# Hydrodynamic models of astrophysical wormholes.
The general concept

Semyon Churilov1, Yury Stepanyants2,3 1Institute of Solar-Terrestrial Physics
of the Siberian Branch of Russian Academy of Sciences, Irkutsk-33, PO Box 291,
664033, Russia, e-mail<EMAIL_ADDRESS>
2School of Sciences, University of Southern Queensland, QLD 4350, Australia;
3Department of Applied Mathematics, Nizhny Novgorod State Technical
University, Nizhny Novgorod, 603950, Russia, e-mail:
<EMAIL_ADDRESS>
###### Abstract

We study hydrodynamic models of astrophysical wormholes when water waves can
be amplified in the course of propagation between two critical points where
wave and current speeds coincide. Such models can be realised in shallow
laboratory ducts with variable width and depths. In this paper, we derive the
basic set of equations for shallow water waves on the spatially variable flow
in the duct of a variable cross-section and present the asymptotic analysis of
solutions in the neighbourhood of the critical points. The critical points
mimic either the black hole (BH) horizon if the flow transits from the
subcritical to the supercritical regime, or the white hole (WH) horizon if the
flow transits from the supercritical to the subcritical regime. We study then
the wave propagation in the flow with two horizons when the flow transits
first the BH horizon and then the WH one. The region between the horizons
mimics a wormhole in general relativity. For the sake of completeness, we also
study the successive transition through WH and BH horizons. The theoretical
results are illustrated by numerical calculations of wave propagation in both
these arrangements. It is shown that the wave amplification after passing the
active zone between the horizons takes place in BH–WH arrangements only and
can occur for different relationships between the subcritical and
supercritical flow velocities. The frequency dependence of the amplification
factor is obtained and quantified in terms of the velocity ratio within and
outside the “wormhole domain”. In the next paper, we plan to present exact
solutions for the specific velocity profile.

###### pacs:

Valid PACS appear here

##  I INTRODUCTION

Almost all scalar wave phenomena that exist in different media can be also
reproduced or modelled by means of water waves. Moreover, the majority of wave
effects can be observed in water by naked eyes. Since that time when Unruh
Unruh81  demonstrated the similarity between the equations of general
relativity describing wave phenomena in the proximity of black holes and
acoustic waves in non-uniform flows, the study of such processes was developed
into a whole scientific direction called the analog gravity (see, e.g. ArtBH ;
AnalogGrav ; Unruh18  and references therein). One of the problems studied
within this direction is modeling the motion of massless particles in the
curved space-time containing event horizons, ergoregions and/or other regions
with special properties inherent in general relativity. The analogy between
the astrophysical phenomena and their counterparts in continuous media is
based on the fact that for some types of waves (their list is permanently
expanding), the equation describing wave propagation in a non-uniform medium
has the form of the d’Alembert equation in the pseudo-Riemannian space with
the metric tensor $G_{\mu\nu}$:

$\frac{1}{\sqrt{-G}}\,\frac{\partial}{\partial
x^{\mu}}\left(\sqrt{-G}\,G^{\mu\nu}\,\frac{\partial\varphi}{\partial
x^{\nu}}\right)=0,$ (1)

where $G=\det G_{\mu\nu}$, $G^{\mu\lambda}G_{\lambda\nu}=\delta^{\mu}_{\nu}$,
$\delta^{\mu}_{\nu}$ is the unit tensor, and Einstein’s rule of summation over
the repeated indices is applied. This fact makes it possible to simulate
various space-time structures by variation the conditions of wave propagation
in laboratory experiments. However, the aforementioned analogy is not too deep
because the dynamical equations in hydrodynamics, which include the
Navier–Stokes equation, and Einstein’s equations for space-time are not
analogues. Nevertheless, many researchers employ the kinematic analogy to
demonstrate that long water waves propagating on a flow can be treated as the
gravitational analogue for the propagation of light in the curved space-time
(see, for example, Peloquin ; Euve16 ; Euve17 ; Euve20  and references
therein).

In the pioneering paper by Unruh Unruh81 , for the modeling of a space-time
structure, sound waves were proposed in the potential barotropic flow ${\bf
U}=\nabla\Phi({\bf x},t)$, where $\Phi$ is the hydrodynamic potential of the
fluid velocity ${\bf U}$. Later it was recognised that instead of sound waves,
surface gravity waves can be used on non-uniform flows in shallow water SchUn
. Despite the fact that, over the past three decades, the arsenal of analog
gravity has expanded significantly, the study of the effects associated with
the propagation of sound and water waves continues to receive much attention.
This is confirmed, in particular, by the materials of the recently held
discussion on the current state of research in the field of analog gravity
organised under the auspices of the London Royal Society NextGen . As was
pointed out in the discussion, one of the topical problems in this field is
the necessity of a further study of the effects related to the transitions
between subcritical ($|{\bf U}|<c$) and supercritical ($|{\bf U}|>c$) flows,
where $c$ is the wave speed in the medium (sound speed or speed of long linear
waves on shallow water). The features of the wave dynamics in the regions with
a different character of flows should be also studied in detail. This was one
of the motivations of our work presented here.

Recently we constructed analytical solutions ChErSt17  for the problem of
surface wave transformation in a one-dimensional duct with a smoothly varying
water velocity along the flow direction, $U(x)>0$. All possible relationships
between the water velocity $U(x)$ and wave velocity $c$ have been studied in
that paper for linear waves in the shallow-water approximation. The
transitions of the flow velocity from the subcritical to the supercritical and
vice versa have been studied in detail on the basis of the solutions derived.
The role of positive and negative energy waves in the supercritical flow has
been also noted. However, the exact analytical solutions in Ref. ChErSt17
were based on two assumptions that impose significant restrictions on the
problem formulation. One of the assumptions is related to the piece-wise
linear velocity profile $U(x)$; this simplifies the derivation of analytical
solutions but is hardly achievable in the laboratory set-ups. Another
assumption presumes that the wave velocity $c$ is $x$-independent; such
independence is difficult to achieve in the experiment too. In particular, in
the experiments described in Refs. Euve15 ; Wein17 ; Euve20 , the water depth
$H$ (and the surface wave velocity $c=\sqrt{gH\vphantom{{}^{2}}}$ (with $g$
being the acceleration due to gravity) was not maintained constant throughout
the flow. Moreover, in some recent publications Aur15 ; Aur15a ; Aur19 , for
the simulation of the curved space-time, the new types of flows were proposed
and realised in the laboratory. In those experiments, the current velocities
$U$ remained constant, whereas the effective sound speed varied in space due
to the special construction of the upper duct boundary.

As follows from the aforementioned, to clarify the basic features of the wave-
current interaction, there is a necessity to study analytically wave
propagation in the case when both velocities, $U$ and $c$ are $x$-dependent in
a rather arbitrary way. To this end, we can employ the method originally
developed in the quantum theory of scattering (see, for example, Calo ; Bab ).
The method, in a certain sense, allows one to separate in each point of $x$
the contributions to the total disturbance of waves travelling in the opposite
directions relative to the moving medium, This makes physically clearer both
the calculations and results obtained. As in our previous paper ChErSt17 , we
will consider surface water waves on the gradually varying flow in a shallow
duct assuming that both the fluid depth $H(x)$ and the duct width $W(x)$ are
slow functions of $x$.

Being equipped with the developed powerful mathematical tool, we will tackle
the problem of the laser effect in hydrodynamics which models the wave
penetration through the wormhole in the theory of relativity. The term
“wormhole” was introduced by John Wheeler in 1950th to describe a connection
between the separate parts of the Universe where the black hole (the term
promoted by J. Wheeler) is connected by the tunnel with its counterpart called
white hole (the term introduce by Igor Novikov in 1960th). Despite the
originally suggested wormholes were found unstable, it was shown later that
their modification containing exotic matter possessing negative energy can
stabilise wormholes Morris . In such a case, a wormhole can be traversable in
both directions. The wormhole analogue in hydrodynamics can stably exist in
the portion of a duct where the fluid flow is supercritical as shown in Fig.
1.

$x_{1}$$x_{2}$$U_{1}$$U_{2}$$U_{3}$$T$$I$$R$$P$$N$$x$$z$$0$$W/2$$-W/2$$1$$3$$2$
Figure 1: (Color online) Sketch of a duct of a variable width and depth with a
non-uniform fluid flow. Positions $x_{1}$ and $x_{2}$ designate the
coordinates of black and white hole horizons, respectively, where $U(x)=c(x)$.
Region 2 mimics a model of a wormhole connecting black and white holes. For
other notations see the text.

There is no necessity in the existence of exotic matter, whereas negative
energy waves can exist in this region FabrStep . As the consequence of
existence of stable analogues of wormholes, an analogue of black hole laser
can be realised between the critical points of a current in a supercritical
region of a flow. Note that the black hole laser effect was theoretically
predicted by Corley and Jacobson Corley , see also Leonhardt . It was studied
numerically in optics Faccio  and observed experimentally in the Bose–Einstein
condensate Steinhauer . In Ref. Peloquin  the authors studied numerically
water wave packets on a spatially varying current in the presence of a surface
tension. Using the ad hoc constructed equation describing water waves, they
demonstrated that traversable and bidirectional analogue of wormholes do exist
in fluid mechanics. However, surface waves in fluid with the surface tension
are subject to dispersion which is absent in the general relativity. Therefore
the analogy between water waves and electromagnetic waves becomes too far,
although the physical effect of wave amplification remains qualitatively
similar. In our paper, we consider the amplification of shallow water waves
without a dispersion within the framework of real hydrodynamic equations. We
assume that the flow velocity $U(x)$ can be controlled by a proper variation
of the duct cross-section, both by the variation of width and depth. The water
depth variation solves the problem of a zero-frequency mode (a stationary free
surface deformation) mentioned in Peloquin . Another problem mentioned in
Peloquin  and related to the rapid damping of small-scale capillary waves is
not topical in our case which pertains to long gravity waves. Therefore, the
viscosity effects can be neglected almost everywhere in the flow except the
critical points (BH or WH horizons) where $U(x)=c$. The effect of viscosity in
such points plays an important role and is studied in detail in the Appendix.
We show that the wave amplification is directly related to the existence of
negative energy waves between the analogue horizons. In this paper, we develop
the general approach to the description of linear waves on inhomogeneous
currents and demonstrate the importance of viscous effects near the critical
points. The developed theoretical concept is illustrated by numerical
calculations with some particular profiles of the mean flow $U(x)$ and the
wave speed $c(x)$. In the next paper, we plan to present the analytical
results on wave amplification for the exactly solvable flow model with the
piece-wise linear velocity profile $U(x)$.

The article is organized as follows. The problem formulation is given in
Section II and the basic equations are also derived there. Section III is
devoted to the asymptotic analysis of solutions in the neighbourhood of the
critical points, where the flow and wave velocities coincide. Such a point is
called the BH horizon (“the black hole horizon”) if the flow transits from the
subcritical to the supercritical regime passing through the point. If the flow
transits from the supercritical regime to the subcritical regime passing
through the point, then it is called the WH horizon (“the white hole
horizon”). In Sections IV and V we study the wave propagation in the flow with
two horizons, when the flow transits from BH to WH (BH–WH duct) or from WH to
BH (WH–BH duct). Section VI contains the results of numerical calculations of
wave propagation in both these arrangements illustrating the theoretical
analysis presented in Sections IV and V. It is shown that the wave
amplification can occur when an incident wave produces a higher amplitude
transmitted wave passing through the “wormhole” between the BH and WH
horizons. The amplification factor is estimated in terms of the velocity ratio
within and outside the wormhole. Section VII is devoted to the discussion of
the results obtained. In Appendix, we present calculations demonstrating the
influence of the water viscosity on the wave transformation in the vicinity of
critical points (BH or WH horizons).

##  II GOVERNING EQUATIONS AND THE PROBLEM STATEMENT

In the physical model of the flow used in this study, the velocity of surface
wave propagation relative to moving water is $c(x)=\sqrt{gH(x)}$. To ensure
the independence of $c(x)$ and $U(x)$, we assume that the flow occurs in a
duct of the width $W$ which smoothly varies with $x$ as required by the law of
mass flux conservation,

$U(x)H(x)W(x)={\rm const}.$ (2)

In the hydrostatic approximation describing the propagation of long waves in
shallow water LL , the pressure can be represented as $p=p_{0}+\rho
g(\eta-y)$, where $p_{0}$ is the atmospheric pressure, $\rho=$ const is the
water density, and $\eta(x,y,t)$ is the deflection of the water surface from
the equilibrium position. Present the horizontal fluid velocity as ${\bf
V}=\\{U(x)+u(x,y,t),\,v(x,y,t)\\}$, where $|u(x,y,t)|\ll U(x)$ and substitute
this into the linearized Euler equation:

$\displaystyle\dfrac{\partial u}{\partial t}$ $\displaystyle+$
$\displaystyle\dfrac{\partial(Uu)}{\partial x}=-g\dfrac{\partial\eta}{\partial
x},$ (3) $\displaystyle\dfrac{\partial v}{\partial t}$ $\displaystyle+$
$\displaystyle U\,\dfrac{\partial v}{\partial
x}=-g\dfrac{\partial\eta}{\partial y}.$ (4)

Equation of mass conservation for shallow-water waves is:

$\dfrac{\partial S}{\partial t}+\dfrac{\partial}{\partial
x}\Bigl{[}S(U+u)\Bigr{]}+\dfrac{\partial}{\partial y}\Bigl{(}Sv\Bigr{)}=0,$
(5)

where $S(x,y,t)=[H(x)+\eta(x,y,t)]W(x)$ is the part of the duct cross section
occupied by water. Linearization of Eq. (5) with respect to small
perturbations $u$, $v$, and $\eta$ leads to the equation:

$\dfrac{\partial\eta}{\partial t}+HU\dfrac{\partial}{\partial
x}\left(\dfrac{u}{U}+\dfrac{\eta}{H}\right)+H\,\dfrac{\partial v}{\partial
y}=0.$ (6)

Now let us introduce the potential $\varphi$ for the perturbed fluid velocity
${\bf v}=(\partial\varphi/\partial x,\,\partial\varphi/\partial y)$. Then, we
find from Eqs. (3) and (4):

$-g\eta=\dfrac{\partial\varphi}{\partial t}+U\dfrac{\partial\varphi}{\partial
x}.$ (7)

Combining Eqs. (6) and (7) and bearing in mind that $c^{2}(x)=gH(x)$, we
arrive to the single equation describing long surface wave propagation in
terms of the velocity potential $\varphi$:

$\left(\dfrac{\partial}{\partial t}+U\dfrac{\partial}{\partial
x}-2U\dfrac{c^{\prime}}{c}\right)\left(\dfrac{\partial\varphi}{\partial
t}+U\dfrac{\partial\varphi}{\partial x}\right)=c^{2}U\dfrac{\partial}{\partial
x}\left(\dfrac{1}{U}\dfrac{\partial\varphi}{\partial
x}\right)+c^{2}\dfrac{\partial^{2}\varphi}{\partial y^{2}},$ (8)

where prime stands for the derivative with respect to $x$. Alternatively this
equation can be presented in the ‘covariant’ form:

$\dfrac{\partial}{\partial
t}\left(-\dfrac{1}{c^{2}U}\,\dfrac{\partial\varphi}{\partial
t}-\dfrac{1}{c^{2}}\,\dfrac{\partial\varphi}{\partial
x}\right)+\dfrac{\partial}{\partial
x}\left(-\dfrac{1}{c^{2}}\,\dfrac{\partial\varphi}{\partial
t}+\dfrac{c^{2}-U^{2}}{c^{2}U}\,\dfrac{\partial\varphi}{\partial
x}\right)+\dfrac{\partial}{\partial
y}\left(\dfrac{1}{U}\,\dfrac{\partial\varphi}{\partial y}\right)=0.$ (9)

Comparing this equation with Eq. (1), we obtain for corresponding ‘metric
tensor’ (cf., e.g., SchUn )

$\displaystyle G_{\mu\nu}$ $\displaystyle=$
$\displaystyle\dfrac{G^{2}_{2}}{c^{2}(x)U^{2}(x)}\left(\begin{array}[]{ccc}-\Bigl{[}c^{2}(x)-U^{2}(x)\Bigr{]}&-U(x)&0\\\
-U(x)&1&0\\\ 0&0&1\end{array}\right),$ (13) $\displaystyle G^{\mu\nu}$
$\displaystyle=$
$\displaystyle\left[\dfrac{U(x)}{G_{2}}\right]^{2}\left(\begin{array}[]{ccc}-1&-U(x)&0\\\
-U(x)&c^{2}(x)-U^{2}(x)&0\\\ 0&0&c^{2}(x)\end{array}\right),$ (17)

where $G_{2}>0$ is a constant chosen for the reason of convenience.

For the elementary monochromatic wave which is independent of $y$ and has
frequency $\omega$, $\varphi=\phi(x){\rm e}^{-{\rm i}\,\omega\,t}$, Eq. (9)
reduces to the ODE:

$(c^{2}-U^{2})\dfrac{{\rm d}^{2}\phi}{{\rm
d}x^{2}}+\left[2U^{2}\dfrac{c^{\prime}}{c}-(c^{2}+U^{2})\dfrac{U^{\prime}}{U}+2{\rm
i}\omega U\right]\dfrac{{\rm d}\phi}{{\rm d}x}+\left(\omega^{2}-2{\rm
i}\,\omega\,U\dfrac{c^{\prime}}{c}\right)\phi=0.$ (18)

Assuming that $\omega$ is a big parameter, one can readily construct the JWKB
solution to this equation:

$\phi(x)=\sqrt{c(x)U(x)}\left[A_{+}\exp\left({\rm i}\,\omega\int\dfrac{{\rm
d}x}{c(x)+U(x)}\right)+A_{-}\exp\left(-{\rm i}\,\omega\int\dfrac{{\rm
d}x}{c(x)-U(x)}\right)\right].$ (19)

This solution describes a superposition of two waves propagating in the
opposite directions – co-current and counter-current. Note that the second
term in the right-hand side is singular at the critical points, where
$U(x)=c(x)$. From the physical point of view, the singularities are caused by
the blocking of the wave propagating against the current, so that its phase
velocity and wavelength tend to zero at the critical point. Therefore, the
consideration of the crossing through the critical point requires special
attention.

Following Bab , we look for an exact solution to Eq. (18) in the form similar
to (19):

$\displaystyle\phi(x)$ $\displaystyle=$ $\displaystyle
A(x)\left[\exp\left({\rm i}\,\omega\int\dfrac{{\rm
d}x}{c(x)+U(x)}\right)+R(x)\exp\left(-{\rm i}\,\omega\int\dfrac{{\rm
d}x}{c(x)-U(x)}\right)\right]$ (20) $\displaystyle\equiv$ $\displaystyle
A\Bigl{(}e_{+}+Re_{-}\Bigr{)},$

where functions $A(x)$ and $R(x)$ can be interpreted respectively as the
complex amplitude of a co-current propagating wave and the relative amplitude
of the counter-current propagating wave (i.e., “the local reflection
coefficient”); $e_{\pm}$ denote the corresponding exponential functions.

The method of variation of constants provides the condition which relates the
functions $A(x)$ and $R(x)$ (see, e.g., Bender ). According to this method,
only the exponential functions should be differentiated when we calculate the
derivative ${\rm d}\phi/{\rm d}x$:

$\dfrac{{\rm d}\phi}{{\rm d}x}={\rm i}\omega
A(x)\left[\dfrac{e_{+}}{c+U}-\dfrac{R(x)e_{-}}{c-U}\right].$ (21)

Then, functions $A(x)$ and $R(x)$ must satisfy the equation:

$\dfrac{{\rm d}A}{{\rm d}x}\Bigl{(}e_{+}+Re_{-}\Bigr{)}+A\dfrac{{\rm d}R}{{\rm
d}x}\,e_{-}=0.$ (22)

Calculating ${\rm d}^{2}\phi/{\rm d}x^{2}$ in Eq. (21) and substituting the
result into Eq. (18), we find:

$\dfrac{{\rm d}A}{{\rm d}x}\Bigl{[}(c-U)e_{+}-(c+U)Re_{-}\Bigr{]}-A\dfrac{{\rm
d}R}{{\rm
d}x}(c+U)e_{-}-A\left(c^{\prime}+c\,\dfrac{U^{\prime}}{U}\right)\Bigl{(}e_{+}-Re_{-}\Bigr{)}=0.$
(23)

Equations (22) and (23) completely describe functions $A(x)$ and $R(x)$. By
resolving these equations with respect to the derivatives, we obtain:

$\displaystyle\dfrac{{\rm d}A}{{\rm d}x}$ $\displaystyle=$ $\displaystyle
a(x)\Bigl{(}1-R(x){\rm e}^{-{\rm i}\,\Phi(x)}\Bigr{)}A(x),$ (24)
$\displaystyle\dfrac{{\rm d}R}{{\rm d}x}$ $\displaystyle=$
$\displaystyle-\,a(x)\left({\rm e}^{{\rm i}\,\Phi(x)}-R^{2}(x){\rm e}^{-{\rm
i}\Phi(x)}\right),$ (25)

where

$\displaystyle a(x)$ $\displaystyle=$
$\displaystyle\dfrac{1}{2}\left(\dfrac{c^{\prime}}{c}+\dfrac{U^{\prime}}{U}\right)\equiv\dfrac{1}{2}\,\dfrac{{\rm
d}\ln\Pi(x)}{{\rm d}x}\,,\quad\Pi(x)=c(x)U(x),\quad{\rm e}^{{\rm
i}\,\Phi(x)}=\dfrac{e_{+}(x)}{e_{-}(x)},$ $\displaystyle\Phi(x)$
$\displaystyle=$ $\displaystyle\omega\int\alpha(x){\rm
d}x,\quad\alpha(x)\,=\,\dfrac{1}{c(x)+U(x)}+\dfrac{1}{c(x)-U(x)}=\dfrac{2c(x)}{c^{2}(x)-U^{2}(x)}.$
(26)

The system of equations (24) and (25) has a number of useful properties that
make the analysis of wave propagation simpler and more intuitive. Firstly, it
is clear that if in some flow region function $\Pi(x)=$ const, then both
$A(x)=$ const and $R(x)=$ const, i.e. waves in this region do not experience
reflection despite that $c$ and $U$ depend on $x$.

Secondly, the problem is reduced to the nonlinear first-order ODE – the
Riccati equation (25); after finding a solution to this equation, the
amplitude equation for $A(x)$ (24) is immediately integrated:

$A(x)=A_{0}\sqrt{\Pi(x)}\exp\Bigl{[}-\int{\rm d}x\,a(x)\,R(x){\rm e}^{-{\rm
i}\Phi(x)}\Bigr{]}.$ (27)

Thirdly, it is easy to see that in the flow regions that do not contain
critical points (where $U(x)=c(x)$) Eqs. (24) and (25) have the first
integral:

${\cal E}=\dfrac{|A(x)|^{2}}{\Pi(x)}\,\Bigl{[}1-|R(x)|^{2}\Bigr{]}={\rm
const}.$ (28)

This equation can be treated as the conservation of wave action which is
equivalent in our case to the conservation of the pseudo-energy McIntyre ;
FabrStep . According to this law, the amplitude of the incident wave $|A(x)|$
can be presented through the transformation coefficient $|R(x)|^{2}$ and the
“geometric factor” of the flow $\Pi(x)$.

For the further consideration, it is convenient to introduce the normalised
amplitude that does not depend on the geometric factor,
$D(x)=A(x)/\sqrt{\Pi(x)}$, and the normalised coefficient $r(x)=R(x){\rm
e}^{-{\rm i}\,\Phi(x)}$ (recall that $R(x)$ is a complex-valued function).
These functions satisfy the following equations (see Eqs. (24)–(26)):

$\displaystyle\dfrac{{\rm d}D}{{\rm d}x}$ $\displaystyle=$
$\displaystyle-\,a(x)\,r(x)\,D(x),$ (29) $\displaystyle\dfrac{{\rm d}r}{{\rm
d}x}$ $\displaystyle=$ $\displaystyle-\,a(x)\Bigl{[}1-r^{2}(x)\Bigr{]}-{\rm
i}\,\omega\,\alpha(x)\,r(x).$ (30)

Then, Eqs. (20), (27) and (28) take the form:

$\displaystyle\phi(x)$ $\displaystyle=$ $\displaystyle
A(x)\Bigl{[}1+r(x)\Bigr{]}\exp\left({\rm i}\,\omega\int\dfrac{{\rm
d}x}{c(x)+U(x)}\right),$ (31) $\displaystyle A(x)$ $\displaystyle=$
$\displaystyle A_{0}\sqrt{\Pi(x)}\exp\left(-\int{\rm d}x\,a(x)\,r(x)\right),$
(32) $\displaystyle{\cal E}$ $\displaystyle=$
$\displaystyle\dfrac{|A(x)|^{2}}{\Pi(x)}\,\Bigl{[}1-|r(x)|^{2}\Bigr{]}\equiv|D(x)|^{2}\Bigl{[}1-|r(x)|^{2}\Bigr{]}={\rm
const}.$ (33)

It should be noted that if at some point of flow $|r(x)|<1$, then this
inequality holds throughout the entire flow due to the conservation law (33)
(the same is true for $|r(x)|>1$).

In some cases it is convenient to present a solution for $r(x)$ through the
module and argument $r(x)=|r(x)|\,{\rm e}^{{\rm i}\,\theta(x)}$ (note that
$|r(x)|=|R(x)|$, where $R(x)$ is a complex-valued function); then for these
quantities we have the equations:

$\displaystyle\dfrac{{\rm d}|r|}{{\rm d}x}$ $\displaystyle=$
$\displaystyle-\,a(x)\Bigl{[}1-|r(x)|^{2}\Bigr{]}\cos\theta(x),$ (34)
$\displaystyle\dfrac{{\rm d}\theta}{{\rm d}x}$ $\displaystyle=$
$\displaystyle\dfrac{a(x)}{|r(x)|}\Bigl{[}1+|r(x)|^{2}\Bigr{]}\sin\theta(x)-\omega\,\alpha(x).$
(35)

In the next Section III we will consider in detail the solution in the
vicinity of a critical point and transition through the critical point.

##  III Solution in the Vicinity of a Critical Point

###  III.1 The asymptotic expansion

Let $x=x_{0}$ be the critical point such that $U(x_{0})=c(x_{0})\equiv c_{0}$.
We consider the general case when the curves $c(x)$ and $U(x)$ intersect
without tangency, i.e. when the quantity
$\mu=U^{\prime}(x_{0})-c^{\prime}(x_{0})$ is nonzero and, moreover, not small,
$\mu=O(1)$. Let us introduce a parameter $0<\varepsilon\ll 1$, put
$x-x_{0}=\varepsilon\xi$, and use the notation $f_{0}=f(x_{0})$ for any
function $f(x)$. Expanding functions $c(x)$ and $U(x)$ in the Taylor series in
the vicinity of $x=x_{0}$ and replacing $x$ with $\xi$ in Eq. (30), we obtain:

$\xi\dfrac{{\rm d}r}{{\rm d}\xi}=\dfrac{{\rm
i}\,\omega}{\mu}\,r+\varepsilon\xi\left[\dfrac{c^{\prime}_{0}+U^{\prime}_{0}}{2c_{0}}\Bigl{(}r^{2}-1\Bigr{)}-\dfrac{{\rm
i}\,\omega}{2}\left(\dfrac{1}{c_{0}}-\dfrac{c^{\prime\prime}_{0}-U^{\prime\prime}_{0}}{\mu^{2}}\right)r\right]+\varepsilon^{2}\xi^{2}S+\ldots,$
(36)

where

$S=\dfrac{1}{2}\left[\left(\dfrac{c^{\prime\prime}_{0}+U^{\prime\prime}_{0}}{c_{0}}-\dfrac{{c^{\prime}_{0}}^{2}+{U^{\prime}_{0}}^{2}}{c^{2}_{0}}\right)\Bigl{(}r^{2}-1\Bigr{)}+\dfrac{{\rm
i}\,\omega}{2}\left(\dfrac{c^{\prime}_{0}+U^{\prime}_{0}}{c^{2}_{0}}+\dfrac{2}{3}\,\dfrac{c^{\prime\prime\prime}_{0}-U^{\prime\prime\prime}_{0}}{\mu^{2}}+\dfrac{(c^{\prime\prime}_{0}-U^{\prime\prime}_{0})^{2}}{\mu^{3}}\right)r\right].$

Let us look for a solution to this equation in the form $r=r^{(0)}+\varepsilon
r^{(1)}+\varepsilon^{2}r^{(2)}+\dots$ and introduce a notation
$\beta=\omega/\mu$. In the zero order on the parameter $\varepsilon$, we
obtain:

$\xi\dfrac{{\rm d}r^{(0)}}{{\rm d}\xi}={\rm i}\,\beta\,r^{(0)},\qquad
r^{(0)}=\tilde{B}_{0}\xi^{{\rm i}\,\beta},\ \ \tilde{B}_{0}={\rm const}.$ (37)

In the first order on this parameter, $O(\varepsilon)$, the equation is:

$\xi\dfrac{{\rm d}r^{(1)}}{{\rm d}\xi}={\rm
i}\,\beta\,r^{(1)}+\xi\left[\dfrac{c^{\prime}_{0}+U^{\prime}_{0}}{2c_{0}}\Bigl{(}\tilde{B}^{2}_{0}\xi^{2\,{\rm
i}\,\beta}-1\Bigr{)}-\dfrac{{\rm
i}\,\omega}{2}\left(\dfrac{1}{c_{0}}-\dfrac{c^{\prime\prime}_{0}-U^{\prime\prime}_{0}}{\mu^{2}}\right)\tilde{B}_{0}\xi^{{\rm
i}\,\beta}\right].$ (38)

Solution to this equation can be readily derived:

$r^{(1)}=\xi\Bigl{(}B_{10}+B_{11}\xi^{{\rm i}\,\beta}+B_{12}\xi^{2\,{\rm
i}\,\beta}\Bigr{)},$ (39)

where

$B_{10}=-\dfrac{c^{\prime}_{0}+U^{\prime}_{0}}{2c_{0}(1-{\rm i}\beta)},\quad
B_{11}=-\dfrac{{\rm
i}\,\omega}{2}\left(\dfrac{1}{c_{0}}-\dfrac{c^{\prime\prime}_{0}-U^{\prime\prime}_{0}}{\mu^{2}}\right)\tilde{B}_{0},\quad
B_{12}=\dfrac{c^{\prime}_{0}+U^{\prime}_{0}}{2c_{0}(1+{\rm
i}\,\beta)}\,\tilde{B}^{2}_{0}$ (40)

In the second order on the parameter $\varepsilon$, $O(\varepsilon^{2})$, the
equation is:

$\begin{array}[]{l}\xi\dfrac{{\rm d}r^{(2)}}{{\rm d}\xi}={\rm
i}\,\beta\,r^{(2)}+\xi\left[\dfrac{c^{\prime}_{0}+U^{\prime}_{0}}{c_{0}}r^{(0)}-\dfrac{{\rm
i}\,\omega}{2}\left(\dfrac{1}{c_{0}}-\dfrac{c^{\prime\prime}_{0}-U^{\prime\prime}_{0}}{\mu^{2}}\right)\right]r^{(1)}\\\
\\\
+\,\dfrac{\xi^{2}}{2}\left[\left(\dfrac{c^{\prime\prime}_{0}+U^{\prime\prime}_{0}}{c_{0}}-\dfrac{{c^{\prime}_{0}}^{2}+{U^{\prime}_{0}}^{2}}{c^{2}_{0}}\right)\Bigl{(}{r^{(0)}}^{2}-1\Bigr{)}+\dfrac{{\rm
i}\omega}{2}\left(\dfrac{c^{\prime}_{0}+U^{\prime}_{0}}{c^{2}_{0}}+\dfrac{2}{3}\,\dfrac{c^{\prime\prime\prime}_{0}-U^{\prime\prime\prime}_{0}}{\mu^{2}}+\dfrac{(c^{\prime\prime}_{0}-U^{\prime\prime}_{0})^{2}}{\mu^{3}}\right)r^{(0)}\right].\end{array}$
(41)

Solution to this equation we present in the form:

$r^{(2)}=\xi^{2}\Bigl{(}B_{20}+B_{21}\xi^{{\rm i}\,\beta}+B_{22}\xi^{2{\rm
i}\,\beta}+B_{23}\xi^{3{\rm i}\,\beta}\Bigr{)},$ (42)

where

$\displaystyle B_{20}$ $\displaystyle=$ $\displaystyle-\,\dfrac{1}{2(2-{\rm
i}\,\beta)}\left[\dfrac{c^{\prime\prime}_{0}+U^{\prime\prime}_{0}}{c_{0}}-\dfrac{{c^{\prime}_{0}}^{2}+{U^{\prime}_{0}}^{2}}{c^{2}_{0}}+{\rm
i}\,\omega\left(\dfrac{1}{c_{0}}-\dfrac{c^{\prime\prime}_{0}-U^{\prime\prime}_{0}}{\mu^{2}}\right)B_{10}\right],$
$\displaystyle B_{21}$ $\displaystyle=$
$\displaystyle\left[\dfrac{c^{\prime}_{0}+U^{\prime}_{0}}{2c_{0}}\,B_{10}+\dfrac{{\rm
i}\,\omega}{8}\left(\dfrac{c^{\prime}_{0}+U^{\prime}_{0}}{c^{2}_{0}}+\dfrac{2}{3}\,\dfrac{c^{\prime\prime\prime}_{0}-U^{\prime\prime\prime}_{0}}{\mu^{2}}+\dfrac{(c^{\prime\prime}_{0}-U^{\prime\prime}_{0})^{2}}{\mu^{3}}\right)\right]\tilde{B}_{0}-\dfrac{{\rm
i}\,\omega}{4}\left(\dfrac{1}{c_{0}}-\dfrac{c^{\prime\prime}_{0}-U^{\prime\prime}_{0}}{\mu^{2}}\right)B_{11},$
$\displaystyle B_{22}$ $\displaystyle=$ $\displaystyle\dfrac{1}{2(2+{\rm
i}\beta)}\left[\left(\dfrac{c^{\prime\prime}_{0}+U^{\prime\prime}_{0}}{c_{0}}-\dfrac{{c^{\prime}_{0}}^{2}+{U^{\prime}_{0}}^{2}}{c^{2}_{0}}\right)\tilde{B}^{2}_{0}+2\,\dfrac{c^{\prime}_{0}+U^{\prime}_{0}}{c_{0}}\tilde{B}_{0}B_{11}-\dfrac{{\rm
i}\,\omega}{2}\left(\dfrac{1}{c_{0}}-\dfrac{c^{\prime\prime}_{0}-U^{\prime\prime}_{0}}{\mu^{2}}\right)B_{12}\right],$
$\displaystyle B_{23}$ $\displaystyle=$
$\displaystyle\dfrac{c^{\prime}_{0}+U^{\prime}_{0}}{2c_{0}(1+{\rm
i}\,\beta)}\,\tilde{B}^{2}_{0}B_{12}.$

The solution obtained consists of two parts. One of them is the sum of a
slowly varying terms that vanish at the critical point; this part of the
solution is represented by the Taylor series:

$r_{sl}(x)=B_{10}(x-x_{0})+B_{20}(x-x_{0})^{2}+\ldots.$ (43)

Another part of the solution contains the terms with the coefficients
$\tilde{B}_{0}$ and $B_{nm}$ ($m\geq 1$); these terms are strongly oscillating
when $x\to x_{0}$ (see equations (37), (39), and (42)). Note that
$B_{nm}\sim\tilde{B}^{m}_{0}$ and stress that $B_{nm}$ depend on $B_{i0}$, but
$B_{i0}$ does not depend on $B_{nm}$ with $m\geq 1$ (this property holds in
the higher orders of expansion too). Therefore, the slowly varying part of the
solution is entirely determined by the local behaviour of functions $c(x)$ and
$U(x)$ in the vicinity of the critical point and does not depend on the fast
oscillating contribution. The latter is determined by both the boundary
conditions and behaviour of functions $c(x)$ and $U(x)$ to the right or to the
left of the point $x_{0}$. Note, by the way, that $\tilde{B}_{0}$ and the
entire quickly oscillating contribution can be different at left and right
sides of $x_{0}$, therefore, the found solution is reasonable to represent in
the form:

$r(x)=r_{sl}(x)+B^{(\pm)}_{0}|x-x_{0}|^{{\rm
i}\,\beta}\Bigl{[}1+O(x-x_{0})\Bigr{]},$ (44)

where the signs plus and minus pertain to the regions where $x>x_{0}$ and
$x<x_{0}$, respectively.

The integrals in the equations (31) and (32) converge for $x\to x_{0}\pm 0$,
and the asymptotic expansions for $A(x)$ and $\phi(x)$ can be writen as:

$\displaystyle A(x)$ $\displaystyle=$ $\displaystyle
c_{0}A^{(\pm)}_{0}\left[1+\dfrac{c^{\prime}_{0}\\!+\\!U^{\prime}_{0}}{2c_{0}}\,(x-x_{0})\left(1-\dfrac{B^{(\pm)}_{0}}{1+{\rm
i}\,\beta}\,|x-x_{0}|^{{\rm i}\,\beta}\right)+O(|x-x_{0}|^{2})\right],$ (45)
$\displaystyle\phi(x)$ $\displaystyle=$ $\displaystyle
c_{0}A^{(\pm)}_{0}\Bigl{[}1+B^{(\pm)}_{0}|x-x_{0}|^{{\rm
i}\,\beta}+O(|x-x_{0}|)\Bigr{]}{\rm e}^{{\rm
i}\,\psi},\quad\psi=\omega\\!\int^{x_{0}}\\!\dfrac{{\rm d}x}{c+U}\,.$ (46)

As the next step, we need to investigate how these solutions can be extended
for the transition through the critical point. This will be done in the next
subsection.

###  III.2 Transition through the critical point

Equation (46) contains only the principal expansion terms. The first term in
the square brackets refers to the co-current propagating wave, and the second
term (that is rapidly oscillating) refers to the counter-current propagating
wave. As shown in ChErSt17 , for the correct matching solutions through the
critical point, it is necessary to take into account a small viscosity of the
medium. The details of calculation of the viscous problem are presented in the
Appendix, and here we summarise the final result. (Note in brackets that in
dispersive media there are other possibilities to match the solutions through
the critical point. The dispersion invokes causality arguments similar to
those which were considered by Hawking Hawking-1975  and Unruh Unruh-1976  for
astrophysical black holes. This option requires, however, assuming that the
flow was uniform in the distant past.)

If the current passes from the subcritical regime into supercritical, then

$A^{(+)}_{0}=A^{(-)}_{0},\qquad B^{(+)}_{0}=B^{(-)}_{0}=0.$ (47)

In other words, co-current traveling waves “do not feel” that the point is
critical. The absence of counter-current running waves is due to the fact that
on both sides of the critical point they propagate out of this point and
therefore, cannot reach it. Hence, for such a transition we have (see Eqs.
(43), (44), (33), and (34)):

$r(x_{0})=0\quad\mbox{and}\quad{\cal E}(x_{0}+0)={\cal E}(x_{0}-0).$ (48)

Therefore, we conclude that if $a(x_{0})\neq 0$ in Eqs. (34) and (35), then
functions $r(x)$ and $\cos\theta$ change their signs in the course of
transition through the critical point, i.e.
$\theta(x_{0}+0)=\theta(x_{0}-0)\pm\pi$.

On the contrary, in the course of the transition from the supercritical to the
subcritical regime, the counter-current propagating waves run to the critical
point and, approaching it with a decreasing wavelength, are completely
absorbed in its viscous neighbourhood. Their relative amplitudes $B^{(+)}_{0}$
and $B^{(-)}_{0}$ depend on their propagation prehistory in the different flow
regions and therefore, are not related with each other, whereas
$A^{(+)}_{0}=A^{(-)}_{0}$. The energy flux in the course of the transition
through the critical point is not generally conserved,

${\cal
E}(x_{0}-0)=\dfrac{|A^{(-)}_{0}|^{2}}{\Pi(x_{0})}\Bigl{(}1-|B^{(-)}_{0}|^{2}\Bigr{)}\neq{\cal
E}(x_{0}+0)=\dfrac{|A^{(+)}_{0}|^{2}}{\Pi(x_{0})}\Bigl{(}1-|B^{(+)}_{0}|^{2}\Bigr{)},$

it can either decrease or increase (see ChErSt17 ).

##  IV Wave propagation in the BH–WH duct model

###  IV.1 The flow model and preliminary analysis

Consider the flow on the left ($-\infty<x<x_{1}$) and right
($x_{2}<x<+\infty$) ends of which the flow is subcritical, $0<U(x)<c(x)$, and
within the middle part, for $x_{1}<x<x_{2}$, – supercritical, $U(x)>c(x)$ (see
Fig. 2). Here $x_{1}$ and $x_{2}$ are the critical points such that
$U(x_{1})=c(x_{1})\equiv c_{1}$ and $U(x_{2})=c(x_{2})\equiv c_{2}$. In what
follows, we will use the dimensionless variables such that:

$c_{1}=1,\qquad\mu_{1}=U^{\prime}_{1}-c^{\prime}_{1}=1.$ (49)

Figure 2: (Color online) The flow models with two critical points: (a) – flow
(78), ${\cal M}=1.5$; (b) – flow (81), $c_{2}=2c_{1}=2$, $d=1$; (c) –
reflectionless flow with $\Pi(x)\equiv c(x)U(x)=1$.

Let us assume that a plane wave of frequency $\omega$ arrives from the left.
Its propagation in the inhomogeneous zone will be described in terms of
functions $D(x)$ and $r(x)$. At the critical point $x=x_{1}$ and when
$x\to+\infty$ there is no wave traveling upstream, therefore solution to Eq.
(30) must satisfy the boundary conditions:

$r(x_{1})=0,\qquad r(+\infty)=0.$ (50)

Therefore, we can conclude that, firstly, in the entire flow $|r(x)|<1$ (as it
follows from the conservation law (33)) and, secondly, the equation (30)
should be integrated into the upstream direction in the sub-critical areas and
downstream – on the supercritical section $x_{1}\leq x<x_{2}$.

Further, suppose that for $x\to\pm\infty$ functions $c(x)$ and $U(x)$ tend to
their nonzero limiting values, and let $D(-\infty)=1$. Then from Eq. (32) and
the matching conditions (47), $A^{(+)}_{0}=A^{(-)}_{0}$, we find the amplitude
of the downstream propagating wave as the function of $x$:

$D(x)=\exp\left[-\\!\displaystyle\int\limits_{-\infty}^{x}\\!a(x^{\prime})\,r(x^{\prime})\,{\rm
d}x^{\prime}\,\right],\quad-\infty<x<\infty.$ (51)

Then we can define the transmission ratio $K(x)\\!\equiv\\!|D(x)|$ and present
it as:

$K(x)=\exp\left[-\\!\displaystyle\int\limits_{-\infty}^{x}\\!a(x^{\prime}){\rm
Re}\,\Bigl{(}r(x^{\prime})\Bigr{)}\,{\rm
d}x^{\prime}\,\right]=\exp\left[-\\!\displaystyle\int\limits_{-\infty}^{x}\\!a(x^{\prime})|r(x^{\prime})|\cos\theta(x^{\prime})\,{\rm
d}x^{\prime}\,\right].$ (52)

As can be seen from Eqs. (34) and (52), the reflection coefficient $|r(x)|$
and transmission ratio $K(x)$ both increase or decrease simultaneously
depending on the sign of the product $a(x)\cos{\theta(x)}$. Finally, in those
regions where $a(x)=0$, i.e. $\Pi(x)=$ const, they are constant too and, as
already noted, waves running in the opposite directions do not interact, but
the phase difference between them $\theta(x)$ changes due to the difference in
the directions of propagation and difference in the wavelengths (i.e., because
$\alpha(x)\neq 0$ – see Eq. (35)).

In accordance with the flow structure and conservation law (33), it is
convenient to distinguish three stages for the downstream propagating wave. In
the interval $x<x_{1}$, the transmission ratio $K(x)$ decreases from unity to
$K(x_{1})=\sqrt{1-|r(-\infty)|^{2}}$. In the supercritical interval, the
amplitude increases, and $K(x_{2})=K(x_{1})/\sqrt{1-|r(x_{2}-0)|^{2}}$, and in
the last interval, $x>x_{2}$, $K(x)$ decreases again to
$K(+\infty)=K(x_{2})\sqrt{1-|r(x_{2}+0)|^{2}}$ (recall that in the general
case $|r(x_{2}-0)|\neq|r(x_{2}+0)|$, therefore the increase and subsequent
decrease of $K$ are independent). Thus, we can conclude that $K(x)$ can vary
non-monotonically in any of these three intervals. As the result, in each
interval of the flow, its integral transmission ratios are presented by the
formulae:

$K_{1}=\sqrt{1-|r(-\infty)|^{2}}\leq 1,\quad
K_{2}=\Bigl{[}1-|r(x_{2}-0)|^{2}\Bigr{]}^{-1/2}\geq 1,$
$K_{3}=\Bigl{[}1-|r(x_{2}+0)|^{2}\Bigr{]}^{1/2}\leq 1,$ (53)

and the total transmission ratio (the transmission coefficient) is $T\equiv
K(+\infty)=K_{1}\cdot K_{2}\cdot K_{3}$. It can be increased by (i) reducing
the transformation of the co-current propagating wave into reflected waves in
the sub-critical intervals of the flow and (ii) increasing its transformation
into the negative energy wave in the supercritical interval. Details of the
concept of negative energy waves can be found in FabrStep ; Maissa . Here we
only recall that a wave whose phase velocity is less than the local flow
velocity is attributed as the negative energy wave. Such a wave cannot
propagate against the flow because the flow is so strong that it pulls the
wave in the direction of the flow.

The first condition can be fulfilled at once for all frequencies if in the
subcritical regions the reflectionless wave propagation occurs with
$a(x)\equiv 0$ (i.e. $c(x)U(x)\equiv$ const), then $K_{1}=K_{3}=1$. Note that
this condition does not prevent the change of the Mach number ${\cal
M}(x)\equiv U(x)/c(x)$ along the flow, (see, e.g., Fig. 2(c)). Another way to
do it is based on the effect similar to that of anti-reflective coating in the
optics. Its principle is that waves reflected from the different flow
intervals add up in anti-phase and extinguish each other. As can be seen from
Eq. (34), the quenching is complete if:

$\displaystyle r(\\!-\infty)$ $\displaystyle=$
$\displaystyle\\!\displaystyle\int\limits_{-\infty}^{x_{1}}\\!a(x)\Bigl{[}1\\!-\\!|r(x)|^{2}\Bigr{]}\cos\theta(x)\,{\rm
d}x\,=0,$ (54) $\displaystyle r(x_{2}\\!+\\!0)$ $\displaystyle=$
$\displaystyle\\!\displaystyle\int\limits^{\infty}_{x_{2}}\\!a(x)\Bigl{[}1\\!-\\!|r(x)|^{2}\Bigr{]}\cos\theta(x)\,{\rm
d}x\,=0.$ (55)

Since in those intervals where $a(x)=0$, function $\theta(x)$ continues to
vary, the fulfillment of Eqs. (54) and (55) can be ensured by variation of
$\theta(x)$ due to “inserts” with $a(x)=0$ of the required length in the
proper intervals of the flow. But since for $a=0$ ${\rm d}\theta/{\rm
d}x\sim\omega$ (see Eq. (35)), the choice of the positions of such inserts and
especially their lengths significantly depends on the wave frequency.

As will be shown below, both the selection of inserts and the difference in
the speeds of wave propagation at the ends of the supercritical interval (both
with $c_{1}<c_{2}$ and $c_{1}>c_{2}$) contribute to this, but the increase in
wave frequency prevents to this.

To simplify further analysis, let us strengthen the previously formulated
condition of fast convergence of $c(x)$ and $U(x)$ to their limiting values
when $x\to\pm\infty$, assuming that $c$ and $U$ are constants and,
respectively, $a(x)=0$ for $x<x_{-}<x_{1}$ and $x>x_{+}>x_{2}$. Below we
perform the analysis for two limiting cases, when $\omega\ll 1$ and $\omega\gg
1$.

###  IV.2 The low-frequency limit, $\omega\ll 1$

Setting $\omega=0$ in Eq. (30), we get:

$\dfrac{{\rm d}r_{0}}{{\rm
d}x}=-\dfrac{\Pi^{\prime}(x)}{2\Pi(x)}\,\Bigl{(}1-r^{2}_{0}\Bigr{)}\ \
\Longrightarrow\ \ r_{0}(x)=\dfrac{R_{0}-\Pi(x)}{R_{0}+\Pi(x)},\quad
R_{0}=c_{i}U_{i}\,\dfrac{1+r_{0i}}{1-r_{0i}},$ (56)

where $c_{i}$, $U_{i}$, and $r_{0i}$ are the values of $c(x)$, $U(x)$, and
$r_{0}(x)$ in some starting point $x=x_{i}$. If $x_{i}$ coincides with the
first critical point $x_{1}$, then $r_{0i}=0$, $R_{0}=1$ and

$r_{0}(x)=\dfrac{1-\Pi(x)}{1+\Pi(x)}\equiv\dfrac{1-c(x)U(x)}{1+c(x)U(x)}=1-\dfrac{2c(x)U(x)}{1+c(x)U(x)}.$
(57)

Next, we look for a solution in the form:
$r(x)=r_{0}(x)+\omega\,r_{1}(x)+\omega^{2}r_{2}(x)+\dots$. In the lowest order
on the frequency, $O(\omega)$, we have:

$\dfrac{{\rm d}r_{1}}{{\rm
d}x}=\dfrac{\Pi^{\prime}(x)}{\Pi(x)}\,r_{0}(x)r_{1}(x)-{\rm
i}\,\alpha(x)r_{0}(x).$

Integrating this equation with the initial condition $r_{1}(x_{i})=0$ and
using Eq. (56), we get:

$r_{1}(x)=-\dfrac{2\,{\rm
i}\,\Pi(x)}{[R_{0}+\Pi(x)]^{2}}\displaystyle\int\limits_{x_{i}}^{x}\\!\dfrac{r_{0}(\xi)}{U(\xi)}\,\dfrac{[R_{0}+\Pi(\xi)]^{2}}{c^{2}(\xi)-U^{2}(\xi)}\,{\rm
d}\xi\,.$ (58)

In the next order on the frequency, $O(\omega^{2})$, we have the following
equation:

$\dfrac{{\rm d}r_{2}}{{\rm
d}x}=\dfrac{\Pi^{\prime}(x)}{2\Pi(x)}\Bigl{[}2r_{0}(x)r_{2}(x)+r^{2}_{1}(x)\Bigr{]}-{\rm
i}\,\alpha(x)r_{1}(x).$

Its solution subject to the initial condition $r_{2}(x_{i})=0$ is:

$r_{2}(x)=-\dfrac{4\Pi(x)}{[R_{0}+\Pi(x)]^{3}}\displaystyle\int\limits_{x_{i}}^{x}\\!\dfrac{1}{U(\xi)}\,\dfrac{R_{0}\Pi(x)+\Pi^{2}(\xi)}{c^{2}(\xi)-U^{2}(\xi)}\,{\rm
d}\xi\,\displaystyle\int\limits_{x_{i}}^{\xi}\\!\dfrac{r_{0}(\eta)}{U(\eta)}\,\dfrac{[R_{0}+\Pi(\eta)]^{2}}{c^{2}(\eta)-U^{2}(\eta)}\,{\rm
d}\eta\,.$ (59)

The integral in the right-hand side of Eq. (58) can be evaluated in the
vicinity of the point $x_{2}$; when $x\to x_{2}$ we have:

$I(x)=\displaystyle\int\limits_{x_{i}}^{x}\\!\dfrac{r_{0}(\xi)}{U(\xi)}\,\dfrac{[R_{0}+\Pi(\xi)]^{2}}{c^{2}(\xi)-U^{2}(\xi)}\,{\rm
d}\xi=I_{1i}-\dfrac{R^{2}_{0}-c^{4}_{2}}{2\mu_{2}c^{2}_{2}}\,\ln{|x-x_{2}|}+O(|x-x_{2}|),$

where $I_{1i}=$ const, $\mu_{2}=U^{\prime}_{2}-c^{\prime}_{2}$. The integral
converges if $r_{0}(x_{2})=0$ (i.e. if $R_{0}=c^{2}_{2}$), otherwise it
logarithmically diverges.

Similarly one can evaluate the integral in the right-hand side of Eq. (59):

$\begin{array}[]{l}\displaystyle\int\limits_{x_{i}}^{x}\\!\dfrac{1}{U(\xi)}\,\dfrac{R_{0}\Pi(x)+\Pi^{2}(\xi)}{c^{2}(\xi)-U^{2}(\xi)}\left(\displaystyle\int\limits_{x_{i}}^{\xi}\\!\dfrac{r_{0}(\eta)}{U(\eta)}\,\dfrac{[R_{0}+\Pi(\eta)]^{2}}{c^{2}(\eta)-U^{2}(\eta)}\,{\rm
d}\eta\right)\,{\rm d}\xi\\\ \\\
{}=I_{2i}-\dfrac{R_{0}+c^{2}_{2}}{2\mu_{2}}\left[I_{1i}\ln{|x-x_{2}|}-\dfrac{R^{2}_{0}-c^{4}_{2}}{4\mu_{2}c^{2}_{2}}\,\ln^{2}{|x-x_{2}|}\right]+O(|x-x_{2}|),\end{array}$

where $I_{2i}=$ const. Therefore, when $x\to x_{2}$, the solution is:

$\begin{array}[]{l}r(x)=\dfrac{R_{0}-c^{2}_{2}}{R_{0}+c^{2}_{2}}\left[1+\dfrac{{\rm
i}\,\omega}{\mu_{2}}\,\ln{|x-x_{2}|}-\dfrac{\omega^{2}}{2\mu^{2}_{2}}\,\ln^{2}{|x-x_{2}|}+\dots\right]\\\
\\\ \phantom{wwa}{}-\dfrac{2{\rm
i}\,\omega\,c^{2}_{2}I_{1i}}{(R_{0}+c^{2}_{2})^{2}}\left[1+\dfrac{{\rm
i}\,\omega}{\mu_{2}}\,\ln{|x-x_{2}|}+\dots\right]-\dfrac{4\omega^{2}c^{2}_{2}I_{2i}}{(R_{0}+c^{2}_{2})^{3}}+O(|x-x_{2}|).\end{array}$
(60)

On the other hand, setting in Eq. (44) $x_{0}=x_{2}$ and $\mu=\mu_{2}$, we
obtain for $|x-x_{2}|\ll 1$ and $\omega|\ln|x-x_{2}||\ll 1$:

$\begin{array}[]{l}r(x)=r_{sl}(x)+B^{(\pm)}_{0}|x-x_{2}|^{{\rm
i}\,\omega/\mu_{2}}\Bigl{[}1+O(|x-x_{2}|)\Bigr{]}\\\ \\\
\phantom{wwl}{}=r_{sl}(x)+B^{(\pm)}_{0}\left[1+\dfrac{{\rm
i}\,\omega}{\mu_{2}}\,\ln{|x-x_{2}|}-\dfrac{\omega^{2}}{2\mu^{2}_{2}}\,\ln^{2}{|x-x_{2}|}+\dots\right]\Bigl{[}1+O(|x-x_{2}|)\Bigr{]}.\end{array}$

Matching of this solution with solution (60) shows that when $x\to x_{2}\pm 0$
the solution should be as this:

$r(x)=r_{sl}(x)+\left[\dfrac{R^{(\pm)}_{0}-c^{2}_{2}}{R^{(\pm)}_{0}+c^{2}_{2}}-\dfrac{2\,{\rm
i}\,\omega\,c^{2}_{2}I^{(\pm)}_{1i}}{(R_{0}+c^{2}_{2})^{2}}+O(\omega^{2})\right]|x-x_{2}|^{{\rm
i}\,\omega/\mu_{2}}\Bigl{[}1+O(|x-x_{2}|)\Bigr{]},$ (61)

where $r_{sl}(x)$ is determined by the series (43) with $x_{0}=x_{2}$. It was
taken into account that $R_{0}$ and $I_{1i}$ can be different on the left and
right of point $x_{2}$, because according to the boundary conditions (50),
they are calculated through the integration of Eq. (30) in the different flow
regions. Therefore, if $r_{0}(x_{2}\pm 0)\neq 0$ (i.e. $R^{(\pm)}_{0}\neq
c^{2}_{2}$), then the reflection coefficient and transmission ratios are:

$|r(x)|\to\left|\dfrac{R^{(\pm)}_{0}-c^{2}_{2}}{R^{(\pm)}_{0}+c^{2}_{2}}\right|^{1/2}=O(1),$
$K_{2}\approx\left[1-\left|\dfrac{R^{(-)}_{0}-c^{2}_{2}}{R^{(-)}_{0}+c^{2}_{2}}\right|^{2}\right]^{-1/2},\quad
K_{3}\approx\left[1-\left|\dfrac{R^{(+)}_{0}-c^{2}_{2}}{R^{(+)}_{0}+c^{2}_{2}}\right|^{2}\right]^{1/2}.$
(62)

Otherwise, $|r(x)|$ is of the order of $O(\omega^{2})$.

It should be emphasized that for $x\to x_{2}\pm 0$, the quantity

$r_{0}(x_{2}\pm 0)=\dfrac{R^{(\pm)}_{0}-c^{2}_{2}}{R^{(\pm)}_{0}+c^{2}_{2}}$

determined by Eq. (56) does not describe the limiting value of $r(x)$ (or its
main part), but, as seen from Eq. (61), the amplitude of the rapidly
oscillating part of the solution. Note that oscillations are concentrate in
the exponentially narrow neighborhood of the point $x_{2}$,

$|x-x_{2}|=O({\rm e}^{-1/\omega}).$ (63)

This allows us, firstly, to consider $r_{0}(x_{2}\pm 0)$ as the main part of
the intermediate asymptotic of the solution, and, secondly, greatly
complicates detection of oscillations in the numerical solution even in the
case of not very small $\omega$ (see Section V and Fig. 4).

Based on these results, we will consider wave propagation within each section
of the flow. Let’s start from the left interval, $-\infty<x<x_{1}$, where
$U(x)<c(x)$. Given that $a(x)\neq 0$ only in the interval
$\sigma=(x_{-},\,x_{1})$, we consider two options. The first option is that
$x_{1}-x_{-}=O(1)$ and $a(x)=O(1)$ in the entire interval $\sigma$. Then,
taking into account the boundary condition (50), we obtain (see equations
(57), (34), and (53)):

$r_{-}\equiv r(x_{-})=\dfrac{1-\Pi_{-}}{1+\Pi_{-}}+O(\omega)\quad\mbox{\rm
and}\quad
K_{1}=\sqrt{1-\left(\dfrac{1-\Pi_{-}}{1+\Pi_{-}}\right)^{2}}+O(\omega),\ \
\Pi_{-}=\Pi(x_{-}).$ (64)

The subcritical flow simulates the “ordinary” space-time, with the quite
natural constancy of the wave velocity, $c\equiv 1$. Then
$\Pi_{-}=U(-\infty)<1$ and there is necessarily a reflected wave; the
transmission rate is $K_{1}<1$. The reflectionless propagation with
$\Pi_{-}=1$ is possible only in a more complex model, where $c(x)>1$ for
$x\to-\infty$ and decreases to 1 as we approach the critical point $x_{1}$ (BH
horizon) – see, for example, Fig. 2(c).

In the second option, we compose the interval $\sigma$ from two intervals of
the length $O(1)$ each, $\sigma_{1}=(x_{-},x_{a})$ and
$\sigma_{2}=(x_{b},x_{1})$. In these intervals function $a(x)=O(1)$; the
intervals are separated by the insert $\sigma_{0}=(x_{a},x_{b})$ where
$a(x)\equiv 0$. Then, Eq. (57) gives $r_{0}(x_{b})=(1-\Pi_{0})/(1+\Pi_{0})$,
where $\Pi_{0}=\Pi(x_{a})=\Pi(x_{b})$. As $a\equiv 0$ within the insert,
therefore, $|r(x)|=$ const, and only the phase difference of waves $\theta(x)$
varies with $x$; however, this variation affects the transmission rate.
Indeed, in the case of the ‘phase inverting insert’, when

$\theta(x_{a})-\theta(x_{b})=\omega\\!\displaystyle\int\limits_{x_{a}}^{x_{b}}\\!\alpha(x)\,{\rm
d}x=(2n+1)\pi,$ (65)

where $n$ is natureal, $r_{0}(x_{a})=-r_{0}(x_{b})$, so that in accordance
with Eq. (56), $R_{0}=\Pi^{2}_{0}$ and

$r_{0}(x_{-})=\dfrac{\Pi^{2}_{0}-\Pi_{-}}{\Pi^{2}_{0}+\Pi_{-}},\qquad
K_{1}=\sqrt{1-\left(\dfrac{\Pi^{2}_{0}-\Pi_{-}}{\Pi^{2}_{0}+\Pi_{-}}\right)^{2}}.$
(66)

As a result, the waves reflected in the intervals $\sigma_{1}$ and
$\sigma_{2}$ cancel out each other upon the condition $\Pi^{2}_{0}=\Pi_{-}$,
which can be easily implemented even in traditional flow models with $c\equiv
1$ (see Fig. 2(a)), if in the insert section
$U(x)=U_{a}=U_{b}=\sqrt{U(-\infty)}$ (in this case, $U(-\infty)<U_{a}<1$).
This equality sets the position of the anti-reflective insert; its length is
determined by Eq. (65) and is very long, since it is proportional to
$\omega^{-1}$.

In conclusion, we note that wave propagation in the interval $x_{2}<x<\infty$
does not differ qualitatively from that just described. Indeed, since $a(x)=0$
for $x>x_{+}$, then $\Pi_{+}\equiv\Pi(x_{+})=\Pi(+\infty)$ and integration of
Eq. (30) starts at $x=x_{+}$ with $r(x_{+})=r(+\infty)=0$. If
$x_{+}-x_{2}=O(1)$ and $a(x)=O(1)$, then

$r_{0}(x_{2}+0)=\dfrac{\Pi_{+}-\Pi_{2}}{\Pi_{+}+\Pi_{2}},\qquad
K_{3}=\sqrt{1-\left(\dfrac{\Pi_{+}-\Pi_{2}}{\Pi_{+}+\Pi_{2}}\right)^{2}}+O(\omega),$
(67)

and the condition of suppression of the reflected wave, $\Pi_{+}=\Pi_{2}$, is
satisfied if $c(+\infty)>c_{2}$, i.e. requires an increase in the wave speed
downstream from the critical point $x_{2}$ (see Fig. 2(c)). But it is possible
to suppress the reflected wave using the above-described ‘anti-reflective
optics effect’, i.e. with the help of inverting insert with $a(x)=0$ in such
point $x=x_{a}$ where $\Pi^{2}(x_{a})=\Pi_{2}\Pi_{+}$.

Let us now turn to the region $x_{1}<x<x_{2}$ where the flow is supercritical.
If $x_{2}-x_{1}=O(1)$ and $a(x)=O(1)$, then Eq. (57) gives:

$r_{0}(x_{2}-0)=\dfrac{1-c^{2}_{2}}{1+c^{2}_{2}},\qquad
K_{2}=\left[1-\left(\dfrac{1-c^{2}_{2}}{1+c^{2}_{2}}\right)^{2}\right]^{-1/2}+O(\omega)=\dfrac{1+c^{2}_{2}}{2c_{2}}+O(\omega).$
(68)

In the case when the wave speeds at the ends of the region are the same
($c_{2}=c_{1}=1$), $r_{0}(x_{2}-0)=0$, and $K_{2}=1+O(\omega^{2})$ is almost
not differs from one. Again, one can significantly increase the transmission
ratio using an inverting insert. Indeed, since $\Pi_{1}=\Pi_{2}=1$, there is a
point $x=x_{m}$ at which $\Pi(x)$ reaches extreme (maximum or minimum) value
of $\Pi_{m}$. Here $a(x)$ changes sign, and the growth of $|r(x)|^{2}$ is
replaced by a decrease, which leads to a decrease in the amplitudes of waves
of both positive and negative energy. Changing at this point (more precisely,
on the inverting insert with $a=0$) the sign of $\cos{\theta}$ (and with it
the $r$ sign), we will continue to grow $|r(x)|^{2}$,

$r_{0}(x_{m}-0)=\dfrac{1-\Pi_{m}}{1+\Pi_{m}}\ \longrightarrow\
r_{0}(x_{m}+0)=-r_{0}(x_{m}-0)\ \longrightarrow\
r_{0}(x_{2}-0)=\dfrac{\Pi^{2}_{m}-1}{\Pi^{2}_{m}+1}\,,$ (69)

and amplitudes of both waves. As the result,
$K_{2}\approx(\Pi^{2}_{m}+1)/(2\Pi_{m})>1$, and to the greater extent, the
more strongly $\Pi_{m}$ differs from 1 (to either way, since
$K_{2}(\Pi_{m})\approx K_{2}(\Pi^{-1}_{m})$).

If $c_{2}\neq 1$, then $K_{2}>1$ and without an inverting insert, and, in
addition, $K_{2}(c_{2})=K_{2}(c_{2}^{-1})$ up to $O(\omega)$. In other words,
amplification of a wave of positive energy due to its transformation into a
wave of negative energy is promoted by both a decrease and an increase in
speed of waves in the supercritical section of the flow.

###  IV.3 The high-frequency limit, $\omega\gg 1$

Let us turn now to high frequency waves, $\omega\gg 1$. This limiting case can
be still consistent with the shallow-water approximation which requires in the
dimensional variables that $\omega\gg ck$, whereas $kh\ll 1$ and
$\sigma\ll\rho gh^{2}/3$, where $\sigma$ is the surface tension. As follows
from the analysis carried out in Section III, in the vicinity of the critical
point $x_{1}$, function $r(x)$ is defined by the series (43) the coefficients
of which have the order of $O(\omega^{-1})$. Therefore, it is natural to
assume that solution to Eq. (30) has the same order. Setting $r(x)={\rm
i}P(x)/\omega$ where $|P|=O(1)$, but $|{\rm d}P/{\rm d}x|=O(\omega)$, we get
the equation:

$\dfrac{1}{\omega}\,\dfrac{{\rm d}P}{{\rm d}x}=-{\rm i}\,\alpha(x)P+{\rm
i}\,a(x)\left(1+\dfrac{P^{2}}{\omega^{2}}\right).$

Neglecting the term $O(\omega^{-2})$ in this equation, we arrive at the linear
non-homogeneous equation whose solution subject to the boundary conditions
(50) is:

$P(x)={\rm i}\,\omega\,e^{-{\rm
i}\,\omega\Psi(x)}\displaystyle\int\limits_{x_{0}}^{x}a(\xi)\,e^{{\rm
i}\,\omega\Psi(\xi)}\,{\rm
d}\xi\,,\quad\mbox{where}\quad\Psi(x)=\int\\!\alpha(x)\,{\rm d}x,$ (70)

and $x_{0}=x_{1}$ when $x<x_{2}$ and $x_{0}=x_{+}$ when $x>x_{2}$.

In Eq. (70), when calculating the integral of a rapidly oscillating function,
we take into account that ${\rm d}\Psi/{\rm d}x=\alpha(x)$ has no zeros on the
real axis and therefore, there are no stationary phase points, but $\alpha(x)$
has poles at $x=x_{1,2}$. Hence, the main contribution to the integral comes
from a neighborhood of the upper limit of integration (see, for example, Olver
), then we have:

$\displaystyle\int\limits_{x_{0}}^{x}a(\xi)\,e^{{\rm
i}\,\omega\Psi(\xi)}\,{\rm d}\xi\,\sim\dfrac{-{\rm
i}\,a(x)}{\omega\,\alpha(x)}\,e^{{\rm i}\,\omega\Psi(x)}=O(\omega^{-1}).$ (71)

We see that indeed $P(x)=O(1)$ in general. However, $P(x)=o(1)$ when $x\to
x_{2}\pm 0$ because of presence of $\alpha(x)$ in the denominator. Thus, in
the high-frequency limit $K_{1}$ differs from unity by $O(\omega^{-2})$
whereas $K_{2}$ and $K_{3}$ differ from unity even less, only by
$o(\omega^{-2})$. Therefore, both the wave reflections in the subcritical
regions of the flow and wave amplification in the supercritical region
diminish.

##  V Wave propagation in the WH–BH duct model

Let us consider now a duct with a supercritical flow ($U(x)>c(x)$) in the
outer domains (left: $-\infty<x<x_{1}$), and right: ($x_{2}<x<+\infty$), and
subcritical in the inner domain ($x_{1}<x<x_{2}$). here we will use
dimensionless variables such that (cf. Eq. (49)):

$U_{2}=c_{2}=1,\qquad\mu_{2}=U^{\prime}_{2}-c^{\prime}_{2}=1.$ (72)

In this case, the physical statement of the problem is not so evident as for
the BH–WH duct considered in Section IV; we will present the detail discussion
of this delicate issue in Section VII. Here we study a restricted problem of
wave propagation only in the rightmost domain where $x>x_{1}$. Let us start
again with the BH horizon (at the point $x=x_{2}$ rather than at $x=x_{1}$!)
because at this point both the reflected wave and NEW vanish. Assume that the
co-current propagating wave arriving from the domain where $x<x_{1}$ has the
amplitude $D(x_{2})=1$. Then, in the domain $x_{1}<x<\infty$ (cf. Eqs (51) and
(52)) we have:

$\displaystyle D(x)$ $\displaystyle=$
$\displaystyle\exp\left[-\\!\displaystyle\int\limits_{x_{2}}^{x}a(x^{\prime})\,r(x^{\prime})\,{\rm
d}x^{\prime}\right],$ (73) $\displaystyle K(x)$ $\displaystyle=$
$\displaystyle\\!\exp\left[\\!-\\!\displaystyle\int\limits_{x_{2}}^{x}a(x^{\prime}){\rm
Re}\,\Bigl{(}r(x^{\prime})\Bigr{)}\,{\rm
d}x^{\prime}\right]\\!\equiv\\!\exp\left[-\\!\displaystyle\int\limits_{x_{2}}^{x}a(x^{\prime})|r(x^{\prime})|\cos\theta(x^{\prime})\,{\rm
d}x^{\prime}\right],$ (74)

and the integral transmission ratios of sub- and supercritical domains are
(cf. Eq. (53)):

$K_{2}=\sqrt{1-|r(x_{1})|^{2}}\leq 1,\quad
K_{3}=\Bigl{[}1-|r(+\infty)|^{2}\Bigr{]}^{-1/2}\geq 1,\quad
K(+\infty)=K_{2}\cdot K_{3}.$ (75)

It should be borne in mind that the oscillating component of $r(x)$ develops
with the distance from $x_{2}$ rather than from $x_{1}$.

Scaling as per Eq. (72) yields $\Pi_{2}\equiv c_{2}U_{2}=1$, therefore, the
scattering of low frequency waves in the subcritical domain is approximately
described by Eq. (57). Then, the transmission ratio in this domain is:

$K_{2}=\sqrt{1-\left(\dfrac{1-c^{2}_{1}}{1+c^{2}_{1}}\right)^{2}}+O(\omega)=\dfrac{2c_{1}}{1+c^{2}_{1}}+O(\omega).$
(76)

From this formula, it follows (bearing in mind a small contribution of the
last term $O(\omega)$) that $K_{2}$ is only slightly less than unity if the
wave velocities in the ending points of the domain are the same,
$c_{1}=c_{2}=1$, but $K_{2}$ decreases when $c_{1}$ deviates from $c_{2}$ in
the either side. It should be also noted that $K_{2}(c_{1})\approx
K_{2}(1/c_{1})$.

In the high-frequency limit, one can demonstrated by analogy with Section IV.2
that $K_{2}\to 1-0$ for any $c_{1}$. As a consequence of this, when $c_{1}=1$,
$K_{2}(\omega)$ goes to unity in the both limiting cases, when $\omega\to 0$
or $\omega\to\infty$, and inside this frequency range, it has at least one
minimum. If $c_{1}\neq 1$, then $K_{2}(\omega)$ grows from $K_{2}(0)<1$ up to
1, but not necessarily monotonically (see right panel in Fig. 15).

In the right outer domain $x>x_{2}$, the transformation of a positive-energy
low-frequency wave into a negative energy wave is described by the same
equation (57), and the transmission ratio is:

$K_{3}=\left[1-\left(\dfrac{\Pi(+\infty)-1}{\Pi(+\infty)+1}\right)^{2}\right]^{-1/2}+O(\omega)=\dfrac{\Pi(+\infty)+1}{2\sqrt{\Pi(+\infty)}}+O(\omega).$
(77)

Asymptotically $K_{3}\sim\sqrt{\Pi(+\infty)}/2$ increases with
$\Pi(+\infty)\equiv c(+\infty)U(+\infty)$. However, when the frequency
increases $K_{3}$ approaches unity from the top.

In the conclusion of this section we note that in Ref. Euve17  it was
demonstrated experimentally that analogue traversable and bidirectional
wormholes can exist in dispersive hydrodynamics when the surface tension
effect is taken into consideration. Then the capillary wavelength plays the
role of a Planckian scale below which long gravity waves are transformed into
short capillary waves that are able to move with the “superluminal” speeds.
Whereas the results obtained are not applicable to putative astrophysical
wormholes per se, they are of interest from the hydrodynamic point of view.

##  VI Results of numerical calculations

###  VI.1 BH–WH duct

The analysis presented above shows that the most interesting scenarios of wave
propagation occur in the middle, supercritical, flow region where a positive
energy wave is amplified due to the coupling with the negative energy wave;
then the transmission coefficient can be noticeably greater than one.
Numerical calculations were performed mainly for this region. Recall that at
the far end of the region where $x_{2}-x\sim\exp(-\omega^{-1})$, the function
$r(x)$ rapidly oscillates; therefore, the spatial resolution of the
oscillating solution in the numerical calculations can be performed without
extra complications only for relatively high $\omega$. Below we consider
separately two particular cases when the wave speed is (i) the same at the
ending points of the supercritical interval and (ii) when it is different.

####  VI.1.1 Currents with the equal velocities at the ending points

Bearing in mind the conditions (49), we take for calculations a simple flow
model with $c_{1}=c_{2}=1$ (see Fig. 2(a)):

$c(x)\equiv 1,\qquad U(x)=1+x(2d-x)/(2d),\quad x_{1}=0,\ \ x_{2}=2d.$ (78)

For such a flow, the Mach number attains its maximum, ${\cal M}=1+d/2$, at the
midpoint of the supercritical interval, $x_{m}=d$, and function $a(x)$ changes
its sign. The phase-shifting insert at this point is simulated by changing of
the phase $\theta=\arg(r)$ by the required value $\Delta\theta$. Note that in
the limit $\omega\to 0$, the most effective insert is inverting one, with
$\Delta\theta=\pm\pi$, we have considered it above. However, when the
frequency increases, the phase $\theta$ changes more and more significantly in
that region where $a(x)\neq 0$, therefore the optimal phase shift that
maximizes the transmission ratio $K_{2}$, depends on the frequency. Specific
values of $\Delta\theta(\omega)$ used in our calculations for the flow with
${\cal M}=4$ ($d=6$) are given in Table 1 (see also Fig. 3).

Table 1: Data for the optimal phase shift versus frequency used in the numerical calculations. $\omega$ | 0.2 | 0.3 | 0.5 | 1 | 3 | 5 | 8
---|---|---|---|---|---|---|---
.$\Delta\theta$ . |  . 2.688 . |  . 2.443 . |  . 1.990 . |  . 0.977 . |  . $-1.187$ . |  . $-2.234$ . |  . $-3.002$ .

Figure 3: (color online) Optimal phase shift versus frequency. Dots show the
values used in the numerical calculations.

Figure 4: (Color online) The reflection coefficient (left panels) and function
$\theta(x)$ (right panel) as functions of $x$ for three values of frequency
$\omega\leq 1$ in the flow model (78) with ${\cal M}=4$: line 1 – NI; line 2 –
OI. Dashed lines correspond to the reference case with $\omega=0$ and
$\Delta\theta=\pi$: line 3 – NI; line 4 – OI.

Figure 4 shows the reflection coefficient $|r(x)|$ that describes the
transformation of the co-current propagating wave into the negative energy
wave, as well as $\theta(x)\equiv\arg(r)$ for the frequencies $\omega\leq 1$,
both with the optimal phase-shifting insert (OI), and without it (NI). The
upper part of Fig. 4 shows that even when the frequency is not very small, the
“zero” approximation $|r_{0}(x|$ as defined by the Eq. (56) (and with phase
inversion, if any) describes pretty well the behavior of function $|r(x)|$,
except for the neighborhood of the point $x=x_{2}=2d$ in the NI version when
there is no insert. The difference between the curves is caused by missing in
$r_{0}(x)$, but accumulated in $r(x)$ (due to $\omega\neq 0$) a rapidly
oscillating component, although its amplitude is significantly less than in
the OI version.

Figure 5: (Color online) The reflection coefficient (left panels) and function
$\theta(x)$ (right panel) as functions of $x$ for two values of the frequency
$\omega>1$ in the flow model (78) with ${\cal M}=4$: line 1 – NI; line 2 – OI.

For $\omega=1$, the downstream change in $\theta$ manifests faster, therefore
$|r(x)|$ and $|r_{0}(x)|$ differ notably throughout the entire supercritical
region, in both versions of the flow, with the insert (OI-version) and without
the insert (NI-version). Note that when $x\to x_{2}=2d$, the reflection
coefficients (which are equal here to the amplitudes of fast oscillations of
the functions $r(x)$) corresponding to the different flow versions, approach
each other (cf. lines 1 and 2 in Fig. 4) and slightly decrease compared with
what was for $\omega=0.3$ in the OI-version. Moreover, we draw attention to
the fluctuation of $|r(x)|$ near the point $x_{2}$, which became noticeable in
the result of expansion of the region of oscillations.

Figure 5 shows the reflection coefficient $|r(x)|$ and function $\theta(x)$
for $\omega>1$ for the both flow models, OI (with insert) and NI (no insert).
Here the region of oscillations is even wider, and oscillations of $|r(x)|$
are seen in a notably greater range of $x$. When $x\to x_{2}$, these
oscillations, as expected, decay, and $|r(x)|$ approaches a finite limit which
is equal to the amplitude of oscillations of the function $r(x)$. Comparison
of graphs for $\omega=1,\ 3,\ 8$ confirms the conclusion made at the end of
Section IV that $|r(x)|$ decreases as $\omega^{-1}$, and even faster in the
vicinity of $x_{2}$.

Figure 6: (Color online) The gain coefficient as the function of $x$ for the
several particular frequencies: 1 – $\omega=0.2$, 2 – $\omega=0.5$, 3 –
$\omega=1$, 4 – $\omega=3$, 5 – $\omega=5$, 6 – $\omega=8$. Solid lines
pertain to the NI model, dashed lines – to the OI model.

In the course of propagation from $x_{1}$ to $x_{2}$, waves of positive and
negative energies interact such that the total energy flux (33) conserves. As
the result, their amplitudes synchronously increase or decrease in according
with the change in their phase difference. Quantitative measure of wave
interaction is the gain of the positive energy wave:

$Q(x)=\dfrac{1}{1-|r(x)|^{2}};$ (79)

this quantity is shown in Fig. 6. According to Eq. (34), the growth is
replaced by the decrease when the sign of either $a(x)$ or $\cos{\theta}$
changes.

At low frequencies ($\omega<1$) in the OI model of the flow, the change of the
function $a(x)$ sign is compensated to a large extent by the phase jump,
therefore $Q(x)$ grows monotonically and begins slightly oscillate only near
the point $x_{2}$ (see curves 3). As the frequency increases, $\theta(x)$
varies more and more rapidly, and $Q(x)$ acquires more and more distinct
oscillatory character whereas its value becomes closer and closer to unity.

Figure 7 shows the dependences of the transmission ratio $K_{2}$ of the
supercritical domain as the functions of frequency $\omega$ for the NI (line
1) and OI (line 2) models. As one can see from the comparison of lines 1 and
2, both models, with the optimal phase-shifting insert (OI) and without it
(NI), provide approximately the same transmission ratio for $\omega>1$,
whereas they differ in the low-frequency domain. Whereas the OI model provides
monotonic increase of $K_{2}$ when $\omega\to 0$, in the NI model, $K_{2}$ has
a maximum at $\omega=1$, and then goes to zero when $\omega\to 0$. Even in the
NI model without any insert, the transmission ratio is notably greater than
one, what can provide the laser effect of wave amplification in the active
zone.

Figure 7: (Color online) The transmission ratio $K_{2}$ as functions of
frequency $\omega$ for the NI (line 1) and OI (line 2) models.

The amplitude gain in the supercritical domain may be drastically reduced by
wave reflection in the left and right subcritical domains. To gain a better
insight into the effect of domain competition, we have calculated the wave
transmission through all three domains in the bell-shaped velocity profile
(see Fig. 2a) with following dependences:

$c(x)\equiv 1,\quad{\rm and}\quad U(x)=U_{0}+(M-U_{0})\mathop{\rm
sech}\nolimits{x}.$ (80)

Calculations were performed for the fixed Mach number ${\cal M}=4$ and with
various velocities at the infinity $U_{0}\in(0,\,1)$. The results obtained are
presented in Fig. 8.

Figure 8: (Color online) Top panels: left – velocity profiles (80) for
$U_{0}=0.1$ and $U_{0}=0.8$, and right – the total transmission ratio
$K=K_{1}K_{2}K_{3}$ as function of frequency $\omega$. Bottom panels: left –
transmission ratios of the left ($K_{1}$) domain and right – for the middle
($K_{2}$) domain as functions of frequency $\omega$; 1 – $U_{0}=0.1$, 2 –
$U_{0}=0.4$, 3 – $U_{0}=0.8$.

As one can see, the supercritical flow with any $U_{0}$ acts as a broadband
amplifier (see the top right panel in Fig. 8). This can be explained as
follows: the wave reflection plays a noticeable role only at low frequencies,
whereas the transmission ratio $K_{2}$ responsible for the wave amplification
decreases with $\omega$ rather slowly. It should be noted in passing that due
to the flow symmetry, the transmission ratios in the subcritical domains of
the flow are equal, $K_{1}=K_{3}$, so that the total transmission ratio is
$K=K_{1}^{2}K_{2}$. And finally, when comparing $K_{2}(\omega)$ shown in Fig.
8 with that presented in Fig. 7 by curve 1, it should be born in mind that in
calculations presented in Fig. 8 the scaling (49) was not fulfilled. Due to
this, the dimensionless frequencies of the maximal amplification are
different.

####  VI.1.2 Currents with the different velocities at the ending points

In this section, we consider currents in which wave velocity $c(x)$
monotonically increases or decreases downstream from $c_{1}=1$ to $c_{2}$. For
calculations, we take a modification of the flow model (78) without a phase-
shifting insert (see Fig. 2(b)),

$c(x)=1+\dfrac{c_{2}-1}{2d}\,x,\qquad U(x)=c(x)+\dfrac{x(2d-x)}{2d},\quad
x_{1}=0,\ \ x_{2}=2d.$ (81)

In such a flow, the maximum Mach number is attained at
$x=2d/(1+\sqrt{c_{2}})$:

${\cal M}=1+\dfrac{2d}{(1+\sqrt{c_{2}})^{2}}\,,$ (82)

so that for the given Mach number we have:

$d=\dfrac{{\cal M}-1}{2}\Bigl{(}1+\sqrt{c_{2}}\Bigr{)}^{2}.$ (83)

Figure 9: (Color online) Dependence of $|r(x)|$ in the flows with increasing
wave velocity $c(x)$ for different values of $c_{2}$ and frequencies shown by
the numbers next to the curves. Dotted line pertains to $\omega=0$.

Let us assume that in the entire region function $\Pi(x)=c(x)U(x)$ varies
monotonically, and $a(x)$ does not change its sign. This limits the range of
the flow parameters, so that

$d\leq|c_{2}-1|,\qquad 1<{\cal
M}\leq\left\\{\begin{array}[]{ll}3-\dfrac{4}{1+\sqrt{c_{2}}}\,,\quad\mbox{if}\quad
c_{2}>1,\\\ \dfrac{4}{1+\sqrt{c_{2}}}-1\,,\quad\mbox{if}\quad
c_{2}<1.\end{array}\right.$ (84)

Figure 10: (Color online) The gain coefficient in the flows with increasing
wave velocity $c(x)$ for the different frequencies and ending values of
$c_{2}$ shown by numbers next to the curves).

The parameters $c_{2}$ and ${\cal M}$ used for calculations are shown in Table
2. For a given $c_{2}$, the Mach number ${\cal M}$ was chosen close to the
maximum value, and then the parameter $d$ was calculated using Eq. (83).

Table 2: The Mach numbers used in the numerical calculations with the different values of $c_{2}$. $\phantom{.}c_{2}\phantom{.}$ | 2 | 3 | 5 | 9 | $2/3$ | 0.5 | $1/3$ | $1/6$ | 0.1
---|---|---|---|---|---|---|---|---|---
$\phantom{.}{\cal M}\phantom{.}$ |  . 1.3 . |  . 1.5 . |  . 1.6 . |  . 2 . |  . 1.2 . |  . 1.3 . |  . 1.5 . |  . 1.8 . |  . 2 .

Figure 11: (Color online) Dependence of $|r(x)|$ in the flows with the
decreasing wave velocity $c(x)$ for the different values of $c_{2}$ and
frequencies: 1 – $\omega=0.2$, 2 – $\omega=0.5$, 3 – $\omega=1$, 4 –
$\omega=2$, 5 – $\omega=5$.

For the flows with the increasing velocity $c(x)$, Fig. 9 shows the dependence
$|r(x)|$ for different $c_{2}$ and $\omega$. It is clearly seen that the
transformation into the negative energy wave increases with increasing of
$c_{2}$, but falls abruptly when the frequency growths. This is also exhibits
by the graphs of the gain coefficient $Q$ for the same set of parameters (see
Fig. 10).

Similar dependences for the flows with the decreasing function $c(x)$ are
shown in Figs. 11 and 12. They demonstrate the strengthening of the wave
transformation with the decrease of the ending value $c_{2}$ and its sharp
weakening with the increase of $\omega$. The oscillatory nature of the
transformation near $x_{2}$ is seen even more clearly than for $c_{2}>1$.

Figure 12: (Color online) The gain coefficient in the flows with decreasing
wave velocity $c(x)$ for the different frequencies and ending values of
$c_{2}$ shown by numbers next to the curves in the upper left panel.

Figure 13 shows the transmission ratio $K_{2}$ of the inner domain as function
of frequency $\omega$ for the different values of $c_{2}>1$ (left panel) and
$c_{2}<1$ (right panel). As one can see from the figure, the transmission
ratio in the both panel monotonically decreases from some maximal value
$K_{2}(\omega=0)$, which depends on $c_{2}$, to zero.

Figure 13: (Color online) The dependences of the transmission ratio $K_{2}$ of
the inner domain as functions of frequency $\omega$ for the different values
of $c_{2}>1$ (left panel) and $c_{2}<1$ (right panel). Numbers next to curves
show the values of $c_{2}$.

###  VI.2 WH–BH duct

For this type of ducts, the inner domain, $x_{1}<x<x_{2}$, is subcritical; the
corresponding transmission ratio is less than one, $K_{2}(\omega)\leq 1$. The
frequency dependence of $K_{2}(\omega)$ is determined by the particular
profiles $c(x)$ and $U(x)$ (see Section IV.3). For the numerical calculations,
we chose a family of flows with the linear $c(x)$ and quadratic $U(x)$
profiles (cf. Eqs. (78) and (81)),

$c(x)=1+\dfrac{(c_{1}-1)(2d-x)}{2d},\qquad U(x)=c(x)-\dfrac{x(2d-x)}{2d}.$
(85)

Here $c_{1}=c(0)$, and $d$ is chosen such that $U(x)>0$ in the entire interval
$x_{1}=0\leq x\leq x_{2}=2d$.

If $c_{1}=c_{2}=1$, then $c(x)\equiv 1$, and $\Pi(x)=c(x)U(x)\equiv U(x)$. In
this case, the transmission ratio $K_{2}(\omega)\to 1$ when the frequency goes
either to zero or to infinity, and its minimum decreases with the decreasing
of $U_{min}=(1-d/2)$ (see Fig. 14). If, however, $c_{1}\neq 1$, then
$K_{2}(0)<1$ (see Eq. (76)) and $K_{2}(\omega)$ increases to 1 (not
necessarily monotonically) when frequency grows (see right panel in Fig. 15).

Figure 14: (Color online) Graphics of the flow as per Eq. (85) with
$c_{1}\equiv 1$. Left panel: the flow velocity $U(x)\equiv\Pi(x)$ as the
function of $x$. Right panel: the frequency dependence of the transmission
ratio $K_{2}(\omega)$. Lines 1 are plotted for $d=1$ with $U_{min}=0.5$; lines
2 – for $d=1.5$ with $U_{min}=0.25$.

The positive energy wave is amplified after passing into the supercritical
domain $x>x_{2}$ due to the interaction with the NEW FabrStep , therefore, the
transmission ratio $K_{3}\geq 1$. For the calculations, we choose $c(x)$ and
$U(x)$ in such that the scaling (72) is satisfied and the finite limits exist
when $x\to+\infty$. In particular, if $c(x)\equiv 1$, than we take

$U(x)=1+D\tanh(x/D),\quad x_{2}=0.$ (86)

The results of calculations for $D=1$ ($U_{max}=2$) and $D=3$ ($U_{max}=4$)
are shown in Fig. 16.

Figure 15: (Color online) Graphics of the flow as per Eq. (85) with $c_{1}\neq
1$. Left panel: the flow velocity $U(x)$ (solid lines) and $\Pi(x)$ (dashed
lines) as functions of $x$. Right panel: the frequency dependence of the
transmission ratio $K_{2}(\omega)$. Lines 1 are plotted for $c_{1}=0.5$,
$d=1$; lines 2 – for $c_{1}=2$, $d=1$; lines 3 – for $c_{1}=2$, $d=1.5$.

Figure 16: (Color online) The flow (86), $c(x)\equiv 1$. Left panel: flow
velocity $U\equiv\Pi$ vs $x$. Right panel: frequency dependence of the
transmission ratio $K_{3}$. 1 – $D=1$ ($U_{max}=2$), 2 – $D=3$ ($U_{max}=4$).
For comparison, the dashed line presents the curve 2 plotted in the right
panel of Fig. 17.

In the more interesting case when $c(x)$ and $U(x)$ grow simultaneously with
$x$ (and $U(x)\geq c(x)$ everywhere), the calculations were carried out for
current and wave speed in the following forms, respectively:

$U(x)=1+A_{U}\tanh\left(\dfrac{x}{D_{U}}\right),\quad
c(x)=1+A_{c}\tanh\left(\dfrac{x}{D_{c}}\right),$ (87)

with the various combinations of parameters satisfying the condition:

$\dfrac{A_{U}}{D_{U}}-\dfrac{A_{c}}{D_{c}}=1,$

which follows from the scaling (72). Figure 17 presents the results of
calculations for the three cases:

$\begin{array}[]{lll}{\rm Case\ 1:}&\displaystyle A_{U}=\frac{5}{3},\
D_{U}=\frac{4}{3},&A_{c}=0.5,\ D_{c}=2,\\\ {\rm Case\ 2:}&A_{U}=2,\
D_{U}=1.6,&\displaystyle A_{c}=\frac{1}{3},\ \ \ D_{c}=\frac{4}{3},\\\ {\rm
Case\ 3:}&\displaystyle A_{U}=1,\ D_{U}=0.8,&A_{c}=1,\ \ \
D_{c}=4.\end{array}$ (88)

Figure 17: (Color online) Left panel: $U(x)$ (solid lines), $\Pi(x)$ (dashes),
and $c(x)$ (dash-dotted lines) graphs. Right panel: frequency dependence of
the transmission ratio $K_{3}$. Figures indicate the number of flow, see Eq.
(88).

As one can see from Fig. 17, the dependence $K_{3}(\omega)$ in the Case 3
notably deviates from the dependences $K_{3}(\omega)$ in the Cases 1 and 2.
There are two distinction peculiarities in the Case 3. Firstly, the
characteristic width of variation of the wave speed, $D_{c}$, significantly
exceeds (5 times) that of current variation $D_{U}$. Secondly,
$U(+\infty)=c(+\infty)$, that is there is a third critical point at the
infinity. The former distinction feature seems, however, not very important
since even if $c(x)\equiv 1$, $K_{3}$ depends on $\omega$ in the nearly the
same way as in the Cases 1 and 2 (cf. line 2 and dashed line in the right
panel of Fig. 16). On the contrary, the presence of an additional (albeit
infinitely remote) critical point has a profound impact on the behavior of
function $r(x)$ which determines the progress of the wave transformation.

Figure 18 demonstrates the absolute value and argument of $r(x)$ for the Case
2 and Case 3; it allows us to compare the wave propagation in these two cases
with the greater detail. It is clearly seen that in the Case 3 the
oscillations of $r(x)$ begin to develop noticeably at a much smaller distance
from the critical point $x_{2}=0$, and this results in lesser values and more
pronounced oscillatory character of $|r(x)|$. Moreover, when $x$ increases,
the oscillation period in the Case 3 decreases whereas in the Case 2 it
approaches a constant value.

Figure 18: (Color online) The absolute value and argument of the function
$r(x)$ for the Case 2 (top) and Case 3 (bottom). Different numbered curves
pertain to different frequencies: 1 – $\omega=0$, 2 – $\omega=0.5$, 3 –
$\omega=1$, 4 – $\omega=2$, 5 – $\omega=5$.

##  VII DISCUSSION AND CONCLUSION

We have carried out the analysis of a simple harmonic wave propagation in two
types of ducts, BH–WH and WH–BH, and have illustrated the theoretical results
by numerical calculations. Note that in contrast to Ref. Peloquin  where the
transformation of dispersive gravity-capillary waves were studied, in our
paper devoted to dispersionless purely gravity long waves, each of the ducts
is traversable only in the co-current direction, from left to right in the
considered geometry. However, the problem statements for the aforementioned
types of ducts are different and should be specified.

At the ends of a BH–WH duct far from the middle supercritical domain, the flow
is subcritical (see Fig. 1), and the problem statement is rather traditional.
A wave running from the left in the course of propagation to the critical
point $x_{1}$ is partially reflected due to scattering on the flow
inhomogeneity. A reflected wave runs to the left and, therefore does not have
any influence on the wavefield downstream of the flow. Upon attaining the
point $x_{1}$, the positive energy incident wave passes through this critical
point keeping its amplitude. In the supercritical domain at $x>x_{1}$ the wave
propagates to the second critical point $x_{2}$ being amplified along the path
due to the interaction with the negative energy wave (NEW) which is generated
due to scattering of the positive energy wave on the flow inhomogeneity. In
the result of such coupling, amplitudes of both waves of positive and negative
energies grow simultaneously FabrStep . The NEW completely absorbs in the
vicinity of the critical point $x_{2}$, whereas the wave of a positive energy
passes through this point keeping its amplitude and penetrating into the right
subcritical domain. In this domain, the penetrated wave propagates towards the
infinity losing its energy for generation of reflected wave of positive energy
due to flow inhomogeneity. Calculation of both the transmission ratio $K_{2}$
for the ‘active’ duct domain with the supercritical flow and the transmission
coefficient (ratio of the amplitudes of the transmitted and incident waves) do
not cause any principal problem in this case. Note that the supercritical
domain plays a role of an active zone of the open-boundary laser.

At the ends of a WH–BH duct far from the middle subcritical domain, the flow
in contrast to the previous case is supercritical. Therefore, waves, both with
positive and negative energies, entering the duct from the left end, propagate
to the critical point $x_{1}$ interacting with each other in such a way that
the total energy flux ${\cal E}$ determined by Eq. (33) is conserved, whereas
the amplitudes of both waves increase. In any case, only the finite domain to
the left of the point $x_{1}$ is accessible for a observer, and in any point
of it, one can see a superposition of both waves having no possibility to
separate which of these two waves arrives from the left infinity. In such a
situation, the only doubtless statement is, if only a positive energy wave is
incoming from the left, then ${\cal E}>0$, whereas the converse is not true,
in general. However, as has been shown in the paper, in the neighbourhood of
the critical point $x_{1}$, any NEW is completely absorbed. Therefore, it
seems that the only reasonable statement of the problem in this case is to set
the amplitude of the co-current propagating wave at either end of the
subcritical domain and ignore the prehistory of wave propagation in the
supercritical domain at $x<x_{1}$. Technically, it is more convenient to set
the amplitude of the co-current propagating wave of positive energy at
$x=x_{2}$, because both the reflected wave and NEW vanish in this point. When
this is done, the picture of the wave propagation becomes quite clear.
Integrating Eq. (30) back from $x_{2}$ to $x_{1}$ and using Eq. (33), one can
find the amplitudes of co-current propagating and reflected (counter-current
propagating) waves at $x=x_{1}+0$ and determine what part of the wave energy
penetrated from the left domain $x<x_{1}$ is lost in the middle domain between
$x_{1}$ and $x_{2}$ due to the reflection on the flow inhomogeneity. Further,
when the positive energy wave enters the supercritical right domain $x>x_{2}$,
we can calculate its amplification due to the transformation into a NEW. This
is precisely the statement of the problem that is adopted in the paper.

Another interesting question is the choice of a flow model to simulate a
curved space-time, in particular, the question what should be the effective
metric at the infinity. However, such problem statement is practically
irrelevant for the WH–BH duct models because in this case the space at the
infinity is “unusual”, with the metric signature turned inside out. But for
the BH–WH duct the choice of the model presumes, in general, that the
curvature vanishes at the infinity, and the analog metric reduces to the
Minkowski metric. At a first glance, for the metric as per Eq. (17) this
requirement is equivalent to the condition $U(\pm\infty)=0$, which makes the
problem considered above incorrectly posed. Indeed, let $c(x)$ is finite
everywhere and $U(x)\to 0$ when $x\to\pm\infty$. Then, first of all, we have a
problem with the mass flux conservation Eq. (2). To cope with this problem, we
need to assume that the duct width $W(x)\to\infty$ as $x\to\pm\infty$.
Further, if a wave of a finite amplitude $A(-\infty)=A_{0}\neq 0$ arrives from
the left, then the perturbation (20) remains finite at least up to the first
critical point $x=x_{1}$. The energy flux (33) in this case will be finite
(and constant) in this region too, including the far zone, $x\to-\infty$,
where function $\Pi(x)=c(x)U(x)\to 0$. However, this is possible only in the
case of the total wave reflection, $|r(-\infty)|=1$. It is interesting that
the energy flux, in this case, can be quite different from zero and, being
unchanged, is transferred to the next, supercritical, region. Then, after the
inevitable transformation at the second critical point $x=x_{2}$, the energy
flux can be transferred in the last, subcritical region. Here, in the case
$U(+\infty)=0$ the reflection coefficient must be also equal to one, which
does not prevent the transmitted wave to have a nonzero amplitude at the
infinity $A(+\infty)$. However, it is impossible, in this case, to find the
transmission coefficient (the ratio of the amplitudes of transmitted and
incidents waves) without additional assumptions.

To avoid such a problem, we assume that the flow velocity $U(x)$ is positive
everywhere, and $c(x)$ and $U(x)$ have finite limits $c_{\pm}>U_{\pm}>0$ when
$x\to\pm\infty$, so that the analog space-time is flat at the infinity. The
easiest way to verify this is to make a transformation to the new temporal
coordinate (see, for example, Barc04 ) ${\rm d}\tau={\rm d}t-U{\rm
d}x/(c^{2}-U^{2})$, which reduces the interval to the Schwarzschild-like form:

${\rm
d}s^{2}=\left(\dfrac{G_{2}}{c\,U}\right)^{2}\left[-\Bigl{(}c^{2}-U^{2}\Bigr{)}{\rm
d}\tau^{2}+\dfrac{c^{2}{\rm d}x^{2}}{c^{2}-U^{2}}+c^{2}{\rm d}y^{2}\right].$

Therefore, the flow models considered in this paper are quite suitable for the
modelling of wave transformation in the curved space-time for both BH–WH and
WH–BH duct models.

The performed analysis has highlighted a very significant influence of the
“geometric factor” $\Pi(x)$ on the wave propagation. Firstly, its constancy
provides reflectionless wave propagation regardless of the frequency.
Secondly, in the BH–WH ducts, the reflection coefficient in the subcritical
flow regions depends mainly on the behavior of function $\Pi(x)$ rather than
on the Mach number at the infinity, ${\cal M}_{\pm\infty}=U/c$ . In addition
to that, amplification of positive energy waves in the supercritical region is
also determined mainly by the difference in the values of $\Pi(x)$ at the
ending points of the ‘active’ (supercritical) region. With regard to this, we
recall that at low frequencies, the gain does not depend which of the speeds
is greater, $c_{1}$ or $c_{2}$ but is determined entirely by their ratio.

In a WH–BH duct, the $\Pi(x)$ behavior has a profound impact on the wave
propagation as well. However, the frequency dependence of the transmission
ratio $K_{3}$ of its final (supercritical) part is highly sensitive to that
does the Mach number tend to unity at the infinity or not (see Fig. 17).

In this paper we have considered quite general models of stationary flows
variable in space. Such flows can be realised in laboratory water tanks with
the varying tank width and depth. As has been shown, the metric provided by
the flows for long surface waves is similar to the Schwarzschild-like metric
in the general relativity. In the linear approximation, we have derived a
second-order ODE describing the wave amplitude in the non-uniform flow with
two critical points that mimic the BH and WH horizons in the general
relativity. The derived ODE can be reduced to the set of two first-order ODEs
containing critical points. Solutions to this set of equations can be
investigated analytically in the vicinity of each critical point. This allowed
us to determine the conditions of the optimal wave amplification for low- and
high-frequency waves. The analytical results have been illustrated by the
direct numerical solutions of the derived set of equation.

In the next paper, we plan to present the detailed results on the wave
amplification after passing through the ‘active’ (supercritical) domain using
exact analytical solutions on the wave propagation in the piece-wise linear
background flow in shallow water. We will derive the transformation
coefficients (the transmission and reflection coefficients) in the analytic
form and quantify the results obtained in terms of the wave frequency, current
speeds in the subcritical and supercritical domains, and length of the
supercritical domain. Ultimately this should shed light on the intriguing
problem of wave propagation through the wormholes if they exist in nature
indeed. But, at least, the results obtained can be of interest to the
interpretation of wave amplification in the natural estuaries (rivers and
canals) or laboratory tanks. The validation of the theoretical results in the
laboratory sets up is a challenge for the experimenters.

###### Acknowledgements.

S.C. was financially supported by the Ministry of Science and Higher Education
of the Russian Federation. Y.S acknowledges the funding of this study provided
by the grant No. FSWE-2020-0007 through the State task program in the sphere
of scientific activity of the Ministry of Science and Higher Education of the
Russian Federation, and the grant No. NSH-2685.2018.5 provided by the
President of Russian Federation for the State support of leading Scientific
Schools of the Russian Federation. The final version of this paper was
completed when YS was a guest visitor of the Sydney University working on the
SMRI grant in March–April 2021.

##  Appendix A Influence of the viscosity in the vicinity of critical points

To investigate the details of a solution in the vicinity of a critical point
where $c(x)=U(x)$, let us take into account a small viscosity in the medium.
Supplementing Eq. (3) with a viscous term and introducing the velocity
potential $u=\partial\varphi/\partial x$, we derive:

$\dfrac{\partial u}{\partial t}+\dfrac{\partial(Uu)}{\partial
x}=-g\dfrac{\partial\eta}{\partial x}+\nu\,\dfrac{\partial^{2}u}{\partial
x^{2}}\ \ \ \Longrightarrow\ \ \ \dfrac{\partial\varphi}{\partial
t}+U\,\dfrac{\partial\varphi}{\partial
x}-\nu\,\dfrac{\partial^{2}\varphi}{\partial x^{2}}=-g\eta,$

where $\nu\ll 1$ is the kinematic viscosity. Substitution of this into Eq. (6)
yields:

$\begin{array}[]{r}\nu\left(\dfrac{\partial}{\partial
t}+U\,\dfrac{\partial}{\partial
x}-2U\,\dfrac{c^{\prime}}{c}\right)\dfrac{\partial^{2}\varphi}{\partial
x^{2}}+(c^{2}-U^{2})\dfrac{\partial^{2}\varphi}{\partial
x^{2}}-2U\,\dfrac{\partial^{2}\varphi}{\partial t\partial x}\\\ \\\
+\,\left[2U^{2}\,\dfrac{c^{\prime}}{c}-(c^{2}+U^{2})\,\dfrac{U^{\prime}}{U}\right]\dfrac{\partial\varphi}{\partial
x}-\dfrac{\partial^{2}\varphi}{\partial
t^{2}}+2U\,\dfrac{c^{\prime}}{c}\,\dfrac{\partial\varphi}{\partial
t}=0.\end{array}$

For a single harmonic, $\varphi(x,t)=\phi(x){\rm e}^{-{\rm i}\,\omega\,t}$, we
have:

$\nu U\,\dfrac{{\rm d}^{3}\phi}{{\rm d}x^{3}}+\left(c^{2}-U^{2}-2\nu
U\,\dfrac{c^{\prime}}{c}-{\rm i}\,\omega\,\nu\right)\dfrac{{\rm
d}^{2}\phi}{{\rm d}x^{2}}$
${}+\left[2U^{2}\,\dfrac{c^{\prime}}{c}-(c^{2}+U^{2})\,\dfrac{U^{\prime}}{U}+2\,{\rm
i}\,\omega\,U\right]\dfrac{{\rm d}\phi}{{\rm d}x}+\left(\omega^{2}-2\,{\rm
i}\,\omega\,U\,\dfrac{c^{\prime}}{c}\right)\phi=0.$ (89)

In the neighborhood of the BH horizon (say, $x=x_{1}$), we put
$x-x_{1}=\varepsilon\xi$, where $\varepsilon^{2}=\nu/(2\mu_{1})$,
$\mu_{1}=U^{\prime}_{1}-c^{\prime}_{1}>0$, and expand $c(x)$ and $U(x)$ in the
Taylor series. Then Eq. (89) yields in the leading order:

$\dfrac{{\rm d}^{3}\phi}{{\rm d}\xi^{3}}-\xi\,\dfrac{{\rm d}^{2}\phi}{{\rm
d}\xi^{2}}-(1-\beta_{1})\,\dfrac{{\rm d}\phi}{{\rm
d}\xi}=0,\quad\beta_{1}=\frac{\omega}{\mu_{1}}>0.$ (90)

After integration of this equation over $\xi$, we obtain:

$\dfrac{{\rm d}^{2}\phi_{1}}{{\rm d}\xi^{2}}-\xi\,\dfrac{{\rm d}\phi_{1}}{{\rm
d}\xi}+{\rm i}\,\beta_{1}\,\phi_{1}=0,\qquad\phi_{1}=\phi-\phi_{01},\ \
\phi_{01}={\rm const}.$ (91)

Then, by means of the ansatz $\phi_{1}(\xi)={\rm e}^{\xi^{2}/4}G(\xi)$, we
obtain the equation of the parabolic cylinder for the function $G(\xi)$:

$\frac{d^{2}G}{d\xi^{2}}+\left({\rm
i}\,\beta_{1}+\dfrac{1}{2}-\dfrac{\xi^{2}}{4}\right)G=0.$ (92)

Solution to this equation can be presented in terms of the functions ${\cal
D}_{{\rm i}\,\beta_{1}}(\pm\xi)$ and ${\cal D}_{-{\rm i}\,\beta_{1}-1}(\pm{\rm
i}\,\xi)$ GR-07  so that, we obtain:

$\phi=\phi_{01}+{\rm e}^{\xi^{2}/4}\Bigl{[}C_{1}{\cal D}_{{\rm
i}\,\beta_{1}}(\xi)+C_{2}{\cal D}_{{\rm i}\,\beta_{1}}(-\xi)\Bigr{]}.$ (93)

This solution should be matched with the asymptotic expansions (46) of the
outer solution calculated for $x=x_{1}$. To this end, we use the asymptotics
of the function ${\cal D}_{p}(s)$ for $|s|\gg 1$ (see GR-07 , Eq. 9.246):

$\displaystyle{\cal D}_{p}(s)$ $\displaystyle\sim$ $\displaystyle s^{p}{\rm
e}^{-s^{2}/4}{}_{2}F_{0}\left(-\frac{p}{2}\,,\frac{1-p}{2}\,;-\frac{2}{s^{2}}\right),\quad|\arg{(s)}|<\frac{3\pi}{4},$
(94) $\displaystyle{\cal D}_{p}(s)$ $\displaystyle\sim$ $\displaystyle
s^{p}{\rm
e}^{-s^{2}/4}{}_{2}F_{0}\left(-\frac{p}{2}\,,\frac{1-p}{2}\,;-\frac{2}{s^{2}}\right)-\frac{\sqrt{2\pi}\,{\rm
e}^{{\rm i}\pi p}}{\Gamma(-p)}\,s^{-p-1}{\rm
e}^{s^{2}/4}{}_{2}F_{0}\left(\frac{p}{2}\,,\frac{1+p}{2}\,;\frac{2}{s^{2}}\right),$
(95) $\displaystyle{\cal D}_{p}(s)$ $\displaystyle\sim$ $\displaystyle
s^{p}{\rm
e}^{-s^{2}/4}{}_{2}F_{0}\left(-\frac{p}{2}\,,\frac{1-p}{2}\,;-\frac{2}{s^{2}}\right)-\frac{\sqrt{2\pi}{\rm
e}^{-{\rm i}\pi p}}{\Gamma(-p)}s^{-p-1}{\rm
e}^{s^{2}/4}{}_{2}F_{0}\left(\frac{p}{2}\,,\frac{1+p}{2}\,;\frac{2}{s^{2}}\right),\phantom{WW}$
(96)

where Eq. (95) is valid for $\pi/4<\arg{(s)}<5\pi/4$, Eq. (96) is valid for
$-5\pi/4<\arg{(s)}<-\pi/4$, $\Gamma(x)$ is the gamma-function, and

${}_{2}F_{0}(a,b;z)\sim\sum_{n=0}^{\infty}\dfrac{\Gamma(a+n)\Gamma(b+n)z^{n}}{\Gamma(a)\Gamma(b)\,n!}.$

It can be easily seen that both terms in square brackets in Eq. (93) grow
infinitely (the former grows when $\xi\to-\infty$, and the later – when
$\xi\to+\infty$), therefore the matching with the bounded outer solutions (see
Eq. (46)) requires:

$C_{1}=C_{2}=0\ \ \Longrightarrow\ \ B^{(+)}_{0}=B^{(-)}_{0}=0,\quad
A^{(+)}_{0}=A^{(-)}_{0}=\phi_{01}{\rm e}^{-{\rm i}\,\psi(x_{1})},$ (97)

because in the matching point $c_{1}=1$ (cf. Eq. (47)).

In the neighborhood of the WH horizon (say, $x=x_{2}$), we put
$x-x_{2}=\varepsilon\xi$, where $\varepsilon^{2}=-\nu/(2\mu_{2})$ because
$\mu_{2}=U^{\prime}_{2}-c^{\prime}_{2}<0$. Then, we obtain the equation
similar to Eq. (91):

$\dfrac{{\rm d}^{2}\phi_{2}}{{\rm d}\xi^{2}}+\xi\,\dfrac{{\rm d}\phi_{2}}{{\rm
d}\xi}-{\rm i}\beta_{2}\,\phi_{2}=0,\qquad\phi_{2}=\phi-\phi_{02},\ \
\phi_{02}={\rm const},$

where $\beta_{2}=\omega/\mu_{2}$. The ansatz $\phi_{2}(\xi)={\rm
e}^{-\xi^{2}/4}G(\xi)$ leads to the equation (cf. Eq. (92)):

$\frac{d^{2}G}{d\xi^{2}}-\left({\rm
i}\beta_{2}+\dfrac{1}{2}+\dfrac{\xi^{2}}{4}\right)G=0.$

Then, function $\phi$ can be written as:

$\phi=\phi_{02}+{\rm e}^{-\xi^{2}/4}\Bigl{[}C_{1}{\cal D}_{-{\rm
i}\,\beta_{2}-1}(\xi)+C_{2}{\cal D}_{-{\rm i}\,\beta_{2}-1}(-\xi)\Bigr{]}.$

This solution remains bounded for any finite constants $C_{1}$ and $C_{2}$.
Indeed, assuming that $-\pi\leq\arg{(\xi)}\leq 0$ (and, consequently,
$-\xi=\xi{\rm e}^{{\rm i}\,\pi}$), using then Eqs. (94)–(96), and keeping only
the leading terms in each sum, we obtain:

$\phi\sim\phi_{02}+\left\\{\begin{array}[]{lr}\Bigl{(}C_{1}-C_{2}{\rm
e}^{\pi\,\beta_{2}}\Bigr{)}\xi^{-{\rm i}\,\beta_{2}-1}{\rm
e}^{-\xi^{2}/2}+\dfrac{\sqrt{2\pi}\,C_{2}}{\Gamma(1+{\rm
i}\,\beta_{2})}\,\xi^{{\rm i}\,\beta_{2}},&\xi\to+\infty,\\\ \\\
\Bigl{(}C_{2}-C_{1}{\rm e}^{-\pi\,\beta_{2}}\Bigr{)}|\xi|^{-{\rm
i}\,\beta_{2}-1}{\rm e}^{-\xi^{2}/2}+\dfrac{\sqrt{2\pi}\,C_{1}}{\Gamma(1+{\rm
i}\,\beta_{2})}\,|\xi|^{{\rm i}\,\beta_{2}},&\xi\to-\infty.\end{array}\right.$

Match of this solution with the outer solution (46) yields:

$A^{(+)}_{0}=A^{(-)}_{0}=\frac{\phi_{02}}{c_{2}}\,{\rm e}^{-{\rm
i}\,\psi(x_{2})}$

and the relations between $C_{1}$ and $B^{(-)}_{0}$ on the one hand and
$C_{2}$ and $B^{(+)}_{0}$ on the other demonstrating the fact that
$B^{(\pm)}_{0}$ are mutually independent.

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