qualitatively and quantitatively. The results
indicate the need for evaluating both $\underline{E}$ and $\underline{H}$ to
obtain statistics of random EM power or energy density, unlike for
deterministic plane waves where knowledge of either field suffices.

For Cartesian, tangential and full vectorial power and energy densities near a
PEC boundary, their local standard deviations were obtained as eqs. (20),
(25), (27), (30); eqs. (32), (33), (35), (37); and eqs. (41)–(44),
respectively. Similar to the averages, these standard deviations exhibit
power-law type decay with the electrical distance $kz$, on average, at
different rates. For the EM power, the envelope of $\sigma_{S_{x}}$ converges
most rapidly to the free-space value, decaying as $(kz)^{-3}$. For the energy
densities $U_{{\rm e}}$ and $U_{{\rm m}}$, the most rapid decay as $(kz)^{-2}$
occurs for the normal components $\sigma_{U_{z}}$; cf. (59). For 3-D vector
fields, the standard deviation for power decays according to $(kz)^{-2}$ in
(71), which is more rapidly than $(kz)^{-1}$ for energy in (88). Only the
normal components $\sigma_{S_{z}}$ and $\sigma_{U_{z}}$ exhibit the same
$(kz)^{-2}$ decay. Thus, the standard deviations exhibit an equal or sharper
roll-off and narrower transition zone of $kz$ than the corresponding averages.
Near the boundary, $\sigma_{S_{z}}$ increases linearly with $kz$, while all
other standard deviations vary quadratically, to leading order.

On approaching the PEC boundary, for $kz\ll\sqrt{5}$, the absolute RMS
fluctuations of the normal and full scalar vs. Poynting powers diverge, i.e.,
$\sigma_{{\rm c}U_{{\rm em}(,z)}}>\sigma_{S_{(z)}}$ (Fig. 1). As a
consequence, this may affect criteria for acceptable levels of field
nonuniformity in MSRC validation procedures [9], depending on whether dipole
antennas or sensors vs. aperture antennas are used, when extending the working
volume to include locations within the boundary zone. The relative RMS
fluctuation of $S$ is much larger than for ${\rm c}U_{{\rm em}}$, at any $kz$;
cf. (93). On the boundary, the CV of $S_{z}$ is finite, unlike the CV of $S$;
cf. (91). For $kz\gg 3/2$, the decay of $\zeta_{S_{z}}$ occurs according to
$(kz)^{-1}$ in (96), indicating that the range of nonlocal variation of
$\sigma_{S_{z}}(kz)$ becomes progressively weaker with distance than that for
$\langle{S_{z}}(kz)\rangle$. By contrast, the CRVs for energies in (99)–(102)
are constant and coincide with their respective CVs.

The Poynting theorem, valid for averages of random power flux and random
energy imbalance [1], extends to (106) for their standard deviations. The RMS
fluctuations of the power flux and imbalance are larger than their mean
values, as expressed by (110) and (111).

## References

  * [1] L. R. Arnaut and G. Gradoni, “Average linear and angular momentum and power of random fields near a perfectly conducting boundary,” IEEE Trans. Electromagn. Compat., vol. 62, no. 4, pp. 1118–1127, Aug. 2020.
  * [2] A. De Leo, G. Cerri, P. Russo, and V. Mariani Primiani, “A novel emission test method for multiple monopole source stirred reverberation chambers,” IEEE Trans. Electromagn. Compat., vol. 62, no. 5, pp. 2334–2337, Oct. 2020.
  * [3] S. M. Smith and C. Furse, “Stochastic FDTD for analysis of statistical variation in electromagnetic fields,” IEEE Trans. Ant. Propag., vol. 60, no. 7, pp. 3343–3350, Jul. 2012.
  * [4] D. A. Hill, Electromagnetic Theory of Reverberation Chambers. NIST Techn. Note 1506, U.S. Dep. Commerce, Boulder, CO, Dec. 1998.
  * [5] J. M. Dunn, “Local, high-frequency analysis of the field in a mode-stirred chamber,” IEEE Trans. Electromagn. Compat., vol. 32, no. 1, pp. 53–58, Feb. 1990.
  * [6] D. A. Hill, “Boundary fields in reverberation chamberx,” IEEE Trans. Electromagn. Compat., vol. 47, no. 2, pp. 281–290, May 2005.
  * [7] L. R. Arnaut, “Spatial correlation functions of inhomogeneous random electromagnetic fields,” Phys. Rev. E, vol. 73, no. 3, 036604, Mar. 2006.
  * [8] L. R. Arnaut, “Probability distribution of random electromagnetic fields in the presence of a semi-infinite isotropic medium,” Radio Sci., vol. 42, RS3001, 2007.
  * [9] IEC Joint Task Force CISPR/A-SC77B: IEC 61000-4-21 Electromagnetic Compatibility (EMC) – Part 4-21: Testing and Measurement Techniques – Reverberation Chamber Test Methods, International Electrotechnical Commission, 2nd ed., Jan. 2011.
  * [10] L. R. Arnaut and P. D. West, “Electromagnetic reverberation near a perfectly conducting boundary,” IEEE Trans. Electromagn. Compat., vol. 48, no. 2, pp. 359–371, May 2006.
  * [11] L. R. Arnaut, “Measurement uncertainty in reverberation chambers: I. Sample statistics,” NPL Report TQE2, 2nd ed., Nat. Phys. Lab., Teddington, U.K., Dec. 2008.
  * [12] R. Serra and C. Carobbi, “A probabilistic interpretation of the IEC 61000-4-21 threshold levels for field uniformity in ideal reverberation chambers,” Proc. 2020 EMC Europe Int. Symp. Electromagn. Compat., 23–25 Sep. 2020, Rome, Italy.
  * [13] A. M. Yaglom, An Introduction to the Theory of Stationary Random Functions. Prentice–Hall: Englewood Cliffs, N. J., 1962.
  * [14] F. W. King, Hilbert Transforms. Cambridge University Press: Cambridge, U.K., 2009.
  * [15] L. R. Arnaut, “Compound exponential distributions for undermoded reverberation chambers,” IEEE Trans. Electromagn. Compat., vol. 44, no. 3, pp. 442–457, Aug. 2002.


11institutetext: Departamento de Matemáticas, Facultad de Ciencias,
Universidad Nacional Autónoma de México
11email<EMAIL_ADDRESS>22institutetext: CONAHCYT-Facultad de
Ciencias en Física y Matemáticas, Universidad Autónoma de Chiapas, México
22email<EMAIL_ADDRESS><EMAIL_ADDRESS>33institutetext:
Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional
Autónoma de México
33email<EMAIL_ADDRESS>44institutetext: Facultad de Ciencias
en Física y Matemáticas, Universidad Autónoma de Chiapas, México
44email<EMAIL_ADDRESS>
# A geometric reduction method for some fully nonlinear first-order PDEs on
semi-Riemannian manifolds

Juan Carlos Fernández 11 Eddaly Guerra-Velasco 22 Oscar Palmas 33 Boris A.
Percino-Figueroa 44

###### Abstract

Given a semi-Riemannian manifold $(M,\langle\cdot,\cdot\rangle_{g}),$ we use
the transnormal functions defined on $M$ to reduce fully nonlinear first order
PDEs of the form

$F(x,u,\langle\nabla_{g}u,\nabla_{g}u\rangle_{g})=0,\qquad\text{on }M$

into ODEs and obtain local existence results of solutions which are constant
along the level sets of the transnormal functions. In particular, we apply
this reduction method to obtain new solutions to eikonal equations with a
prescribed geometry.

MSC: 34A05, 35B06, 35F20, 53C21, 58J70.

###### keywords:

eikonal equation, semi-Riemannian manifolds, transnormal functions,
isoparametric functions, fully nonlinear first order PDE

##  1 Introduction

Let $(M,\langle\cdot,\cdot\rangle_{g})$ be a semi-Riemannian manifold without
boundary of dimension $n\geq 3$. In this paper, we are interested in solutions
to a fully nonlinear partial differential equation of the form

$F(x,u(x),\left\langle\nabla_{g}u(x),\nabla_{g}u(x)\right\rangle_{g})=0,\qquad\text{on
}M.$ (1)

where $u\colon M\to\mathbb{R}$ is a differentiable function. Notice that, when
$F$ is independent of $u$, (1) becomes

$F(x,\left\langle\nabla_{g}u(x),\nabla_{g}u(x)\right\rangle_{g})=0,\qquad\text{on
}M.$

which is a generalization of the eikonal equation

$\langle\nabla_{g(x)}u(x),\nabla_{g(x)}u(x)\rangle_{g}=U(x),\quad U\in
C^{\infty}(M).$ (2)

In this paper, we will prove the existence of local solutions to problem (1)
with a prescribed geometry on the level sets in two different ways. To
establish our main results, we define and recall some concepts.

For a smooth function $f\colon M\to\mathbb{R}^{k}$, we will say that $F\colon
M\times\mathbb{R}^{2}\to\mathbb{R}$ is _$f$ -invariant in the first variable_
if for any $x_{1},x_{2}\in M$ such that $f(x_{1})=f(x_{2})$, we have that

$F(x_{1},r,p)=F(x_{2},r,p),$

for all $(r,p)\in\mathbb{R}^{2}$. This is equivalent to saying that there is a
unique smooth function
$\widehat{F}\colon\mathrm{Im}\,f\times\mathbb{R}^{2}\subset\mathbb{R}^{k+2}\to\mathbb{R}$
such that for all $(r,p)\in\mathbb{R}^{2}$,

$F(x,r,p)=\widehat{F}(f(x),r,p),\qquad x\in M.$

We say that a neighborhood $\Omega\subset M$ is $f$-invariant if $x\in\Omega$
implies $y\in\Omega$ for any $y\in M$ such that $f(x)=f(y)$. In other words,
$\Omega$ is the union of level sets of $f$.

We make extensive use of transnormal functions. Given a semi-Riemannian
manifold $(M,g)$, we say that $f\colon M\rightarrow\mathbb{R}$ is a
_transnormal_ function if there exists a smooth function
$a\colon\mathbb{R}\rightarrow\mathbb{R}$ such that

$\langle\nabla_{g}f,\nabla_{g}f\rangle_{g}=a\circ f.$

Examples of transnormal functions include isoparametric functions (see, for
example, [4, 7, 8] for the semi-Riemannian setting and the book [1] in the
Riemannian setting). In the Riemannian setting, there are explicit examples of
transnormal functions which are not isoparametric (cf. [9]), but almost
nothing is known in the semi-Riemannian case.

Isoparametric functions have proven to be useful to find solutions for many
PDEs on manifolds; for example, in nonlinear second order elliptic and
ultrahyperbolic PDEs [3, 4], nonlinear higher order elliptic PDEs [2, 5, 6]
and mean curvature flow equations [8, 10]. The main idea in this context is to
use isoparametric functions to reduce the PDEs into ODEs which are simpler to
analyze, generalizing the idea of looking for radial solutions in
$\mathbb{R}^{n}$ or in a ball in $\mathbb{R}^{n}$. However, for many fully
nonlinear first order PDEs, as equation (1), it suffices to perform the
reduction method by using only transnormal functions.

In this paper we show two instances of this method, reducing problem (1) to an
ODE or a two-dimensional PDE defined in a rectangle in $\mathbb{R}^{2}$. Let
us describe briefly the method, main results and applications for each
instance. In first place, for the reduction into an ODE, the key result is the
following one.

######  Proposition 1.1.

Let $f\colon M\rightarrow\mathbb{R}$ be a transnormal function,
$w\colon\mathbb{R}\rightarrow\mathbb{R}$ smooth and suppose $F\colon
M\times\mathbb{R}^{2}\to\mathbb{R}$ is $f$-invariant in the first variable.
Then $u=w\circ f$ is a solution to the equation

$F(x,u,\langle\nabla_{g}u,\nabla_{g}u\rangle_{g})=0,\qquad\text{on }M$

if and only if $w$ is a solution to

$\widehat{F}\left(t,w(t),a(t)|w^{\prime}(t)|^{2}\right)=0,\quad\text{in
}\mathrm{Im}\,f\subset\mathbb{R}.$ (3)

The proof of this result is a straightforward computation. As a consequence,
we obtain our first main result, a local existence result for equation (1):

######  Theorem 1.2.

Let $(M,g)$ be a semi-Riemannian manifold, $f\colon M\rightarrow\mathbb{R}$ be
a transnormal function with $\langle\nabla_{g}f,\nabla_{g}f\rangle_{g}=a\circ
f$ and let $t_{0}\in\mathbb{R}$ be a regular value of $f$. Define
$M_{0}:=f^{-1}(t_{0}).$ Then, for any $C^{1}$ function $F\colon
M\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ which is $f$-invariant in the
first variable, and for any $(x_{0},r_{0},p_{0})\in
M_{0}\times\mathbb{R}\times(\mathbb{R}\smallsetminus\\{0\\})$ satisfying

$F(x_{0},r_{0},p_{0})=0,\quad\frac{\partial F}{\partial
p}(x_{0},r_{0},p_{0})\neq 0\quad\text{and}\quad a(t_{0})p_{0}>0,$ (4)

equation (1) posseses a unique $f$-invariant solution defined in an
$f$-invariant domain $\Omega\subset M$ such that $M_{0}\subset\Omega$.

We prove this result in Section 2, which is an easy consequence of the Theorem
of Existence and Uniqueness of ODEs. In the same section, and in order to
illustrate the method behind the proof, we give some applications in case of
the eikonal equation (2) in semi-Riemannian and Riemannian manifolds. In
particular, using isoparametric functions defined in the semi-Euclidean space
and in the pseudo-sphere [4, 7], we construct, in Examples 2.1 and 2.2,
several solutions to this problem in both manifolds, with prescribed
nontrivial geometry on the level sets. In Example 2.3, we do the same
reduction for cohomogeneity-one Riemannian manifolds [5]. Moreover, in Example
2.4, we give a concrete and interesting example of a solution to equation (2)
defined in the hyperbolic Riemannian space, which is constant along the level
sets of a transnormal function having non-isoparametric level sets [9]. In
particular, we have the following result for the Riemannian eikonal equation.
We denote by $\mathrm{Isom}(M,g)$ the group of isometries of $M$.

######  Corollary 1.3.

Let $c>0$ and consider the eikonal equation

$\langle\nabla_{g}u,\nabla_{g}u\rangle_{g}=c,\qquad\text{on }M,$ (5)

where $(M,g)$ is Riemannian.

  1. (i)

If $(M,g)$ is a closed Riemannian manifold and $\Gamma$ is a closed subgroup
of $\mathrm{Isom}(M,g)$ acting by cohomogeneity one on $M$, then there exists
a $\Gamma$-invariant solution to (5), defined in $M\smallsetminus\mathcal{Z}$,
where $\mathcal{Z}$ is the union of the singular orbits of the action.

  2. (ii)

If $(M,g)$ is the sphere with its canonical metric $(\mathbb{S}^{n},g_{0})$,
then, for any isoparametric hypersurface $M_{0}\subset\mathbb{S}^{n}$, there
exists a solution to equation (5), constant on $M_{0}$ and defined in
$\mathbb{S}^{n}\smallsetminus\mathcal{Z}$, where $\mathcal{Z}$ is a union of
two submanifolds of $M$ of codimension $\geq 2$.

  3. (iii)

If $(M,g)$ is the hyperbolic space with its canonical metric
$(\mathbb{H}^{n},g_{0})$, there exists a non isoparametric hypersurface
$M_{0}$ in $\mathbb{H}^{n}$ and a globally defined solution
$u:\mathbb{H}^{n}\rightarrow\mathbb{R}$ such that $u$ is constant on $M_{0}$.

We prove this Corollary at the end of Section 2.

In Section 3 we will explore the local existence of solutions to problem (1)
with a different qualitative behavior. To describe it, let $(L,g_{L})$ and
$(N,g_{N})$ be two connected and complete Riemannian manifolds, without
boundary, of dimensions $s$ and $k$, respectively, and consider the semi-
Riemannian warped product

$L\times_{\alpha}N:=(L\times N,g:=-g_{L}+\alpha^{2}g_{N}),$ (6)

where $\alpha\colon L\to\mathbb{R}$ is smooth and positive.

Let $f_{L}\colon L\to\mathbb{R}$ and $f_{N}\colon N\to\mathbb{R}$ be
transnormal functions and consider a $C^{2}$ function $F\colon(L\times
N)\times\mathbb{R}^{2}\to\mathbb{R}$ which is $(f_{L}\times f_{N})$-invariant
in the first variable, that is,

$F(x_{1},z_{1},r,\tau)=F(x_{2},z_{2},r,\tau),$

whenever $f_{L}(x_{1})=f_{L}(x_{2})$ and $f_{N}(z_{1})=f_{N}(z_{2})$. As
before, this implies the existence of a smooth function
$\widehat{F}\colon\mathbb{R}^{2}\times\mathbb{R}^{2}\to\mathbb{R}$ such that

$F(x,z,r,\tau)=\widehat{F}(f_{L}(x),f_{N}(z),r,\tau).$ (7)

In this setting, problem (1) can be written as:

$F(x,y,u,\langle\nabla_{g}u,\nabla_{g}u\rangle_{g})=0\qquad\text{on }\
L\times_{\alpha}N.$ (8)

In Section 3 we will show the local existence of functions $u\in
C^{\infty}(L\times N)$ satisfying (8) and such that, for fixed $x\in L$ and
$y\in N$, the functions $u(\cdot,y)$ and $u(x,\cdot)$ are constant on the
level sets of $f_{L}$ and $f_{N}$, respectively. The key step is to reduce the
problem (8) into a $2$-dimensional PDE of the first order in a rectangle in
$\mathbb{R}^{2}$, easier to handle and visualize, for instance, by numerical
methods.

We state the reduction method in the following proposition. In what follows,
we define the intervals $I_{f_{L}}:=\mathrm{Im}\,f_{L}$ and
$I_{f_{N}}:=\mathrm{Im}\,f_{N}$, the parameters $t:=f_{L}(x)\in I_{f_{N}}$ and
$s:=f_{N}(y)\in I_{f_{N}}$, so that $\widehat{F}=\widehat{F}(t,s,r,\tau)$ is
defined in $I_{f_{L}}\times I_{f_{N}}\times\mathbb{R}^{2}$, and the functions
$a_{f_{L}}$ and $a_{f_{N}}$ are such that
$|\nabla_{g_{L}}f_{L}|^{2}_{g}=a_{f_{L}}\circ f$ and
$|\nabla_{g_{N}}f_{N}|^{2}_{g}=a_{f_{N}}\circ f$.

######  Proposition 1.4.

Let $F$ be $(f_{L}\times f_{N})$-invariant in the first variable as in (7),
and let $w\colon\mathbb{R}^{2}\rightarrow\mathbb{R}$ be smooth. If
$\alpha=\widehat{\alpha}\circ f_{L}$ with
$\widehat{\alpha}\colon\mathbb{R}\rightarrow\mathbb{R}$ smooth, then
$u=w(f_{L},f_{N})$ is a solution of (8) if and only if $w$ satisfies

$\widehat{F}\left(t,s,w(t,s),-a_{f_{L}}(t)w_{t}^{2}(t,s)+\frac{a_{f_{N}}(s)}{\widehat{\alpha}^{2}(t)}w_{s}^{2}(t,s)\right)=0,\quad\text{in
}I_{f_{L}}\times I_{f_{N}},$ (9)

where $w_{t}:=\frac{\partial w}{\partial t}$ and $w_{s}:=\frac{\partial
w}{\partial s}$.

We prove this proposition in Section 3.1. As before, this proposition also
implies a local existence result of $(f_{L}\times f_{N})$-invariant solutions.
In order to state it, define the singular Riemannian foliations in $L$ and
$N$, induced by the transnormal functions $f_{L}$ and $f_{N}$,

$\mathcal{F}_{f_{L}}:=\\{f_{L}^{-1}(t):t\in\mathbb{R}\\}\qquad\text{and}\qquad\mathcal{F}_{f_{N}}:=\\{f_{N}^{-1}(s):s\in\mathbb{R}\\},$

respectively. Since $(L,g_{L})$ and $(N,g_{N})$ are Riemannian manifolds, the
leaves of both foliations are also smooth manifolds (see [12]). Notice also
that $f_{L}$ and $f_{N}$ induce functions
$C_{f_{L}}\colon\mathcal{F}_{f_{L}}\to\mathbb{R}$ and
$C_{f_{N}}\colon\mathcal{F}_{f_{N}}\to\mathbb{R}$ given as
$C_{f_{L}}(f_{L}^{-1}(t))=f_{L}(x)$ for any $x\in f_{L}^{-1}(t)$ and
$C_{f_{N}}(f_{N}^{-1}(s))=f_{N}(z)$ for any $z\in f^{-1}(s)$. We say that a
function $\Sigma_{f_{L}}\colon
I\subset\mathbb{R}\rightarrow\mathcal{F}_{f_{L}}$ is a _smooth uniparametric
family of leaves in $L$_ if $C_{f_{L}}\circ\Sigma_{f_{L}}\colon
I\subset\mathbb{R}\rightarrow\mathbb{R}$ is smooth. In the same way, we define
a smooth uniparametric family of leaves in $N$. As a consequence, we have the
following local existence result.

######  Theorem 1.5.

Let $f_{L}$ and $f_{N}$ be transnormal in $L$ and $N$, and let $t_{0}$ and
$s_{0}$ be regular values of $f_{L}$ and $f_{N}$, respectively. Define
$L_{0}:=f_{L}^{-1}(t_{0})$ and $N_{0}:=f_{N}^{-1}(s_{0}).$ Let
$F\colon(L\times N)\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ be a $C^{2}$,
$(f_{L}\times f_{N})$-invariant function on the first variable. Suppose that
$(x_{0},z_{0},r_{0},p_{0},q_{0})\in L_{0}\times N_{0}\times\mathbb{R}^{3}$ and

$\tau_{0}:=-a_{f_{L}}(t_{0})p_{0}^{2}+\frac{a_{f_{N}}(s_{0})}{\widehat{\alpha}^{2}(t_{0})}q_{0}^{2}$

are such that $(p_{0},q_{0})\neq(0,0)$,

$F(x_{0},z_{0},r_{0},\tau_{0})=0,$ (10)

and suppose further that

$\frac{\partial F}{\partial\tau}(x,z,r,\tau)\neq 0$ (11)

in a neighborhood of $(t_{0},s_{0},r_{0},\tau_{0})$. Then, for any given
smooth function $R\colon(-\varepsilon,\varepsilon)\rightarrow\mathbb{R}$ and
any pair of smooth one parameter families $\Sigma_{f_{L}},\Sigma_{f_{N}}$
defined in $(-\varepsilon,\varepsilon)$ such that

  1. _( $\Sigma.1$)_

$\Sigma_{f_{L}}(0)=L_{0}\ $ and $\ \Sigma_{f_{N}}(0)=N_{0}$,

  2. _( $\Sigma.2$)_

$q_{0}\;\frac{d}{d\zeta}\big{|}_{\zeta=0}\left(\Sigma_{f_{L}}\circ
C_{f_{L}}\right)\neq\
p_{0}\;\frac{d}{d\zeta}\big{|}_{\zeta=0}\left(\Sigma_{f_{N}}\circ
C_{f_{N}}\right)$, and

  3. _( $\Sigma.3$)_

$R^{\prime}(0)=p_{0}\;\frac{d}{d\zeta}\big{|}_{\zeta=0}\left(\Sigma_{f_{L}}\circ
C_{f_{L}}\right)+q_{0}\;\frac{d}{d\zeta}\big{|}_{\zeta=0}\left(\Sigma_{f_{N}}\circ
C_{f_{N}}\right)$,

the problem (8) has an $(f_{L}\times f_{N})$-invariant solution $u$ defined in
an $(f_{L}\times f_{N})$-invariant open neighborhood of $L_{0}\times N_{0}$,
satisfying the condition

$u(\Sigma_{f_{L}}(\zeta),\Sigma_{f_{N}}(\zeta))=R(\zeta),\quad\zeta\in(-\varepsilon,\varepsilon).$

We prove this Theorem in Section 3.2. In the same section, we give a concrete
example in the case of the de Sitter space in order to sketch the idea behind
the method.

Our paper is organized as follows. In Section 2 we prove Theorem 1.2, give
concrete examples where the one dimensional reduction method holds true and
prove Corollary 1.3. In Section 3.1, we compute several identities involving
norms of gradients of functions defined in warped products of the form (6) and
prove Proposition 1.4. Finally, in Section 3.2, we prove Theorem 1.5 and give
a concrete example of the two-dimensional reduction method. In this same
section, in order to simplify equation (9), we briefly introduce the theory of
distance functions induced by transnormal functions on Riemannian manifolds.

##  2 The one dimensional reduction method

In this section, we prove Theorem 1.2 and apply it in several examples.

_Proof of Theorem 1.2._ Let $(x_{0},r_{0},p_{0})\in
M_{0}\times\mathbb{R}\times\left(\mathbb{R}\smallsetminus\\{0\\}\right)$ be a
point satisfying the hypothesis (4). Without loss of generality, suppose that
$p_{0}>0$, so that $a(t_{0})>0$. By Proposition 1.1, solving (1) is equivalent
to solving (3). The hypotheses imply that

$\widehat{F}(t_{0},r_{0},p_{0})=0,\quad\text{and}\quad\frac{\partial\widehat{F}}{\partial
p}(t_{0},r_{0},p_{0})\neq 0.$

By the Implicit Function Theorem there exists an open set
$U\subset\mathbb{R}^{2}$ and a $C^{1}$ function $H\colon
U\rightarrow\mathbb{R}$ such that $H(t_{0},r_{0})=p_{0}$,
$\widehat{F}(t,r,H(t,r))=0$ and $H(t,r)>0$ for any $(t,r)\in U$. Consider now
the Cauchy problem

$w^{\prime}(t)=\sqrt{\frac{H(t,w(t))}{a(t)}},\qquad w(t_{0})=r_{0}.$

As $H$ and $a$ have the same sign near $t_{0}$, and as $a(t_{0})\neq 0$, by
the Theorem of Existence and Uniqueness of solutions to initial value ODEs,
there exists $\varepsilon>0$ and a unique $C^{1}$ solution
$w\colon(t_{0}-\varepsilon,t_{0}+\varepsilon)\rightarrow\mathbb{R}$ to this
problem. Hence $a(t)w^{\prime}(t)^{2}=H(t,w(t))$ and $w$ solves the problem
(3) in the same interval. By Proposition 1.1, the function $u:=w\circ f$ is
$f$-invariant and satisfies equation (1) in
$\Omega:=f^{-1}(t_{0}-\varepsilon,t_{0}+\varepsilon)$. By construction,
$\Omega$ is an $f$-invariant open set and $M_{0}\subset\Omega$. $\square$

In what follows, we develop some examples to sketch the reduction method and
see how the local existence holds true.

######  Example 2.1.

Let $M=\mathbb{R}^{n}_{s}$ be the semi-Euclidean space of signature $s<n$. For
any $A\in\mathrm{Sym}(\mathbb{R}^{n}_{s})$, $a\in\mathbb{R}^{n}_{s}$ and
$\alpha\in\mathbb{R}$ such that $(A-\alpha I)A=0$, it is shown in [7,
Proposition 2.3] that the function

$f\colon\mathbb{R}^{m}_{s}\rightarrow\mathbb{R},\quad f(x)=\langle
Ax,x\rangle+2\langle a,x\rangle$

is transnormal, in fact isoparametric, and satisfies that

$\langle\nabla f,\nabla f\rangle=4\alpha f+\beta,$

where $\beta=4\langle a,a\rangle$. Consider the eikonal equation (2) and
suppose $U$ is $f$-invariant. Let
$\widehat{U}\colon\mathbb{R}\rightarrow\mathbb{R}$ be a smooth function
satisfying $U=\widehat{U}\circ f$. Hence, (2) reduces to

$(4\alpha t+\beta)|w^{\prime}(t)|^{2}=\widehat{U}(t),\quad
t\in\mathrm{Im}\,f,$

where $\mathrm{Im}\,f$ depends on the parameters $\alpha$ and $\beta$; see,
for instance [4, Appendix A]. If $\alpha\neq 0$, this equation has a unique
singularity at $t_{\ast}:=-\frac{\beta}{4\alpha}$. Note that, for any $t\neq
t^{*}$, $4\alpha t+\beta$ and $\widehat{U}(t)$ must have the same sign. Hence,
for any $t\neq t_{\ast}$, we can write

$w^{\prime}(t)=\pm\sqrt{\frac{\widehat{U}(t)}{4\alpha t+\beta}}.$

For instance, if $t_{0}<t_{\ast}$, we have an explicit expression for $w$
given as follows:

$w(t)=\pm\int_{t_{0}}^{t}\sqrt{\frac{\widehat{U}(s)}{4\alpha s+\beta}}\
ds,\qquad t\in(-\infty,t_{\ast}).$

Hence, we can explicitly define the function $u\colon
f^{-1}(-\infty,t_{\ast})\rightarrow\mathbb{R}$ by $u(x)=w\circ f(x)$, which is
$f$-invariant by definition. The open set $\Omega:=f^{-1}(-\infty,t_{\ast})$
is just an $f$-invariant neighborhood of the regular level set
$M_{0}:=f^{-1}(t_{0})$. $\square$

######  Example 2.2.

Consider the pseudo-sphere $(\mathbb{S}^{n}_{s},g_{0})$ of dimension $n\geq 3$
and signature $0\leq s<n$, and let
$f\colon\mathbb{S}^{n}_{s}\rightarrow\mathbb{R}$ be an isoparametric function
defined by a Cartan-Münzner polynomial in $\mathbb{R}^{n}_{s}$, see [4,
Section 2.2]. Thus, $f$ is transnormal and satisfies

$\langle\nabla_{g_{0}}f,\nabla_{g_{0}}f\rangle_{g_{0}}=a\circ f,\quad
a(t)=\ell^{2}(1-t^{2}),$

where $\ell\in\\{1,2,3,4,6\\}$ are the number of distinct principal curvatures
of the hypersurfaces induced by $f$. In this case, $\mathrm{Im}\,f$ is
$[-1,1]$ in the Riemannian case ($s=0$) and could be either $\mathbb{R}$,
$(-\infty,-1],(-\infty,1],[-1,\infty)$ or $[1,\infty)$ in the semi-Riemannian
case ($s>0$), see [4, Appendix B]. Let $U\in C^{\infty}(\mathbb{S}_{s}^{n})$
be $f$-invariant, so that $U=\widehat{U}\circ f$ and suppose further that
$\widehat{U}$ is positive in $(-1,1)$ and negative in
$\mathbb{R}\smallsetminus[-1,1]$. Then, by Proposition 1.1, a function
$u=w\circ f$ is a solution to the eikonal equation (2) in $\mathbb{S}_{s}^{n}$
if and only if $w$ satisfies the equation

$\ell^{2}(1-t^{2})|w^{\prime}(t)|^{2}=\widehat{U}(t),\qquad
t\in\mathrm{Im}\,f.$ (12)

Observe that this equation has one or two singularities at $t=\pm 1$,
corresponding to the focal manifolds $M_{\pm}:=f^{-1}(\pm 1)$, where,
possibly, one is empty, depending on $\mathrm{Im}\,f$. We next solve
separately the cases $t\in(-1,1)$ and $t\in\mathbb{R}\smallsetminus[-1,1]$.

For $t\in(-1,1)$, consider the change of variables $t=\cos s$ and
$v(s):=w(t)$, $s\in(0,\pi)$; then $w$ solves equation (12) if and only if $v$
solves

$|v^{\prime}(s)|^{2}=\frac{1}{\ell}\widehat{U}(\cos s),\qquad s\in(0,\pi).$

As $\widehat{U}>0$ in $[-1,1]$, we can solve explicitly for any
$s_{0},s\in(0,\pi)$ and obtain

$v(s)=\frac{1}{\ell}\int_{s_{0}}^{s}\sqrt{\widehat{U}(\cos(\zeta))}d\zeta.$

Hence, $u=v\circ\arccos f$ solves (2) in $\Omega:=f^{-1}(-1,1)$ and
$M_{0}:=f^{-1}(t_{0})\subset\Omega$, where $t_{0}=\arccos s_{0}$.

Next, observe that $w(t)$ is a solution to (12) for $t\in(1,\infty)$ if and
only if $\widetilde{w}(t)=w(-t)$ is a solution to the same equation for
$t\in(-\infty,1)$. So, it suffices to solve equation (12) for $t>1$. To do
this, consider the change of variables $t=\cosh s$ and let $v(s):=w(t)$,
$s\in(0,\infty)$. Then $w$ solves (12) in $(1,\infty)$ if and only if $v$
solves

$|v^{\prime}(s)|^{2}=-\frac{1}{\ell^{2}}\widehat{U}(\cosh s),\qquad
s\in(0,\infty).$

As $\widehat{U}<0$ in $(1,\infty)$, this equation can be solved explicitly for
any $s_{0},s\in(0,\infty)$, giving that

$v(s)=\frac{1}{\ell}\int_{s_{0}}^{s}\sqrt{-\widehat{U}(\cosh\zeta)}\;d\zeta.$

This yields the $f$-invariant solution $u:=v\circ\operatorname{arccosh}f$,
defined in $\Omega:=f^{-1}(1,\infty)$, which is constant on the hypersurface
$M_{0}=f^{-1}(t_{0})$, with $t_{0}:=\operatorname{arccosh}s_{0}$. $\square$

######  Example 2.3.

Let $(M,g)$ be a closed Riemannian manifold and let $\Gamma$ be a closed
subgroup of $\mathrm{Isom}(M,g)$. By the theory of cohomogeneity one actions
(Cf. [5, Section 4]), there are exactly two singular orbits $\Gamma x_{1}$ and
$\Gamma x_{2}$, for some $x_{1},x_{2}\in M$. If $\mathrm{dist}_{g}$ denotes
the geodesic distance in $M$, the distance function
$d_{\Gamma}:M\rightarrow\mathbb{R}$ given by

$d_{\Gamma}(x):=\mathrm{dist}_{g}(x,\Gamma x_{1})$

is constant along each $\Gamma$-orbit and satisfies

$\langle\nabla_{g}d_{\Gamma},\nabla_{g}d_{\Gamma}\rangle_{g}=1,\quad\text{in
}M\smallsetminus\mathcal{Z},\quad\text{and}\quad\mathrm{Im}\,d_{\Gamma}=[0,\delta_{\Gamma}]$

with $\mathcal{Z}:=\Gamma x_{1}\cup\Gamma x_{2}$ and
$\delta_{\Gamma}:=\mathrm{dist}_{g}(\Gamma x_{1},\Gamma x_{2})$. Hence, for
any smooth and positive $\Gamma$-invariant function
$U:M\rightarrow\mathbb{R}$, by Proposition 1.1, a $\Gamma$-invariant function
$u:M\smallsetminus\mathcal{Z}\rightarrow\mathbb{R}$ of the form $u=w\circ
d_{\Gamma}$ is a solution to the eikonal equation

$|\nabla_{g}u|_{g}=U,\qquad\text{on }M\smallsetminus\mathcal{Z}$ (13)

if and only if $w$ is a solution to the problem

$|w^{\prime}(t)|^{2}=\widehat{U}(t)\qquad\text{in }(0,\delta_{\Gamma}).$

where $\widehat{U}:[0,\delta_{\Gamma}]\rightarrow\mathbb{R}$ is smooth and
such that $U=\widehat{U}\circ d_{\gamma}$. As $U>0$, $\widehat{U}$ is also
positive and for any $t_{0},t\in(0,\delta_{\Gamma})$, the solution to this
problem is given explicitly by

$w(t):=\int_{t_{0}}^{t}\sqrt{\widehat{U}(\zeta)}\;d\zeta.$

Thus, the function $u:=w\circ d_{\Gamma}$ is a $\Gamma$-invariant function
satisfying equation (13) in
$d_{\Gamma}^{-1}(0,\delta_{\Gamma})=M\smallsetminus\mathcal{Z}$. $\square$

######  Example 2.4.

Consider the Riemannian hyperbolic space $\mathbb{H}^{n}$ with its canonical
metric $g_{0}$. Following Miyaoka’s example given in [9, Theorem 1.5], we
construct a transnormal, non-isoparametric function with foliation
$\mathcal{F}_{f}$. To this end, take any totally geodesic $\mathbb{H}^{n-1}$
in $\mathbb{H}^{n}$ and let $M_{0}$ be a slight deformation of
$\mathbb{H}^{n-1}$ in such a way that the principal curvatures
$\lambda_{i}\colon M_{0}\rightarrow\mathbb{R}$ are not constant and
$|\lambda_{i}|<1$, $i=1,\ldots,n-1.$ Then, there exists a transnormal function
$f\colon\mathbb{H}^{n-1}\rightarrow\mathbb{R}$ such that
$M_{0}\in\mathcal{F}_{f}$, $\mathrm{Im}\,f=\mathbb{R}$, all the leaves of
$\mathbb{F}_{f}$ are regular level sets of $f$ and

$|\nabla_{g_{0}}f(x)|_{g_{0}}=1\qquad\text{for every }x\in\mathbb{H}^{n}.$

Hence, as in Example 2.3, if $M_{0}=f^{-1}(t_{0})$ for some
$t_{0}\in\mathbb{R}$, the function
$u\colon\mathbb{H}^{n}\rightarrow\mathbb{R}$ given by $u:=w\circ f$, where

$w(t):=\int_{t_{0}}^{t}\sqrt{\widehat{U}(\zeta)}d\zeta$

is a globally defined $f$-invariant solution to the eikonal equation

$|\nabla_{g_{0}}u|_{g_{0}}=U\qquad\text{in }\mathbb{H}^{n},$

for any $f$-invariant function $U=\widehat{U}\circ f$, with
$\widehat{U}\colon\mathbb{R}\rightarrow\mathbb{R}$ positive. In particular,
$u$ is constant on $M_{0}$ and $M_{0}$ is not isoparametric, because
isoparametric hypersurfaces in $\mathbb{H}^{n}$ have constant principal
curvatures, see, for instance [1, Corollary 3.7]. $\square$

_Proof of Corollary 1.3._ Just fix $s=0$ in Example 2.2 and take $U\equiv c>0$
in Examples 2.2, 2.3 and 2.4. $\square$

##  3 The two dimensional reduction method

###  3.1 Some computations on warped products

In this section, we prove Proposition 1.4, beginning with some elementary
computations for the warped product manifold
$(L^{s}\times_{\alpha}N^{k},g=-g_{L}+\alpha^{2}g_{N}).$ First, notice that in
local coordinates, the metric $g$ and its inverse have the following
expressions

$g_{ij}=\begin{cases}-(g_{L})_{ij},&1\leq i,j\leq s,\\\
\alpha^{2}(g_{N})_{ij},&s+1\leq i,j\leq s+k,\\\
0,&\text{otherwise},\end{cases}$ (14)

and

$g^{ij}=\begin{cases}-g_{L}^{ij},&1\leq i,j\leq s,\\\
\alpha^{-2}g_{N}^{ij},&s+1\leq i,j\leq s+k,\\\
0,&\text{otherwise},\end{cases}$ (15)

respectively. Now, recall that the gradient of a function $u$ is given in
coordinates $y=(x,z)$ as

$\nabla_{g}u=\sum_{j}\left(\sum_{k}g^{jk}\partial_{y_{k}}u\right)\partial_{y_{j}}.$

We have the following computations for the gradient and its interior products.

######  Lemma 3.1.

For any $u\in C^{\infty}(L\times N)$ we have that

$\displaystyle\nabla_{g}u$
$\displaystyle=-\nabla_{g_{L}}u+\alpha^{-2}\nabla_{g_{N}}u,$
$\displaystyle\langle\nabla_{g}u,\nabla_{g}u\rangle_{g}$
$\displaystyle=-|\nabla_{g_{L}}u|^{2}_{g_{L}}+\alpha^{-2}|\nabla_{g_{N}}u|^{2}_{g_{N}}.$

######  Proof 3.2.

For a point $y=(x,z)$ in a coordinate chart, from (14) and (15), we have that

$\displaystyle\nabla_{g}u$
$\displaystyle=\sum_{j}^{s+k}\left(\sum_{i}^{s+k}g^{ji}\partial_{y_{i}}u\right)\partial_{y_{j}}$
$\displaystyle=\sum_{j=1}^{s}\left(\sum_{i=1}^{s+k}g^{ji}\partial_{y_{i}}u\right)\partial_{x_{j}}+\sum_{j=s+1}^{s+k}\left(\sum_{i=1}^{s+k}g^{ji}\partial_{y_{i}}u\right)\partial_{z_{j}}$
$\displaystyle=\sum_{j=1}^{s}\left(\sum_{i=1}^{s}-g_{L}^{ji}\partial_{x_{i}}u\right)\partial_{x_{j}}+\sum_{j=s+1}^{s+k}\left(\sum_{i=s+1}^{s+k}\alpha^{-2}g_{N}^{ji}\partial_{z_{i}}u\right)\partial_{z_{j}}$
$\displaystyle=-\nabla_{g_{L}}u+\alpha^{-2}\nabla_{g_{N}}u.$

Since $\nabla_{g_{L}}u\in\Gamma(TL)$ and $\nabla_{g_{N}}u\in\Gamma(TN)$, if
$\nabla_{g_{L}}u=\sum_{i=1}^{s}a_{i}\partial_{x_{i}}$ and
$\nabla_{g_{N}}u=\sum_{i=s+1}^{s+k}b_{i}\partial_{z_{i}}$, for some smooth
coefficients $a_{i}$ and $b_{i}$, then

$\displaystyle\langle\nabla_{g}u,$
$\displaystyle\nabla_{g}u\rangle_{g}=\langle-\nabla_{g_{L}}u+\alpha^{-2}\nabla_{g_{N}}u,-\nabla_{g_{L}}u+\alpha^{-2}\nabla_{g_{N}}u\rangle$
$\displaystyle=\langle\nabla_{g_{L}}u,\nabla_{g_{L}}u\rangle_{g}+2\langle-\nabla_{g_{L}}u,\alpha^{-2}\nabla_{g_{N}}u\rangle+\langle\alpha^{-2}\nabla_{g_{N}}u,\alpha^{-2}\nabla_{g_{N}}u\rangle$
$\displaystyle=\left\langle\sum_{i=1}^{s}a_{i}\partial_{x_{i}},\sum_{i=1}^{s}a_{i}\partial_{x_{i}}\right\rangle_{g}+\alpha^{-4}\left\langle\sum_{i=s+1}^{s+k}b_{i}\partial_{z_{i}},\sum_{i=s+1}^{s+k}b_{i}\partial_{z_{i}}\right\rangle_{g}$
$\displaystyle=\sum_{i,j=1}^{s}a_{i}a_{j}\langle{\partial}_{x_{i}},{\partial}_{x_{j}}\rangle_{g}+\alpha^{-4}\sum_{i,j=s+1}^{s+k}b_{i}b_{j}\langle{\partial}_{z_{i}},{\partial}_{z_{j}}\rangle_{g}$
$\displaystyle=-|\nabla_{g_{L}}u|^{2}+\alpha^{-2}|\nabla_{g_{N}}u|^{2}.$

For later use, we compute the norms of the gradients of $(f_{L}\times
f_{N})$-invariant functions, not necessarily transnormal. In what follows,
$t:=f_{L}(x),s:=f_{N}(z).$

######  Lemma 3.3.

Let $w\in C^{1}(\mathbb{R}^{2})$ and let $f_{L}\colon L\rightarrow\mathbb{R}$
and $f_{N}\colon N\rightarrow\mathbb{R}$ be smooth. If $u:=w(f_{L},f_{N})$,
then

$|\nabla_{g_{L}}u|^{2}_{g_{L}}=w_{t}^{2}(f_{L},f_{N})|\nabla_{g_{L}}f_{L}|^{2}_{g_{L}},\quad\text{and}\quad|\nabla_{g_{N}}u|^{2}_{g_{N}}=w_{s}^{2}(f_{L},f_{N})|\nabla_{g_{L}}f_{L}|^{2}_{g_{L}}.$

######  Proof 3.4.

In local coordinates $y=(x,z)$, the differential of $u=w(f_{L},f_{N})$ is
given by

$\displaystyle d(u)$
$\displaystyle=d(w\circ(f_{L},f_{N}))=dw(f_{L},f_{N})\circ d(f_{L}\times
f_{N})$ $\displaystyle=dw\circ(df_{L}\times df_{N})$
$\displaystyle=\big{(}w_{t}(f_{L},f_{N}),w_{s}(f_{L},f_{N})\big{)}\begin{bmatrix}\nabla_{g_{L}}f_{L}&0\\\
0&\nabla_{g_{N}}f_{N}\end{bmatrix}$
$\displaystyle=\big{(}w_{t}(f_{L},f_{N})\nabla_{g_{L}}f_{L},w_{s}(f_{L},f_{N})\nabla_{g_{N}}f_{N}\big{)}.$

Hence

$\partial_{x_{i}}u=w_{t}(f_{L},f_{N})\partial_{x_{i}}f_{L},\quad\text{and}\quad\partial_{z_{i}}u=w_{s}(f_{L},f_{N})\partial_{z_{i}}f_{N}.$

Therefore,

$\displaystyle\nabla_{g_{L}}(w(f_{L},f_{N}))$
$\displaystyle=\sum_{j=1}^{s}\left(\sum_{i=1}^{s}g_{L}^{ij}\partial_{x_{i}}(w(f_{L},f_{N}))\right)\partial_{x_{j}}$
$\displaystyle=\sum_{j=1}^{s}\left(\sum_{i=1}^{s}g_{L}^{ij}w_{t}(f_{L},f_{N})\partial_{x_{i}}f_{L}\right)\partial_{x_{j}}$
$\displaystyle=w_{t}(f_{L},f_{N})\sum_{j=1}^{s}\left(\sum_{i=1}^{s}g_{L}^{ij}\partial_{x_{i}}f_{L}\right)\partial_{x_{j}}$
$\displaystyle=w_{t}(f_{L},f_{N})\nabla_{g_{L}}f_{L},$

where we conclude that

$\displaystyle|\nabla_{g_{L}}u|^{2}$ $\displaystyle=\langle
w_{t}(f_{L},f_{N})\nabla_{g_{L}}f_{L},w_{t}(f_{L},f_{N})\nabla_{g_{L}}f_{L}\rangle_{g_{L}}$
$\displaystyle=w_{t}^{2}(f_{L},f_{N})|\nabla_{g_{L}}f_{L}|^{2}_{g_{L}}.$

Similar computations give

$\nabla_{g_{N}}u=w_{s}(f_{L},f_{N})\nabla_{g_{N}}f_{N}\quad\text{ and
}\quad|\nabla_{g_{L}}u|^{2}=w_{s}^{2}(f_{L},f_{N})|\nabla_{g_{L}}f_{L}|^{2}_{g_{L}},$

as we wanted.

Now suppose that $f_{L}\colon L\to\mathbb{R}$ and $f_{N}\colon N\to\mathbb{R}$
are transnormal functions satisfying

$|\nabla_{g_{L}}f_{L}|_{g_{L}}^{2}=a_{L}\circ
f_{L}\quad\text{and}\quad|\nabla_{g_{N}}f_{N}|_{g_{N}}^{2}=a_{N}\circ f_{N},$

for some smooth functions $a_{L},a_{N}\colon\mathbb{R}\to\mathbb{R}$. We have
the following immediate consequence of the previous two lemmas.

######  Corollary 3.5.

For $f_{L},f_{N}$ transnormal, $u=(f_{L},f_{N})$ and
$\alpha=\widehat{\alpha}\circ f_{L}$,

$\langle\nabla_{g}u,\nabla_{g}u\rangle_{g}=-w_{t}^{2}(f_{L},f_{N})a_{L}\circ
f_{L}+\frac{a_{N}\circ f_{N}}{\widehat{\alpha}^{2}\circ
f_{L}}w_{s}^{2}(f_{L},f_{N}).$

_Proof of Proposition 1.4_. Using the identity in Corollary 3.5, the proof is
straightforward. $\square$

###  3.2 Reduction using distance functions

In this subsection, we simplify (9) via distance functions, and prove Theorem
1.5. By the theory of transnormal functions for Riemannian manifolds (see [9]
and [12]), the connected components of the critical sets of $f_{L}$ and
$f_{N}$ are submanifolds of $L$ and $N$ respectively, called _focal
submanifolds_. There are, at most, two focal submanifolds for a given
transnormal function, and, in some cases, this set is empty. In what follows,
we will suppose further that $f_{L}$ and $f_{N}$ admit, at least, one focal
manifold, say $L_{0}$ and $N_{0}$, and we will denote by $L_{1}$ and $N_{1}$
the second, possibly empty, focal submanifold. In this case, we can define the
distance functions $d_{L}\colon L\rightarrow\mathbb{R}$ and $d_{N}\colon
N\rightarrow\mathbb{R}$ given by

$d_{L}(x):=\mathrm{dist}_{g_{L}}(x,L_{0})\quad\text{and}\quad
d_{N}(y):=\mathrm{dist}_{g_{N}}(y,N_{0}).$

Let $\delta_{L},\delta_{N}\in(0,\infty]$ be the supremum of these functions in
$L$ and $N$, respectively. The case of infinite distance corresponds to the
non-compact case of $L$ or $N$, while, in the case of compact manifolds,
$\delta_{L}=\mathrm{dist}_{g_{L}}(L_{1},L_{0})$ and
$\delta_{N}=\mathrm{dist}_{g_{N}}(N_{1},N_{0})$. Moreover, for any $x\in
L\smallsetminus(L_{0}\cup L_{1})$ and any $z\in N\smallsetminus(N_{0}\cup
N_{1})$, the distance functions satisfy

$\langle\nabla_{g_{L}}d_{L}(x),\nabla_{g_{L}}d_{L}(x)\rangle_{g_{L}}=1\quad\text{and}\quad\langle\nabla_{g_{N}}d_{N}(z),\nabla_{g_{N}}d_{N}(z)\rangle_{g_{N}}=1$

and, so, they are transnormal in $L\smallsetminus(L_{0}\cup L_{1})$ and in
$N\smallsetminus(N_{0}\cup N_{1})$ respectively, with $a_{d_{L}}=1$ and
$a_{d_{N}}=1$, and $I_{d_{L}}=[0,\delta_{L}]$ and $I_{d_{N}}=[0,\delta_{N}]$.

Now, observe that $d_{L}$ and $f_{L}$ have the same level sets and, hence,
$f_{L}$-invariant functions are $d_{L}$-invariant functions and vice-versa.
The same is true for $d_{N}$ and $f_{N}$. In fact, $d_{L}$ and $d_{N}$ are
just reparametrizations of $f_{L}$ and $f_{N}$. For this reason, and without
loss of generality, we will consider directly that $F$ is $(d_{L}\times
d_{N})$-invariant in its first variable and that $\alpha$ is
$d_{L}$-invariant. Moreover, since $\mathcal{F}_{f_{L}}=\mathcal{F}_{d_{L}}$
and $\mathcal{F}_{f_{N}}=\mathcal{F}_{d_{N}}$, the one parameter families of
hypersurfaces satisfy that $\Sigma_{f_{L}}=\Sigma_{d_{L}}$ and
$\Sigma_{f_{N}}=\Sigma_{d_{N}}$.

_Proof of Theorem 1.5._ By the previous remarks, it suffices to prove the
result in the $d_{L}$ and $d_{N}$-invariant setting, that is, suppose that $F$
is $(d_{L}\times d_{N})$-invariant in the first variable, and let
$\widehat{F}\colon I_{d_{L}}\times
I_{d_{N}}\times\mathbb{R}^{2}\rightarrow\mathbb{R}$ such that
$F=\widehat{F}(d_{L}\times d_{N},\cdot,\cdot)$ in $L\times
N\times\mathbb{R}^{2}$. In the same way, as $\alpha$ is $f_{L}$-invariant, it
is $d_{L}$ invariant and $\alpha=\widehat{\alpha}\circ d_{L}$ in
$L\smallsetminus(L_{0}\cup L_{1})$. Accordingly, we consider the one parameter
families $\Sigma_{d_{L}}$ and $\Sigma_{d_{N}}$ such that
$\Sigma_{d_{L}}(0)=L_{0}=d_{L}^{-1}(t_{0})$ and
$\Sigma_{d_{N}}(0)=N_{0}=d_{L}^{-1}(s_{0})$, for regular values
$t_{0}\in(0,\delta_{L})$ and $s_{0}\in(0,\delta_{N})$ of $d_{L}$ and $d_{N}$,
respectively. To simplify notation, let $T:=C_{d_{L}}\circ\Sigma_{d_{L}}$ and
$S:=C_{d_{N}}\circ\Sigma_{d_{N}}$.

Since $d_{L}$ and $d_{N}$ are transnormal in $L\smallsetminus(L_{0}\cup
L_{1})$ and $N\smallsetminus(N_{0}\cup N_{1})$, respectively, by Proposition
1.4, to solve equation (1) with $(d_{L}\times d_{N})$-invariance, it suffices
to find $w\in C^{1}\left((0,\delta_{L})\times(0,\delta_{N})\right)$ solving
the reduced equation

$\widehat{F}\left(t,s,w(t,s),-w_{t}^{2}(t,s)+\frac{1}{\widehat{\alpha}^{2}(t)}w_{s}^{2}(t,s)\right)=0,\quad\emph{\text{
in }}\ (0,\delta_{L})\times(0,\delta_{N}).$

Observe this equation can be written as a first order fully nonlinear PDE of
the form

$H(t,s,w,w_{t},w_{s})=0\qquad\text{ in }(0,\delta_{L})\times(0,\delta_{N}),$

where
$H\colon(0,\delta_{L})\times(0,\delta_{N})\times\mathbb{R}^{3}\to\mathbb{R}$
is given by

$H(t,s,r,p,q)=\widehat{F}\left(t,s,r,-p^{2}+\widehat{\alpha}^{-2}(t)q^{2}\right).$

As

$\frac{\partial H}{\partial
p}(t,s,r,p,q)=2p\frac{\partial\widehat{F}}{\partial\tau}(t,s,r,-p^{2}+\widehat{\alpha}^{-2}q^{2})$

and

$\frac{\partial H}{\partial
q}(t,s,r,p,q)=2q\frac{\partial\widehat{F}}{\partial\tau}(t,s,r,-p^{2}+\widehat{\alpha}^{-2}q^{2}),$

then, for any $(x_{0},z_{0})\in L_{0}\times N_{0}$, by the hypotheses (10),
(11), _$(\Sigma.2)$ _ and _$(\Sigma.3)$ _, we have that

$H(t_{0},s_{0},r_{0},p_{0},q_{0})=0,\qquad
p_{0}T^{\prime}(0)+q_{0}S^{\prime}(0)=R^{\prime}(0),$

that

$T^{\prime}(0)\frac{\partial H}{\partial
p}(t_{0},s_{0},r_{0},p_{0},q_{0})-S^{\prime}(0)\frac{\partial H}{\partial
q}(t_{0},s_{0},r_{0},p_{0},q_{0})\neq 0,$

and that

$\left(\frac{\partial H}{\partial
p}\right)^{2}(t,s,r,p,q)+\left(\frac{\partial H}{\partial
q}\right)^{2}(t,s,r,p,q)\neq 0$

in a small neighborhood of $(t_{0},s_{0},r_{0},p_{0},q_{0})$. Hence, by the
standard theory of first-order fully nonlinear equations (see, for instance,
[11, Section 4.6]), the Cauchy problem

$\begin{cases}\widehat{F}\left(t,s,w(t,s),w_{t}^{2}(t,s)+\widehat{\alpha}^{2}w_{s}^{2}(t,s)\right)=0,\\\
w(T(\zeta),S(\zeta))=R(\zeta),\end{cases}$

has a unique solution $w$ of class $C^{1}$ defined in a small neighborhood of
$(t_{0},s_{0})$, and again, using Proposition 1.4, the function
$u:=w(d_{L},d_{N})$ is $(d_{L}\times d_{N})$-invariant and satisfies equation
(8) in a small $(f_{L}\times f_{N})$-invariant neighborhood of $L_{0}\times
N_{0}$, as we wanted to prove. $\square$

To conclude this paper, let us analyze a simple application of the method in
case of the de Sitter space.

######  Example 3.6.

Let $n\geq 3$ and let $(\mathbb{S}^{n},g_{0})$ be the sphere with the usual
metric and consider the de Sitter space

$(M,g)=(\mathbb{R}\times\mathbb{S}^{n},dt^{2}+\cosh^{2}t\,g_{0}).$

For $n_{1},n_{2}\in\mathbb{N}$ such that $n_{1}+n_{2}=n+1,\,n_{1},n_{2}\geq
2$, take

$\mathbb{S}^{n}\subset\mathbb{R}^{n+1}=\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}}$

and for $z=(z_{1},z_{2})\in\mathbb{S}^{n}$ define the function

$f(z)=\arccos(|z_{1}|^{2}-|z_{2}|^{2}).$

This function is transnormal and satisfies that $\mathrm{Im}\,f=[0,\pi]$. The
sets

$f^{-1}(0)=\mathbb{S}^{n_{1}-1}\times\\{0\\}\quad\text{and}\quad
f^{-1}(\pi)=\\{0\\}\times\mathbb{S}^{n_{2}-1}$

are the focal submanifolds, while

$f^{-1}(s)=\mathbb{S}^{n_{1}-1}_{\cos s}\times\mathbb{S}^{n_{2}-1}_{\sin s}$

are regular hypersurfaces for $s\in(0,\pi)$ and

$|\nabla_{g_{0}}f|^{2}_{g_{0}}=4,\quad\text{ in }\mathbb{S}^{n}\smallsetminus
f^{-1}\\{0,\pi\\}.$

For any $(t_{0},z_{0})\in\mathbb{R}\times\left(\mathbb{S}^{n}\smallsetminus
f^{-1}\\{0,\pi\\}\right),$ and any smooth function
$\widehat{U}\colon\mathbb{R}\times[0,\pi]\to\mathbb{R}$, we will show next
that the eikonal equation on $M$

$\langle\nabla_{g}u,\nabla_{g}u\rangle=\widehat{U}(t,z),$ (16)

has a solution which is $f$-invariant in the $z$-variable and is defined in a
neighborhood of $\\{t_{0}\\}\times\mathbb{S}_{\cos
s_{0}}^{n_{1}-1}\times\mathbb{S}_{\sin s_{0}}^{n_{2}-1}$ for suitable one-
parameter families $\Sigma_{\mathbb{R}}$ and $\Sigma_{\mathbb{S}^{n}}$, where
$s_{0}=f(z_{0})$.

First, notice that $\mathrm{Id}\colon\mathbb{R}\to\mathbb{R}$ is transnormal,
since $|\nabla\mathrm{Id}(t)|=1$ for any $t\in\mathbb{R}$. So the function
defining the eikonal equation (16) is just

$F(t,z,r,\tau)=\tau-U(t,z).$

This function is $(\mathrm{Id}\times f)$-invariant on the $(t,z)$-variables.
Therefore, by Proposition 1.4, any function of the form
$u=w\circ(\mathrm{Id}\times f)$ solving (16) must satisfy the equation

$-w_{t}^{2}(t,s)+4\cosh^{-2}(t)w_{s}^{2}(t,s)=\widehat{U}(t,s)$ (17)

in $\mathbb{R}\times[0,\pi]$ which can be written as

$\widehat{F}\left(t,s,w(t,s),-w_{t}^{2}(t,s))+4\cosh^{-2}(t)w_{s}^{2}(t,s)\right)=0$

where $\widehat{F}(t,s,w,\tau)=\tau-\widehat{U}(t,s)$. We proceed to solve
this equation with the theory of fully nonlinear equations; cf. [11].

Fix $(t_{0},z_{0})\in\mathbb{R}\times\mathbb{S}^{n}\smallsetminus
f^{-1}(\\{0\\})$ and let $s_{0}=f(z_{0}).$ Take
$(p_{0},q_{0})\in\mathbb{R}^{2}\smallsetminus\\{(0,0)\\}$ such that

$-p_{0}^{2}+4\cosh^{-2}(t_{0})q_{0}^{2}-\widehat{U}(t_{0},s_{0})=0.$ (18)

Notice that such $(p_{0},q_{0})\neq(0,0)$ exists because (18) defines a
hyperbola in the $(p_{0},q_{0})$-variables.

Then, the function
$H\colon\mathbb{R}\times[0,\pi]\times\mathbb{R}^{3}\to\mathbb{R}$ given by

$\displaystyle H(t,s,r,p,q)$
$\displaystyle=\widehat{F}\left(t,s,r,-p^{2}+4\cosh^{-2}(t)q^{2}\right)$
$\displaystyle=-p^{2}+4\cosh^{-2}(t)q^{2}-\widehat{U}(t,s)$

is used to rewrite (17) as

$H(t,s,w(t,s),w_{t}(t,s),w_{s}(t,s))=0.$

So, for any $r_{0}\in\mathbb{R}$, the point
$(t_{0},z_{0},r_{0},p_{0},q_{0})\in\mathbb{R}\times\left(\mathbb{S}^{n}\smallsetminus
f^{-1}(\\{0,\pi\\})\right)\times\mathbb{R}^{3}$ is such that
$F(t_{0},z_{0},\tau_{0})=0,$ where
$\tau_{0}=-p_{0}^{2}+4\cosh^{-2}(t_{0})q_{0}^{2},$ and, therefore

$H(t_{0},s_{0},r_{0},p_{0},q_{0})=0.$

Since $\frac{\partial F}{\partial\tau}\equiv 1\neq 0,$

$H_{p}(t,s,r,p,q)=-2p,\quad H_{q}(t,s,r,p,q)=8\cosh^{-2}(t)q,$

and the nontriviality hypothesis

$H_{p}^{2}+H_{q}^{2}=4p^{2}+64\cosh^{-2}(t)q^{2}$

is satisfied for any $(p,q)\neq(0,0)$.

For $\varepsilon>0$, let
$T,S\colon(-\varepsilon,\varepsilon)\to\mathbb{R}^{3}$ be such that

$(T(\zeta),S(\zeta))\in\mathbb{R}\times(0,\pi),\quad\zeta\in(-\varepsilon,\varepsilon),$

with

$T(0)=t_{0},\qquad S(0)=s_{0}\quad\text{ and }\quad
T^{\prime}(0)q_{0},S^{\prime}(0)p_{0}>0.$

Then, for any smooth function $R\colon(-\varepsilon,\varepsilon)\to\mathbb{R}$
satisfying

$R^{\prime}(0)=p_{0}T^{\prime}(0)+q_{0}S^{\prime}(0),$

the initial value problem

$\begin{cases}H(t,s,w(t,s),w_{t}(t.s),w_{s}(t,s))=0,\\\
w(T(\zeta),S(\zeta))=R(\zeta),\end{cases}$

admits a $C^{2}$ solution in a neighborhood of $(t_{0},s_{0}).$ Defining
$\Sigma_{\mathrm{Id}}(\zeta):=\\{T(\zeta)\\}$ and
$\Sigma_{f}(\zeta):=f^{-1}(S(\zeta))\,$ we have that

$\Sigma_{\mathrm{Id}}(\zeta)\circ
C_{\mathrm{Id}}=T,\quad\Sigma_{f}(\zeta)\circ C_{f}=S,$

and $u=w\circ(\mathrm{Id}\times f)$ is a $(\mathrm{Id}\times f)$-invariant
solution defined in a neighborhood of $\\{t_{0}\\}\times\mathbb{S}_{\cos
s_{0}}^{n_{1}-1}\times\mathbb{S}_{\sin s_{0}}^{n_{2}-1}$ of the form

$\Omega=(t_{0}-\varepsilon_{1},t_{0}+\varepsilon_{1})\times
f^{-1}(s_{0}-\varepsilon_{2},s_{0}+\varepsilon_{2}),$

for some $\varepsilon_{1},\varepsilon_{2}>0$ small enough. Moreover, $u$
satisfies that

$u(\Sigma_{Id}(\zeta),\Sigma_{f}(\zeta))=R(\zeta)\quad\text{ in
}(-\varepsilon_{3},\varepsilon_{3}),$

for some $\varepsilon_{3}>0$ small enough. $\square$

######  Remark 3.7.

More examples where this method applies are given by considering warped
products of Riemannian space forms of the form $L\times_{\alpha}N$, where $L$
is either the Euclidean space $\mathbb{R}^{s}$ or the hyperbolic space
$\mathbb{H}^{s}$, and $N$ may be chosen as $\mathbb{R}^{k}$, $\mathbb{H}^{k}$
or the sphere $\mathbb{S}^{k}$. As mentioned before, the case of the
hyperbolic space is of particular interest due to the existence of transnormal
functions having non isoparametric foliations [9], see Example 2.4 above.
$\square$

## References

  * [1] T. E. Cecil and P. J. Ryan.  Geometry of Hypersurfaces.  Springer Monographs in Mathematics. Springer New York Heidelberg Dordrecht London, 2015.  doi:0.1007/978-1-4939-3246-7.
  * [2] M. Clapp, J. C. Fernández, and A. Saldaña.  Critical polyharmonic systems and optimal partitions.  Commun. Pure Appl. Anal., 20(11):4007–4023, 2021.  doi:10.3934/cpaa.2021141.
  * [3] J.C. Fernández and J. Petean.  Low energy nodal solutions to the Yamabe equation.  J. Differential Equations, 268(11):6576–6597, 2020.  doi:10.1016/j.jde.2019.11.043.
  * [4] J. C. Fernández and O. Palmas.  An ODE reduction method for the semi-Riemannian Yamabe problem on space forms.  Nonlinear Analysis, 209:112342, 2021.  URL: https://www.sciencedirect.com/science/article/pii/S0362546X21000626, doi:10.1016/j.na.2021.112342.
  * [5] J. C. Fernández, O. Palmas, and J. Torres Orozco.  Equivariant solutions to the optimal partition problem for the prescribed Q-curvature equation.  J. Geom. Anal., 34, 2024.  doi:10.1007/s12220-024-01554-4.
  * [6] H.-C. Grunau, M. O. Ahmedou, and W. Reichel.  The Paneitz equation in hyperbolic space.  Ann. I. H. Poincaré, 25:847–864, 2008.  doi:10.1016/j.anihpc.2007.05.001.
  * [7] J. Hahn.  Isoparametric hypersurfaces in the pseudo-Riemannian space forms.  Math. Z., 187:195–208, 1984.  doi:10.1007/BF01161704.
  * [8] M.-A. Lawn and M. Ortega.  Translating solitons in a Lorentzian setting, submersions and cohomogeneity one actions.  Mediterr. J. Math., 19, 2022.  doi:10.1007/s00009-022-02020-7.
  * [9] R. Miyaoka.  Transnormal functions on a Riemannian manifold.  Differential Geom. Appl., 31:130–139, 2013.
  * [10] M. Ortega and H. Yildirim.  Rotationally invariant translators of the mean curvature flow in Einstein’s static universe.  Differential Geometry and its Applications, 95:102153, 2024.  URL: https://www.sciencedirect.com/science/article/pii/S0926224524000469, doi:10.1016/j.difgeo.2024.102153.
  * [11] S. Salsa.  Partial Differential Equations in Action: From modelling to Theory.  Universitext. Springer-Verlag Italia, 2008.
  * [12] Q. M. Wang.  Isoparametric functions on Riemannian manifolds I.  Math. Ann., 277:639–646, 1987.