# Black hole analogues in two-dimensional flows with constant shear

Alessia Biondi, Scott Robertson, Germain Rousseaux

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Alessia Biondi, Scott Robertson, Germain Rousseaux. Black hole analogues in two-dimensional flows with constant shear. 2026. hal-05534357

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Black hole analogues in two-dimensional flows with constant shear

Alessia Biondi,¹ Scott Robertson,¹ and Germain Rousseaux¹

¹Institut Pprime, CNRS-Université de Poitiers-ISAE-ENSMA. TSA 51124, 86073 Poitiers Cedex 9, France

(Dated: March 3, 2026)

We review the Analogue Gravity description of a unidirectional water wave system, assuming no prior knowledge of General Relativity or differential geometry. In so doing, we generalize established results concerning an effective curved spacetime for surface waves on irrotational 2D flows, by including flows with constant shear. We show that such flows remain perfectly compatible with the existence of an effective curved spacetime and, in particular, of a metric description.

# I. INTRODUCTION

Analogue Gravity [1] is a transdisciplinary field that aims to mimic field propagation in curved spacetime using waves propagating in a laboratory-based system. It exploits a mathematical analogy between the wave equations applicable to these vastly different cases, which tends to hold in the long-wavelength regime where the short-scale physics is lost from view. Unruh was the first to suggest a concrete use of the analogy [2] by pointing out the possibility of analogue Hawking radiation from “horizons” in the effective spacetime. In the past two decades, Analogue Gravity has become an experimental field, with realizations in systems such as Bose-Einstein condensates [3–7], optical fibers and waveguides [8, 9], polariton fluids [10–12], and surface waves on water [13–17].

The surface wave analogy was first expounded by Schützhold and Unruh [13] and has since led to several experimental studies, including the observation of an analogue of the Hawking effect in subcritical flows [14, 15], rotational super-radiance at the ergoregion of a rotating flow [16], and scattering of an infalling wave at a black-hole horizon [17]. It is thus a very relevant Analogue Gravity system, despite its classical, macroscopic nature. But it tends to be far more complicated in practice than the simple theoretical descriptions would suggest, being subject to vorticity, viscosity, friction, nonlinearities, and turbulence. All of these may be expected to distort the wave behavior away from the gravitational analogy, potentially rendering the system less relevant. It thus behooves us to investigate these features and to assess whether they might be incorporated into an Analogue Gravity framework.

Furthermore, it is important to ensure that the Analogue Gravity description is accessible to those not already familiar with it, because the analogy at the heart of the subject is meant to be bidirectional: water wave physics can help us to explore certain concepts from gravity, but at the same time, the application of such concepts to water waves can help us to explore new and under-explored processes and regimes in hydrodynamics. Indeed, the regime of interest to Analogue Gravity is typically quite different from that normally studied in hydrodynamics: the waves considered are in the linear regime; their scattering is primarily induced by inhomogeneities in the background flow (rather than by nonlinear wave interactions); and there exist waves of negative energy [17, 18], a crucial ingredient in the analogues of the Hawking effect [19] and super-radiance [20, 21].

The purpose of this paper is two-fold. Its main purpose is to present, in a pedagogical manner aimed at the hydrodynamics community, the derivation of the wave equation for long-wavelength surface waves on a two-dimensional, unidirectional flow, and to explain why it is of interest from an Analogue Gravity perspective. In this sense it is mostly review. We also generalize the usual treatment to the case where the background flow has a constant shear vorticity, and we will see that this case is still perfectly compatible with the existence of an analogue spacetime. Previous works have found the dispersion relation of waves on a linear shear flow [22–25]. Here, our Analogue Gravity viewpoint takes a different approach: we recover their long-wavelength behaviour, but we do so in the context of a wave equation, which allows a quantitative analysis of wave scattering on inhomogeneous flows. Within Analogue Gravity, we note that there exist previous studies that have examined the effects of vorticity on analogue rotating black holes [26–30], focusing primarily on cases where the vorticity is oriented perpendicular to the fluid plane. Our interest here, however, lies in a “hidden” vorticity arising from shear flow – one that leaves no obvious imprint on the effective metric.

The paper is organized as follows. In Sec. II, the essential idea behind an effective spacetime is introduced, as well as how it is expected to apply to waves in a moving medium. Section III is devoted to a derivation of the wave equation on a channel flow in the appropriate limit of long-wavelength, which is identified with that expected of a scalar field in an effective spacetime. In Sec. IV, the physical properties of the solutions of the wave equation are examined, and an abstract field theory description is briefly considered. Section V considers the solution of the stationary background which determines the effective spacetime, paying particular attention to the case where this spacetime contains the analogue of a black-hole horizon, and applying the standard Hawking-Unruh prediction. Finally, in Sec. VI we look at the scattering induced by inhomogeneities in the conformal factor, the main quantity affected by the presence of vorticity. We conclude in Sec. VII.
## II. EFFECTIVE METRIC FOR WAVES IN MOVING MEDIUM

### A. Idea

The idea underpinning Analogue Gravity is that waves in a moving medium behave like fields propagating in a curved spacetime. Apart from providing a mathematically analogous system that can be exploited experimentally, it also provides a more intuitive picture to help understand the behavior of gravitational systems. Historically, it has been especially oriented towards black holes, and we shall focus on black-hole analogues in this work.

To be more concrete, imagine an accelerating flow which, in the context of water waves, can be experimentally realized by having it pass over an obstacle. In the upstream region, the speed of the flow is smaller in magnitude than the speed of waves with respect to the flow. This means that waves may propagate in either direction, though with different total speeds: the flow will boost the speed of waves co-propagating with the flow, and conversely it will diminish the speed of waves counter-propagating against the flow. As the flow accelerates, this change of the wave speed will become gradually more extreme. Eventually, we may reach a situation where, on the downstream side, the flow speed is faster than the intrinsic wave speed. In this regime, all waves will appear to propagate with the flow, even those that are counter-propagating because the flow speed completely dominates over the wave speed. All waves are dragged inexorably downstream on this side, so that no waves can escape from this region. This is analogous to the interior of a black hole, and the point at which the flow speed is critical – where it is equal to the wave speed – is the analogue of the black hole horizon.

It is clear from the above picture that the analogue black hole arises through setting up a competition between two velocities: the wave speed and the flow speed. The wave speed measured in the rest frame of the fluid is independent of the propagation direction, and is related to the properties of the background flow. It plays the role of the speed of light in General Relativity, except that it will generally depend on position due to inhomogeneities in the flow profile. The flow speed has no clear analogue in General Relativity, because it relates to the motion of a preferred reference frame (that in which the fluid is at rest), while it is a foundational principle of relativity theory that there exists no preferred reference frame. Nevertheless, we will use the existence of the fluid frame to our advantage in identifying how the analogue of the “spacetime interval” is expected to behave, and we will make the analogy with a particular description of the gravitational black-hole spacetime that happens to have a similar form.<fnref n="1" />

### B. The spacetime metric

The local structure of spacetime is mathematically described by the *metric* [32–34], which relates small displacements in coordinates to physical distances and durations. This is done by introducing the infinitesimal spacetime interval, $ds^2$, and writing it as a quadratic form in the infinitesimal coordinate displacements $dq^\mu$. The spacetime interval is written as

$$ds^2 = g_{\mu\nu} dq^\mu dq^\nu, \qquad (1)$$

where $g_{\mu\nu}$ is the *metric tensor* (and can be thought of as a matrix). The repeated indices $\mu$ and $\nu$ imply a summation over all spacetime coordinates according to the Einstein summation convention. In the special case of *flat spacetime* in Cartesian coordinates $x^\mu = (t, x, y)$ – we take two spatial coordinates to match the dimensionality of the surface of water – this becomes

$$ds^2 = \eta_{\mu\nu} dx^\mu dx^\nu = -c^2 dt^2 + dx^2 + dy^2, \qquad (2)$$

so that the flat-space metric tensor $\eta_{\mu\nu}$ would be, in matrix form,

$$\eta_{\mu\nu} = \begin{bmatrix} -c^2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}. \qquad (3)$$

If in Eq. (2) time is held constant (so $dt = 0$) and only spatial displacements are involved, then it simply expresses the Pythagorean theorem of Euclidean geometry. The presence of the time coordinate, which is multiplied by the speed of light $c$ and appears with a minus sign, allows $ds^2$ to be negative. Displacements are then characterized as

<footnote n="1">¹ In fact, and in spite of the principle of relativity, it can prove useful to assume the existence of a preferred frame in deriving the black-hole spacetime; see [31].</footnote>
space-like ($ds^2 > 0$), time-like ($ds^2 < 0$), or light-like ($ds^2 = 0$).<fnref n="2" /> As suggested by the name, light propagates along light-like intervals, which is another way of saying that it always propagates at speed $c$.

The generalization to Eq. (1) encompasses two changes. One is a change of coordinates while leaving the underlying spacetime fixed; for example, we may re-express Eq. (2) using cylindrical or spherical coordinates, and the metric tensor would change accordingly even though the spacetime remains exactly the same. But it is also possible for the generalized metric to describe a different spacetime, which in general will be *curved*. A simple example is the surface of a sphere, which is different from the plane and on which geometry becomes non-Euclidean.

### C. Effective metric of “flowing” spacetime

Let us motivate the expected form of the effective metric describing the behavior of surface waves.

Imagine waves on *static* water. As already mentioned, we expect the wave speed to play the role of the speed of light, and we would thus expect that the effective metric describing this system is exactly (2) with $c$ now being the speed of surface waves. If the system is inhomogeneous (due, say, to the presence of an obstacle on the bottom of the channel), $c$ may become position-dependent.

The inclusion of flow can be thought of as performing a Galilean boost on the system. The key point is that the spatial contribution to the spacetime interval in Eq. (2) is measured *with respect to the fluid* – that is, in the preferred reference frame where the fluid is at rest. This remains true when the fluid is moving, except that the relevant displacement is no longer simply $dx$, but $dx - u dt$ where $u$ is the fluid velocity in the $x$-direction. (To see this, note that if $dx = u dt$, it means that we simply follow the fluid and the spatial displacement is zero.) In general we can do the same for the $y$-direction, though as mentioned above we will restrict ourselves to flows that are unidirectional (oriented along $x$) and two-dimensional (having no dependence on $y$). Finally, we may allow the flow velocity $u$ to depend on position, just like the wave speed $c$. Then the effective metric associated with waves in a flowing fluid might be expected to take the form

$$ds^2 = \Lambda^2(x) \left[ -c^2(x) dt^2 + (dx - u(x)dt)^2 + dy^2 \right]. \qquad (4)$$

This is exactly as justified in the text above, with one additional ingredient: the overall prefactor $\Lambda^2(x)$. This is introduced for generality because waves propagate along light-like intervals with $ds^2 = 0$, and these are unaffected by the presence of this prefactor. We will refer to it as the *conformal factor*, since multiplication of the metric by an overall factor is known as a *conformal transformation*. For now it is left rather abstract, but later we will see the physical importance of $\Lambda$ in determining how waves scatter into each other.

There is an exact gravitational analogue of the metric (4). In General Relativity [32, 33], the simplest black hole is the Schwarzschild black hole (which has no electric charge and no spin). While it is typically described using the Schwarzschild metric, a change of coordinates yields the Painlevé-Gullstrand metric, which has the following form [31, 35]:

$$ds^2 = -c^2 dt^2 + \left( dr + \sqrt{\frac{r_s}{r}} c dt \right)^2 + r^2 d\Omega^2. \qquad (5)$$

Here, $r_s$ is the Schwarzschild radius of the black hole, and $d\Omega^2 = d\theta^2 + \sin^2\theta d\phi^2$ is the infinitesimal angular displacement on the surface of a sphere of constant radius $r$. The key point is that, as in the “flowing” spacetime, the line element is as *in* flat space except for the form of the radial displacement, which appears in the form $dr - v dt$ where $v = -c\sqrt{r_s/r}$. In light of the intuitive picture provided by the “flowing” spacetime, this behaves exactly as if there were a “true” spatial displacement with respect to some preferred frame, described by a fluid flowing radially inwards at velocity $v$. Moreover, this imagined fluid crosses the speed of light at radius $r = r_s$. Therefore, light waves which are attempting to propagate outwards will succeed in doing so for radius larger than $r_s$, but will be dragged inexorably inwards for $r < r_s$. This is the most direct gravitational analogue of a black-hole flow.

### D. Wave equation

The analogy is concerned with the behavior of waves, and so we are particularly interested in wave propagation. To this end, we focus our attention on a *massless scalar* field; here, *scalar* means that the field has no directionality

<footnote n="2">² The overall sign of $ds^2$ is arbitrary and represents a choice known as the *metric signature*.</footnote>
to it (unlike, say, a vector field), while massless can be taken to mean that the wave speed with respect to the fluid is independent of the wavelength. In the flat spacetime with Cartesian coordinates described by the metric (2), a massless scalar field $\phi$ obeys the d'Alembert equation

$$ \Box\phi = -\eta^{\mu\nu}\partial_{\mu}\partial_{\nu}\phi = \left(\frac{1}{c^2}\partial_t^2 - \nabla^2\right)\phi = \left(\frac{1}{c^2}\partial_t^2 - \partial_x^2 - \partial_y^2\right)\phi = 0, \qquad (6) $$

where $\eta^{\mu\nu}$, with upper indices, is the inverse of the metric tensor. (The reason for the appearance of the inverse is the dual nature of infinitesimal displacements and derivatives, which transform in complementary ways: the metric tensor applies to the former, while the inverse metric tensor applies to the latter [32, 33].) It can be shown that, starting from Eq. (6) and making an arbitrary coordinate transformation, the wave equation in the new coordinate system becomes [34]

$$ \Box\phi = \frac{-1}{\sqrt{-g}}\partial_{\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\phi) = 0. \qquad (7) $$

Here, $g$ is the determinant of the metric tensor; it is negative because of the negative contribution of temporal displacements to the spacetime interval. Equation (7) reduces to Eq. (6) in Cartesian coordinates because there the metric components are constant. Finally, it is generalized to curved spacetime by assuming the wave equation remains in exactly the form of Eq. (7).<fnref n="3" /> We thus adopt it here as our fundamental wave equation.

Equation (7) captures the essence of Analogue Gravity and encompasses what we mean by the effective metric description of wave propagation. If we may identify a field $\phi$ that obeys (at least approximately, in the long-wavelength limit) a wave equation of this form, then we may read off an effective metric tensor and identify the effective spacetime that is “felt” by the field $\phi$. By direct substitution into Eq. (7), the wave equation for massless fields in a spacetime with metric (4) (and having, like the flow, no dependence on the $y$-direction) is found to be

$$ \left[ (\partial_t + \partial_x u) \frac{\Lambda}{c} (\partial_t + u \partial_x) - \partial_x \Lambda c \partial_x \right] \phi = 0. \qquad (8) $$

Here, partial derivatives act on everything to their right (so, for example, $\partial_x \Lambda c \partial_x \phi = \partial_x (\Lambda c \partial_x \phi)$). Our aim is to show that, in the long-wavelength limit, surface waves on flows with constant shear obey a wave equation of the form (8). The flow can thus be associated with an effective metric of the form (4), with corresponding values for $u(x)$, $c(x)$ and $\Lambda(x)$.

### III. DERIVATION OF THE WAVE EQUATION IN A CHANNEL FLOW

In this section, we derive the equation for surface waves in the long-wavelength/shallow-water limit, often described by the Saint-Venant equations and characterized by the small parameter $kh \ll 1$ where $k$ is the wave number and $h$ is the water depth. Moreover, the equations shall be linearized, so we also assume $ka \ll 1$ where $a$ is the wave amplitude. In this linear, non-dispersive regime, we can expect to find an equation of the form (8).

#### A. Basic equations

We begin by specifying the complete set of fundamental equations describing our system. We let $x$ be the longitudinal coordinate along the flume, $y$ the transverse coordinate, and $z$ the vertical coordinate (required here to describe the flow beneath the free surface, and in contrast to Eq. (8) which is meant to describe waves on the free surface). As in the previous section, the flow is assumed to have no dependence on and no component along $y$. We thus consider a two-dimensional flow, such as might be found in an elongated water channel.

##### 1. Equations in the bulk

We assume an inviscid flow, subject to no viscous damping or friction at the boundaries. Such a flow is governed by the Euler equations [36]

$$ \frac{d\mathbf{V}}{dt} = \partial_t \mathbf{V} + (\mathbf{V} \cdot \nabla)\mathbf{V} = -\frac{1}{\rho}\nabla P + \mathbf{g}, \qquad (9) $$

<footnote n="3">³ This means that there is no direct interaction between the field and the spacetime curvature. Such a field is said to be minimally coupled to gravity.</footnote>
where $\mathbf{V} = u\mathbf{x} + w\mathbf{z}$ is the local velocity of the fluid, $P$ is the pressure, $\rho$ the density, and $\mathbf{g} = -g\hat{\mathbf{z}}$ is the acceleration due to gravity.

We assume the flow to be *incompressible*, with the density $\rho$ remaining constant and uniform throughout the fluid. Then the continuity equation for mass conservation becomes a constraint on the flow velocity $\mathbf{V}$:

$$ \nabla \cdot \mathbf{V} = \partial_x u + \partial_z w = 0. \qquad (10) $$

The *vorticity* is defined as $\Omega = \nabla \times \mathbf{V}$, and a governing equation for $\Omega$ can be derived from (9). As our flow is in the two-dimensional plane $(x, z)$, the vector $\Omega$ always points in the $y$-direction and can thus be considered as a scalar: $\Omega = \Omega \cdot \mathbf{y} = \partial_z u - \partial_x w$. Its governing equation is one of straightforward advection:

$$ (\partial_t + \mathbf{V} \cdot \nabla) \Omega = 0. \qquad (11) $$

This is trivially satisfied in a flow with constant $\Omega$. We restrict our attention to this case it what follows.

An important point to raise here is that, although we will set $\Omega$ to a constant value so that it becomes a parameter of the fixed background flow, this does not automatically imply that the *total* vorticity will be constant, for we also wish to consider perturbations on top of this background which, *a priori*, may carry some non-zero vorticity $\delta\Omega$. However, when linearizing Eq. (11) to get the governing equation for the vorticity of the perturbation, we find simply

$$ (\partial_t + \mathbf{V}_0 \cdot \nabla) \delta\Omega = 0, \qquad (12) $$

where $\mathbf{V}_0$ is the background flow velocity. This implies that any non-zero perturbation of the vorticity will simply be advected by the flow, being motionless in the rest frame of the fluid (and is thus best considered as a perturbation of the background flow). As we are interested in propagating wave solutions, we remove this possibility by restricting to the case $\delta\Omega = 0$ when linearizing the equations below.

## 2. Boundary conditions

The boundaries of the flow are:

* the bottom, which we shall assume fixed: $z = b(x)$, representing some arbitrary but fixed shape of the bottom of the channel; and

* the free surface, which can be written as $z = b(x) + h(t, x)$; $h$ is then the instantaneous depth of the fluid at a particular position.

At each boundary, the flow velocity can be decomposed into components tangent and orthogonal to the boundary, where the orthogonal component will be accompanied by a change in the position of the boundary with time. Since the bottom is assumed fixed, the no-penetration condition requires that the orthogonal component there must vanish, while at the free surface it will engender a time-derivative of $h$. We thus have the following *kinematical* boundary conditions:

$$ \begin{aligned} [w - u \partial_x b]_{z=b} &= 0, \\ [w - u \partial_x (b + h)]_{z=b+h} &= \partial_t h. \end{aligned} \qquad (13) $$

Finally, there is a *dynamical* boundary condition in the form of the continuity of the pressure at the free surface. For simplicity, we may neglect any variations of the air pressure and simply set this to a constant atmospheric pressure, which in turn may be set to zero because only pressure gradients enter the governing equations. So our final boundary condition is

$$ [P]_{z=b+h} = 0. \qquad (14) $$

## 3. Effective continuity equation

Let us define the depth-averaged velocity

$$ \bar{u} = \frac{1}{h} \int_{b}^{b+h} u \, dz. \qquad (15) $$
Then, through a combination of the incompressibility condition (10) and the kinematical boundary conditions (13), it is straightforward to show that

$$ \partial_t h + \partial_x (h \bar{u}) = 0. \qquad (16) $$

This is intuitive: since the flow is incompressible with constant 3D density, $h$ plays the role of the local 1D “density” of the fluid, while $h \bar{u}$ is the local “current”. Equation (16) is simply an expression of the local conservation of the mass (or volume) of the fluid.

Equation (16) is one of the two governing equations for the fields $\bar{u}$ and $h$. It must be complemented by an equation for $\partial_t \bar{u}$, but to get such an equation in a tractable form, we first make a “slowly-varying” approximation.

## B. Slowly-varying limit

We now take the limit in which variations in the longitudinal ($x$) direction (as well as in time) take place on much longer scales than variations in the vertical ($z$) direction. To implement this, we introduce a dimensionless scaling parameter $\epsilon \ll 1$ that acts as if the system were being stretched along $x$: the smaller is $\epsilon$, the more stretching occurs. We thus make the following replacements:

$$ \partial_t, \partial_x, w \rightarrow \epsilon \partial_t, \epsilon \partial_x, \epsilon w. \qquad (17) $$

These are then substituted into the above equations, and we keep only the lowest-order terms in $\epsilon$. In this way, we focus on the long-wavelength, long-period behavior, and the short-scale physics associated with the “extra dimension” $z$ is lost from view.<fnref n="4" />

The above procedure simplifies two of the governing equations. First, the vorticity now takes the form

$$ \Omega = \partial_z u, \qquad (18) $$

with no remaining contribution from the vertical component $w$. Since we take $\Omega$ to be strictly constant, this implies that, at fixed $x$, $u$ varies linearly with $z$. It is useful to introduce the values of $u$ at the bottom and at the free surface:

$$ u^{(b)} = u|_{z=b}, \quad u^{(s)} = u|_{z=b+h}. \qquad (19) $$

Then Eq. (18) implies that

$$ u^{(s)} - u^{(b)} = \Omega h, \qquad (20) $$

while the depth-averaged velocity entering Eq. (16) is

$$ \bar{u} = \frac{1}{2} (u^{(b)} + u^{(s)}) = u^{(b)} + \frac{1}{2} \Omega h. \qquad (21) $$

The second change effected by taking the slowly-varying limit is in the $z$-component of the Euler equations (9), which becomes

$$ \frac{1}{\rho} \partial_z P + g = 0. \qquad (22) $$

This means that the pressure is hydrostatic, and since we also have the boundary condition (14), we immediately find

$$ P = \rho g (b + h - z). $$

Substituting this result, as well as Eq. (18), back into the $x$-component of the Euler equations, we have

$$ \partial_t u + \partial_x \left[ \frac{u^2}{2} + g(h+b) \right] + \Omega w = 0. \qquad (23) $$

<footnote n="4">⁴ The first dispersive corrections occur for $kh \sim 1$, where $h$ is the depth of the fluid. Hence we take a limit here in which $kh \ll 1$.</footnote>
This is not an effective 1D equation in the manner of Eq. (16), because both $u$ and $w$ depend on $z$. To remove any $z$-dependence, let us evaluate it at the fixed boundary on the bottom, where $z = b$. There, $w$ is determined by $u$ because of the boundary condition (13), and we are able to combine two terms:

$$ 
\begin{aligned}
\left[ \partial_x \left( \frac{u^2}{2} \right) + \Omega w \right]_{z=b} &= [u(\partial_x u + \partial_z u \partial_x b)]_{z=b} \\
&= u(x, z=b) \frac{d}{dx} [u(x, z=b)] \\
&= \partial_x \left( \frac{u^{(b)2}}{2} \right).
\end{aligned} 
\qquad (24) $$

Equation (23) then becomes

$$ \partial_t u^{(b)} + \partial_x \left[ \frac{u^{(b)2}}{2} + g(h+b) \right] = 0. \qquad (25) $$

This is written in terms of the flow velocity on the bottom, but, following Eq. (16) where $\bar{u}$ plays a natural role as the effective flow velocity, it will be most convenient to express everything in terms of $\bar{u} = u^{(b)} + \Omega h/2$. We simply plug this expression into the above equation and combine with Eq. (16) to remove the term in the time-derivative of $h$. In this way we arrive at the following equation:

$$ \partial_t \bar{u} + \partial_x \left[ \frac{\bar{u}^2}{2} + g(h+b) + \frac{\Omega^2 h^2}{8} \right] = 0. \qquad (26) $$

Equations (16) and (26) represent the main governing equations of our system, which is fully characterized by the one-dimensional fields $h(t, x)$ and $\bar{u}(t, x)$, with $b(x)$ representing an externally imposed constraint and $\Omega$ a parameter of the flow.

### C. Linearization

The Analogue Gravity viewpoint relies on our ability to separate the system into two components of very different strengths [37]. One is the strong background, which plays the role of the effective spacetime. The other is the perturbations on top of this background, which are supposed to be sufficiently small that they do not affect the background significantly. This is equivalent to the idea of a “test particle” in gravity theory, which responds to the presence of a gravitational field but whose contribution to the gravitational field need not be taken into account [32, 33].

Here, we separate the system into a time-independent background flow (quantities associated with the background will be indicated with the subscript 0), plus a time-dependent perturbation of very small amplitude (these will be indicated by the use of the prefix $\delta$). Explicitly,

$$ h(t, x) = h_0(x) + \delta h(t, x), \qquad \bar{u}(t, x) = \bar{u}_0(x) + \delta \bar{u}(t, x). \qquad (27) $$

These are plugged into Eqs. (16) and (26), which are then linearized in the perturbations $\delta h$ and $\delta \bar{u}$. This procedure yields two distinct sets of equations: a pair of nonlinear equations for the background, and a pair of linear equations for the perturbations. We will return to the nonlinear background equations in the following section; for now, let us assume $h_0(x)$ and $\bar{u}_0(x)$ are given, and see how the perturbations behave (since it is this that determines, through Eq. (8), the form of the effective metric).

As noted above, since we consider perturbations that carry no vorticity of their own, the form of the governing equations is left unchanged by their presence and we can thus describe them by direct linearization of Eqs. (16) and (26). This gives

$$ (\partial_t + \partial_x \bar{u}_0) \delta h + \partial_x (h_0 \delta \bar{u}) = 0, \qquad (28) $$

$$ (\partial_t + \partial_x \bar{u}_0) \delta \bar{u} + \partial_x \left[ \left( g + \frac{\Omega^2 h_0}{4} \right) \delta h \right] = 0. \qquad (29) $$

To proceed, we invoke the fact that, as the dynamical fields $\delta h$ and $\delta \bar{u}$ depend only on $t$ and $x$, they can be written as the spatial derivative of a potential. We may choose a potential for either; these will lead to different but equivalent formulations. Typically, because for irrotational fields one may introduce a velocity potential such that $\mathbf{u} = -\nabla \varphi$, it
is $\delta \bar{u}$ that is written as $\delta \bar{u} = -\partial_x (\delta \varphi)$. Then Eq. (29) may be integrated with respect to $x$, and the above equations become

$$ (\partial_t + \partial_x \bar{u}_0) \delta h - \partial_x (h_0 \partial_x \delta \varphi) = 0, \qquad (30) $$

$$ -(\partial_t + \bar{u}_0 \partial_x) \delta\varphi + \left(g + \frac{\Omega^2 h_0}{4}\right) \delta h = 0. \qquad (31) $$

These may now be combined into a single equation for $\delta\varphi$:

$$ \left[ (\partial_t + \partial_x \bar{u}_0) \frac{1}{g + \frac{\Omega^2 h_0}{4}} (\partial_t + \bar{u}_0 \partial_x) - \partial_x h_0 \partial_x \right] \delta\varphi = 0. \qquad (32) $$

This second-order wave equation is exactly of the form (8), which governs the propagation of a massless scalar field in the “flowing” spacetime metric (4). We may read off the three quantities entering this metric:

$$ u = \bar{u}_0 = \frac{q}{h_0}, \quad c^2 = gh_0 + \frac{\Omega^2 h_0^2}{4}, \quad \Lambda^2 = \frac{h_0/h_{up}}{1 + \frac{\Omega^2 h_0}{4g}} = \frac{g}{h_{up}} \frac{h_0^2}{c^2} = \frac{g q^2}{h_{up} u^2 c^2}. \qquad (33) $$

We use the upstream water depth $h_{up}$ to adimensionalize $\Lambda^2$ (though the overall normalization of this quantity is irrelevant to the physics). The value of $c$ derived here agrees with the long-wavelength limit of the dispersion relation derived in [22, 24, 25]. Note that, when $\Omega = 0$, we have $c^2 = gh_{up} \Lambda^2 = gh_0$, which is the case used in the theoretical description of [17]. Furthermore, we may define the Brard number, Br, as the following quantity:

$$ \mathrm{Br} = \frac{\Omega(\Omega h_0)}{4g} = \frac{\Omega^2 h_0^2}{4gh_0}, \qquad (34) $$

which compares the rotational correction term to the gravitational wave speed $c^2 = gh_0$ characteristic of an irrotational flow. This dimensionless number provides insight into the dominant physical mechanisms within the system. Specifically, for $\mathrm{Br} \ll 1$, we recover the limit where the flow is essentially irrotational; conversely, for $\mathrm{Br} \gg 1$, the rotational vorticity effects become the governing dynamics of the system, and $\Lambda^2$ becomes almost independent of $h_0$ and hence of the position. This has important consequences for wave scattering, as we will see in Sec. VI.

Finally, we note that this immediately yields a generalized form of the Froude number, which is simply the ratio $u/c$:

$$ \mathrm{Fr} = \frac{u}{c} = \frac{q}{\sqrt{gh_0^3 + \frac{1}{4}\Omega^2 h_0^4}} = \frac{\mathrm{Fr}_{\Omega=0}}{\sqrt{1 + \mathrm{Br}}}, \qquad (35) $$

where $\mathrm{Fr}_{\Omega=0}$ is the Froude number for given $q$ and $h_0$ when the flow is assumed irrotational. This dimensionless number distinguishes subcritical ($\mathrm{Fr} < 1$) from supercritical regions ($\mathrm{Fr} > 1$), with the horizon occurring at $\mathrm{Fr} = 1$.

## IV. WAVE PROPERTIES

In this section, we consider some physical properties of the wave solutions of Eq. (32) on top of a given flow. Here, we are mainly concerned with properties of plane waves. Therefore, we pay particular attention to the case where the background is uniform so that these properties are not varying continuously in space. After the dispersion relation, we go on to calculate (in the spirit of [38]) the momentum, energy and norm (wave action) [39] associated with plane waves, noting that the energy can be negative when the flow is supercritical. Much of these results are reproduced in Appendix A from a more abstract field theory point of view.

### A. Dispersion relation

Assume that $u$, $c$ and $\Lambda$ are all uniform in space, and that we have a plane wave solution $\delta\varphi \propto \exp(ikx - i\omega t)$. Then Eq. (32) implies

$$ (\omega - uk)^2 = c^2 k^2 \quad \Longleftrightarrow \quad \omega = (u \pm c) k. \qquad (36) $$
This is the dispersion relation, and its (real) roots identify all independent plane wave solutions of the wave equation. We thus find two independent solutions, corresponding to one wave traveling in each direction with respect to the flow. The $-uk$ appearing on the left-hand side is indicative of the Doppler effect: the measured frequency $\omega$ is different from the frequency $\omega - uk$ that would be measured in a frame at rest with respect to the fluid. The total speeds of the two waves are $u \pm c$, corresponding to the co-propagating and counter-propagating waves depending on whether their intrinsic velocity is aligned or anti-aligned with the flow. The combination of the two velocities shows explicitly the competition that is set up between the flow velocity and the wave speed, especially for the counter-propagating waves whose total velocity, $u-c$, can be positive or negative depending on whether the flow speed or the wave speed is faster.

Note that only $u$ and $c$ enter the dispersion relation, not $\Lambda$. This is to be expected as the dispersion relation provides a description of the wave kinematics, and as stated in Sec. II, this is given by $ds^2 = 0$ which is unaffected by $\Lambda$.

## B. Momentum

The $x$-component of the momentum density is just $\rho u$, where $\rho$ is the fluid density and $u$ is the total fluid velocity in the $x$-direction. This needs to be integrated over space to get the total momentum:

$$P = \int_{x_1}^{x_2} dx \int_{y_1}^{y_2} dy \int_{b}^{b+h} dz \rho u = \rho W \int_{x_1}^{x_2} dx \bar{u} h, \quad (37)$$

where we have used the definition of the depth-averaged velocity to remove the integration over $z$, and the independence of $y$ to replace the $y$-integration by multiplication by the channel width $W$. The fields $\bar{u}$ and $h$ will oscillate in $x$ due to the presence of waves, but these oscillations will average to zero when integrated over directly. This means that terms linear in the perturbations $\delta u$ and $\delta h$ can be neglected if the $x$-integral is taken as an average in $x$. Consequently, the effective momentum density of the wave will be quadratic in the perturbation fields, and thus quadratic in the wave amplitude (as might have been expected). Then $P$ can be split into two contributions, one from the background and one from the wave:

$$P = \rho W \left\{ \int_{x_1}^{x_2} dx \bar{u}_0 h_0 + \int_{x_1}^{x_2} dx \delta \bar{u} \delta h \right\} \equiv P_0 + \delta P. \quad (38)$$

An important consideration is to derive the transformation law governing how the momentum changes when making a Galilean boost into another reference frame. Such a boost affects $u$ uniformly by the addition of a constant speed: $u \to u + U$. Because the boost velocity $U$ is constant (and can be arbitrarily large), it will be absorbed into the background speed rather than the perturbation speed, i.e., $\bar{u}_0 \to \bar{u}_0 + U$, $\delta \bar{u} \to \delta \bar{u}$. That is, the perturbation velocity $\delta \bar{u}$ is invariant under boosts, and so is the wave momentum $\delta P$.

To be explicit, let us write $\delta \varphi$ as a plane wave of constant amplitude $A$: $\delta \varphi = A \sin(kx - \omega t)$. From this we may derive $\delta \bar{u}$ and $\delta h$ using previously derived equations:

$$\begin{aligned} \delta \bar{u} &= -\partial_x (\delta \varphi) = -k A \cos(kx - \omega t), \\ \delta h &= \frac{h_0}{c^2} (\partial_t + \bar{u}_0 \partial_x) \delta \varphi = -\frac{h_0}{c^2} (\omega - \bar{u}_0 k) A \cos(kx - \omega t). \end{aligned} \quad (39)$$

Then the average wave momentum per unit length is

$$\delta P = \frac{\rho W h_0}{2 c^2} k (\omega - \bar{u}_0 k) A^2. \quad (40)$$

Notice that $\omega - \bar{u}_0 k$ is the frequency in the rest frame of the fluid (see dispersion relation (36)). It is therefore invariant under boosts, and so is $\delta P$, as expected.

## C. Energy

We can perform a similar analysis for the energy of the system. The energy density is given by $\rho (u^2/2 + gz)$ (where, in line with the long-wavelength limit described in Sec. III B, we neglect the kinetic energy due to the vertical motion). Therefore, the total energy is

$$E = \int_{x_1}^{x_2} dx \int_{y_1}^{y_2} dy \int_{b}^{b+h} dz \rho \left( \frac{u^2}{2} + gz \right) = \rho W \int_{x_1}^{x_2} dx \left( \frac{\bar{u}^2 h}{2} + \frac{\Omega^2 h^3}{24} + \frac{gh^2}{2} \right), \quad (41)$$
where we have used the constancy of the vorticity and the definition of the depth-averaged velocity to write $u = \bar{u} + \Omega(z - b - h/2)$, before proceeding with the integration over $z$. As before, we wish to split this into two contributions, one from the background and one from the wave, neglecting terms linear in the perturbations as they will average away when integrating over $x$. We also note that the first two terms in the integrand will generate third-order terms in the perturbations; these will oscillate in $x$ and can be neglected for the same reason. So, as before, the wave energy will be quadratic in the perturbation fields and in the wave amplitude. We have

$$ 
\begin{aligned}
E &= \rho W \left\{ \int_{x_1}^{x_2} dx \left( \frac{\bar{u}_0^2 h_0}{2} + \frac{\Omega^2 h_0^3}{24} + \frac{gh_0^2}{2} \right) + \int_{x_1}^{x_2} dx \left[ \frac{h_0}{2} (\delta \bar{u})^2 + \bar{u}_0 \delta \bar{u} \delta h + \left( \frac{\Omega^2 h_0}{8} + \frac{g}{2} \right) (\delta h)^2 \right] \right\} \\
&\equiv E_0 + \delta E.
\end{aligned} 
\quad (42) $$

Consider now the transformation of the energy under a boost. As was stated above, the boost velocity $U$ is absorbed into the background flow velocity rather than the perturbation, but we see that the wave energy $\delta E$ includes one term that is proportional to $\bar{u}_0$. In fact, this term is nothing but $\bar{u}_0$ multiplied by the wave momentum $\delta P$. Therefore, if under a boost we have $\bar{u}_0 \rightarrow \bar{u}_0 + U$, then we also have $\delta E \rightarrow \delta E + U \delta P$. So the energy is not invariant under boosts, though it can be written as $\delta E = \delta E^{(0)} + \bar{u}_0 \delta P$, where $\delta E^{(0)}$ is the energy in the “rest frame” of the fluid (where $\bar{u}_0 = 0$). When assuming the same plane wave as in Eqs. (39), this gives for the average wave energy per unit length, in the “rest frame” of the fluid:

$$ 
\begin{aligned}
\delta E^{(0)} &= \frac{\rho W}{2} \left[ h_0 (\delta \bar{u})^2 + \frac{c^2}{h_0} (\delta h)^2 \right] \\
&= \frac{\rho W}{4} \left[ h_0 k^2 + \frac{h_0}{c^2} \omega^{(0)2} \right] A^2 \\
&= \frac{\rho W}{2} \frac{h_0}{c^2} \omega^{(0)2} A^2 \\
&= \frac{\rho W}{2} \frac{h_0}{c^2} (\omega - \bar{u}_0 k)^2 A^2,
\end{aligned} 
\quad (43) $$

where $\omega^{(0)} = \omega - \bar{u}_0 k$ is the wave frequency in the “rest frame” of the fluid. In the frame with background flow velocity $\bar{u}_0$, this becomes

$$ 
\begin{aligned}
\delta E &= \delta E^{(0)} + \bar{u}_0 dP \\
&= \frac{\rho W}{2} \frac{h_0}{c^2} [(\omega - \bar{u}_0 k) + \bar{u}_0 k] (\omega - \bar{u}_0 k) A^2 \\
&= \frac{\rho W}{2} \frac{h_0}{c^2} \omega (\omega - \bar{u}_0 k) A^2.
\end{aligned} 
\quad (44) $$

Note that $\delta E/\delta P = \omega/k$, the phase velocity of the wave. $\delta E$ and $\delta P$ also transform in the same way as $\omega$ and $k$ under boosts.

A final point to emphasize here is that, because of how it transforms under boosts, it is possible for the wave energy $\delta E$ to become negative [38]. We see that this occurs when $\delta E^{(0)} + \bar{u}_0 \delta P < 0$, which means that $\bar{u}_0$ is sufficiently fast to change the sign of the phase velocity of the wave. In other words, it occurs for counter-propagating waves whenever $|\bar{u}_0| > c$. This is very important in the context of Analogue Gravity, because this is what happens in the interior region of a black hole [19], which is indeed associated with the existence of negative-energy excitations. Here we see the same phenomenon emerging within a non-relativistic framework. It is clear from the analysis that the total energy of the system is always positive, and it is only the wave energy, defined as a difference with respect to the energy of the background, that can be negative.

### D. Norm and wave action

We may write the momentum and energy densities in the form [38]

$$ \delta P = k \delta N, \quad \delta E = \omega \delta N, \quad (45) $$

where we have defined

$$ \delta N = \frac{\rho W}{2} \frac{h_0}{c^2} (\omega - \bar{u}_0 k) A^2. \quad (46) $$
By analogy with quantum mechanics – where excitations come in quanta whose momentum and energy are proportional to $k$ and $\omega$, respectively – we see that $\delta N$ plays the role of “number of quanta”. This is a conserved quantity. In field theory it is often referred to abstractly as the “norm”, though when the theory is quantized it becomes precisely the number of quanta. In fluid mechanics, it corresponds to the *wave action*. In a stationary background (such as those considered in this paper), where the frequency $\omega$ is itself a conserved quantity, the conservation of norm becomes equivalent to the conservation of energy. However, norm conservation is more general.

For a stationary mode whose time-dependence is given by an oscillating phase $e^{-i\omega t}$, conservation of norm is equivalent to the constancy of the norm flux, $\delta N \times d\omega/dk$, which becomes $\delta N \times \omega/k$ in a non-dispersive limit. Recognizing that $h_0/c^2 \propto \Lambda/c$, and using the fact that $\rho$ and $W$ are also constant in our setup, this becomes

$$ \partial_x \left[ \frac{\Lambda}{c} (\omega - \bar{u}_0 k) \frac{\omega}{k} A^2 \right] = \partial_x [\pm \omega \Lambda A^2] = 0, \quad (47) $$

where $+(-)$ refers to the co- (counter-)propagating wave. These modes always have opposite norm flux, whether the flow is sub- or supercritical. Moreover, for both waves, we have

$$ \partial_x [\Lambda A^2] = 0. \quad (48) $$

We thus find that, in a stationary inhomogeneous background, the wave amplitude varies as $1/\sqrt{\Lambda}$ as a result of norm conservation.

There is a caveat to be mentioned here: Eqs. (47) and (48) hold for a single mode that varies continuously in space but does not scatter into other modes. In holds in an adiabatic limit, and the factor of $1/\sqrt{\Lambda}$ can be conveniently included in the mode definition so that the adiabatic limit corresponds to a strictly constant amplitude. In general, this amplitude will change through wave scattering. However, it remains true in general that the *total* norm, summed over all products of the scattering process, is conserved. (For the case of a stationary mode with fixed $\omega$, we see from Eqs. (45) that this is equivalent to energy conservation.) Since, as mentioned above, the co- and counter-propagating waves always have opposite norm flux, this means that Eq. (48) can be generalized to

$$ \partial_x [\Lambda (A_+^2 - A_-^2)] = 0, \quad (49) $$

or, absorbing a factor of $1/\sqrt{\Lambda}$ into the amplitude by defining $A_\pm = A_\pm/\sqrt{\Lambda}$, and complexifying the field so that the phase of the wave is captured by the complex phase of $A_\pm$ (i.e., writing $\delta\varphi = \text{Re}\{A_\pm e^{ikx-i\omega t}\}$), this becomes

$$ \partial_x [|A_+|^2 - |A_-|^2] = 0. \quad (50) $$

This is a more exact expression of norm or wave action conservation when wave scattering is taken into account.

Importantly, the norm $\delta N$ has no fixed sign; indeed, since $\omega - \bar{u}_0 k = \omega^{(0)}$ is the frequency in the “rest frame” of the fluid (by which we mean the frame in which the depth-averaged velocity vanishes), then its sign is precisely the sign of this “co-moving” frequency. Moreover, Eq. (44) indicates that the energy is negative whenever the co-moving frequency $\omega - \bar{u}_0 k$ and the lab frequency $\omega$ have opposite signs, i.e, when the flow is sufficiently strong to reverse the sign of the frequency with respect to its intrinsic sign. In scattering on a stationary background, it is the lab frequency $\omega$ that is conserved. This means that, on a black-hole flow where the flow passes from subcritical to supercritical, it is possible for scattering to mix modes of opposite energy (equivalently, opposite norm).

Finally, we note that scattering is typically described using an “in-out” formalism. That is, each of the asymptotic plane waves are characterized as either an *ingoing* or an *outgoing* wave, depending on whether their velocity points towards or away from the scattering region where the flow is inhomogeneous. The scattering process can then be mathematically described as a linear transformation between a set of ingoing amplitudes and a corresponding set of outgoing amplitudes. It is captured by the *scattering matrix*, $S$, whose elements are the normalized amplitudes of all outgoing waves produced by a single ingoing wave of unit normalized amplitude. The conservation of norm implies that

$$ \sum_{\text{out,norm}>0} |\mathcal{A}_{\text{norm}>0}^{\text{out}}|^2 - \sum_{\text{out,norm}<0} |\mathcal{A}_{\text{norm}<0}^{\text{out}}|^2 = 1, \quad (51) $$

where $\mathcal{A}^{\text{out}}$ are the asymptotic amplitudes of the outgoing waves. Equation (51) is often referred to as the *unitarity relation*. If none of the outgoing waves into which scattering can occur has negative norm, then only the first term on the left-hand side of Eq. (51) survives, and we are in a situation of *normal* scattering where the incident energy is simply redistributed among the outgoing waves. However, whenever outgoing waves of different sign of norm are present, we have a situation of *anomalous* scattering, where the total energy of outgoing positive-norm waves is larger than the energy of the incident wave in order to compensate for the negative-norm contribution. This can be viewed as the extraction of energy from the background.
V. EFFECTIVE BLACK-HOLE SPACETIME

Having explored some of the properties of the wave-like perturbations that are solutions of Eq. (32) on a fixed background, let us return to a consideration of the governing equations for the background flow. In particular, we will examine how, among the solutions of these governing equations, there exist those that are analogous to a black-hole spacetime. We also introduce the analogue Hawking effect, and calculate its associated “temperature” in the context of the black-hole flow.

A. Black-hole flow

We treat the background as a stationary flow. Therefore, the equations describing it are found by setting all time derivatives in the governing equations to zero. Equations (16) and (26) then immediately yield

$$ \partial_x [\bar{u}_0 h_0] = 0, \qquad \partial_x \left[ \frac{\bar{u}_0^2}{2g} + \frac{\Omega^2 h_0^2}{8g} + h_0 + b \right] = 0. \qquad (52) $$

The stationary background flow is thus characterized by two conserved quantities. One is the flow rate, $q = \bar{u}_0 h_0$. We may write $\bar{u}_0 = q/h_0$ and plug this into the second conservation equation, which will now be expressed solely in terms of the water depth $h_0$. It can be written in the form

$$ e(h_0) + b = \text{const.}, \qquad e(h_0) = \frac{q^2}{2gh_0^2} + \frac{\Omega^2 h_0^2}{8g} + h_0. \qquad (53) $$

Up to a term proportional to $q$, $e(h_0) + b$ may be thought of as the energy of a fluid parcel on the surface of the flow. Restricting to positive (i.e., physical) values of $h_0$, the function $e(h_0)$ has a single minimum and goes to infinity for $h_0 \to 0$ and $h_0 \to \infty$. The minimum occurs at the zero of the derivative:

$$ e'(h_0) = -\frac{q^2}{gh_0^3} + \frac{\Omega^2 h_0}{4g} + 1 = \frac{1}{gh_0} \left( -\frac{q^2}{h_0^2} + gh_0 + \frac{\Omega^2 h_0^2}{4} \right) = \frac{1}{gh_0} (-u^2 + c^2). \qquad (54) $$

Therefore, $e(h_0)$ achieves its minimum value wherever the flow velocity is equal in magnitude to the wave speed. For a given value of $q$, this has a well-defined value of $h_0$, which we shall call $h_{crit}$. For $h_0 > h_{crit}$, $e(h_0)$ is an increasing function of $h_0$, so we have $u^2 < c^2$; we refer to this as a *subcritical* flow. Conversely, for $h_0 < h_{crit}$, $e(h_0)$ is a decreasing function of $h_0$ and therefore we have $u^2 > c^2$, which we refer to as a *supercritical* flow.

While $q$ and $b$ are experimentally controlled, the value of $e+b$ will typically be determined by other considerations. Here, we are most interested in *transcritical* flows, which are those that pass from a subcritical region (the “black hole exterior”) to a supercritical one (the “black hole interior”). For this to occur smoothly requires passing through the critical depth $h_{crit}$ in the middle of the flow. Since $e(h_0) + b$ is constant and $e(h_{crit}) = e_{min}(q, \Omega)$ is the minimum possible value of $e(h_0)$, it follows that it is reached only when $b$ achieves its *maximum* value, which we shall call $b_{max}$. Therefore, for a transcritical flow,

$$ e(h_0) + b = e(h_{crit}) + b_{max} = e_{min}(q, \Omega) + b_{max}. \qquad (55) $$

It is fully determined by the values of $q$, $\Omega$, and $b_{max}$. The analogue horizon, where $Fr = 1$, occurs exactly on the top of the obstacle, where $b = b_{max}$.

B. Analogue Hawking effect

The passage from a subcritical region to a supercritical one means that the nature of the counter-propagating mode changes, depending on where we are within the flow. On the subcritical side, it is able to propagate upstream, and since $u^2 < c^2$ this wave must have positive energy. On the supercritical side, the flow overcomes the intrinsic wave speed, carrying it downstream even though it is attempting to propagate upstream. As discussed in Sec. IV C, this wave has negative energy. So the black-hole horizon splits the counter-propagating mode into two disconnected pieces with opposite energies, each propagating away from the horizon. It thus appears energetically possible for pairs of such modes to be continuously emitted from the horizon, since this would incur no overall energy cost.

The Analogue Gravity program was born from Unruh’s realization [2] that such emission of opposite-energy pairs is indeed expected to occur in such a scenario, for reasons very similar to those applicable to gravitational black
holes [19]. The simplest way to derive this result is to note that, as we approach the horizon, the wave number of the counter-propagating modes diverges, which engenders a logarithmic phase singularity for those modes in the vicinity of the horizon. In the near-horizon region, $u(x) - c(x) \approx \kappa(x - x_{\text{hor}})$, so applying the dispersion relation (36) to the counter-propagating mode with frequency $\omega$ gives a wave number $k = \omega / (u - c)$, and so the waveform is

$$ \begin{aligned} \phi(x,t) = \exp \left( i \int^{x} k(x') dx' - i\omega t \right) &= \exp \left( i \int^{x} \frac{\omega}{u(x') - c(x')} dx' - i\omega t \right) \\ &\approx \exp \left( i \int^{x} \frac{\omega}{\kappa(x' - x_{\text{hor}})} dx' - i\omega t \right) \\ &\approx \exp \left( i \frac{\omega}{\kappa} \log (x - x_{\text{hor}}) - i\omega t \right) \end{aligned} \tag{56} $$

This behavior of the phase encodes the fact that the rays of this mode can be traced back in time towards the horizon ad infinitum (see Figure 1). While this singular behavior is unavoidable, the field $\phi(x,t)$ can be made “well-behaved” if it can be continued smoothly onto the complex plane of $x$. As the logarithm necessarily has a branch cut ending at the point $x = x_{\text{hor}}$, this continuity can be achieved in either the upper- or lower-half complex plane (i.e., with $x$ obtaining a positive or negative imaginary part), but not in both directions simultaneously. So we get two independent modes that obey this continuity condition. Moreover, the continuation of $\log(x - x_{\text{hor}})$ is easily accomplished as $x - x_{\text{hor}}$ simply picks up a complex phase $e^{i\theta}$, with $\theta$ varying from 0 to $\pm\pi$ depending on which half-plane continuity is established. This gives the relative amplitude of the modes localized on either side of the horizon as $\exp(\mp\pi\omega/\kappa)$. Letting their amplitudes be $\alpha$ and $\beta$ with $|\beta| < |\alpha|$, this gives $|\beta/\alpha| = \exp(-\pi\omega/\kappa)$. Finally, upon normalizing the amplitudes so that $|\alpha|^2 - |\beta|^2 = 1$ (in line with Eq. (51) – note the minus sign, necessary because these two modes have opposite energy), we find

$$ |\beta|^2 = \frac{1}{e^{2\pi\omega/\kappa} - 1}, \quad \text{where} \quad \kappa = \partial_x (u-c)|_{\text{hor}} = \frac{1}{2u} \partial_x (u^2 - c^2)|_{\text{hor}}. \tag{57} $$

This quantifies the amount of “pair creation” – emission of opposite-energy waves on either side of the horizon – that must accompany an outgoing counter-propagating wave of frequency $\omega$ in order for the continuity condition to be respected. It has exactly the form of a Planck spectrum, which quantifies the occupation number of a bosonic mode at a certain temperature, and we can thus read off the corresponding temperature as $k_B T = \hbar\kappa/2\pi$. The occurrence of $\hbar$ indicates that this temperature is a quantum mechanical result, and indeed, this thermal emission is expected when it is triggered by quantum vacuum fluctuations. But we see from the above formula that it is more fundamentally described by a characteristic frequency $\kappa/2\pi$, which enters the expression for the scattering coefficients and is perfectly applicable to our classical system.

In the above description, we have glossed over the physical meaning behind the continuity in “complex” $x$, and the argument is thus quite mathematical in nature. There are more physical ways of regularizing the singular behavior at the horizon. In his original work, Hawking considered the formation of a black hole rather than a truly stationary background, then focused on the behavior in the final “stationary” regime to show that a steady thermal flux was present there. In the Analogue Gravity context, the regularization is typically achieved by the inclusion of dispersion, which implies that the outgoing modes described above are actually the result of scattering of an incident short-wavelength dispersive mode. It is a remarkable fact that the above argument, which is mathematically simple but physically obscure, manages to reproduce the essential behavior of more complicated treatments that take into account time-dependence or dispersion. So the scenario described above, and illustrated in Figure 1, can be considered as the late-time regime of a more complete scattering process, and the continuity condition for complex $x$ encodes the “smoothness” of the early-time behavior which has been lost from view by the simplicity of the model.

### C. The “surface gravity” of the black-hole flow

The Hawking-Unruh prediction of Eq. (57) applies to any system capable of realizing an analogue black-hole horizon in the manner described in this paper. Let us then apply it to the black-hole flow described in Sec. V A.

Both $u$ and $c$ depend on the water depth $h_0$, so we need to know how the derivative $dh_0/dx$ behaves at the horizon. We have already seen how the depth profile $h_0(x)$ is determined for a given obstacle profile $b(x)$ and flow rate $q$, through the constancy of $e(h_0) + b$. Differentiating with respect to $x$, this gives

$$ e'(h_0) \frac{dh_0}{dx} + \frac{db}{dx} = 0. \tag{58} $$
<figure><img src="image_2.png" /><figcaption>Figure 1. Space-time diagrams of two scattering processes occurring in a black-hole spacetime. The flow is from left to right, with the horizon occurring at the center. Red (blue) lines correspond to the co- (counter-) propagating mode. The negative-energy mode is shown in dashed line. (left) The Hawking effect: Governed by the bifurcation of the counter-propagating mode at the horizon, the relative amplitude between the two follows an exponential law that can be interpreted as a Boltzmann factor. (right) Scattering of an incident mode by inhomogeneities in the metric: The co-propagating mode simply crosses the horizon as it experiences nothing unusual there, and it is only on either side that it scatters into each of the counter-propagating modes.</figcaption></figure>

We have seen in the previous subsection that, at the horizon, $e'(h_0) = e'(h_{crit}) = 0$ and $db/dx = 0$, so (58) is still satisfied even though $dh_0/dx$ can remain finite there. Differentiating a second time, we have

$$e''(h_0) \left(\frac{dh_0}{dx}\right)^2 + e'(h_0) \frac{d^2h_0}{dx^2} + \frac{d^2b}{dx^2} = 0, \qquad (59)$$

and at the horizon, using again $e'(h_{crit}) = 0$, this becomes

$$\left|\frac{dh_0}{dx}\right|_{\text{hor}} = \sqrt{\frac{-d^2b/dx^2|_{\text{hor}}}{e''(h_{\text{crit}})}}, \qquad e''(h_{\text{crit}}) = \frac{3q^2}{gh_{\text{crit}}^4} + \frac{\Omega^2}{4g}. \qquad (60)$$

We have seen in Eq. (54) that $u^2 - c^2 = -gh_0 e'(h_0)$, and so

$$\kappa = \left| \frac{h_0}{2q} gh_0 e''(h_0) \frac{dh_0}{dx} \right|_{\text{hor}} = \frac{gh_{\text{crit}}^2}{2q} \sqrt{e''(h_{\text{crit}})} \sqrt{-\frac{d^2b}{dx^2}} \Big|_{\text{hor}} \qquad (61)$$

Plugging in $e''(h_{crit})$, this becomes

$$\kappa = \sqrt{1 - \frac{gh_{\text{crit}}^3}{4q^2}} \sqrt{-g \frac{d^2b}{dx^2}} \Big|_{\text{hor}}. \qquad (62)$$

The term $\sqrt{1 - gh_{crit}^3/4q^2}$ indicates how $\kappa$ depends on the parameters of the flow, in particular on $\Omega$ and $q$ (noting that $h_{crit}$, the water depth at the horizon, is a function of $\Omega$ and $q$ through Eq. (54)). Interestingly, when $\Omega = 0$ we have $gh_{crit}^3/q^2 = 1$, and $\kappa$ is seen to be independent of $q$. We also note that, from Eq. (54), $q^2/gh_{crit}^3$ becomes larger than 1 when $\Omega \neq 0$, so $gh_{crit}^3/q^2$ will thus lie between zero and one. Therefore, the factor $\sqrt{1 - gh_{crit}^3/4q^2}$ must lie in a narrow window between $\sqrt{3}/2 \approx 0.866$ and 1.

In short, the flow parameters $q$ and $\Omega$ can have only a very mild effect on the prediction for the “surface gravity” $\kappa$. The parameter of most direct relevance is the curvature of the obstacle given by $d^2b/dx^2$ at the horizon.

## VI. WAVE SCATTERING

The Hawking effect described in the previous section is essentially a form of wave scattering, but of a very particular kind as it is induced by the presence of the horizon, and is sensitive only to the local behavior of the flow in the
vicinity of the horizon. Here, we turn our attention to a more general kind of scattering that is not so local, but occurs continuously throughout the inhomogeneous region of the flow. More precisely, and in line with comments made in earlier sections, it is the inhomogeneities in the conformal factor $\Lambda^2(x)$ that will be responsible for this type of scattering.

## A. Implications of the conformal factor

The conformal factor $\Lambda^2$, which appears as an overall prefactor in the effective metric, is the key governing quantity in determining how waves scatter into each other during propagation. In particular, when this term is constant, it implies that no scattering occurs between the co- and counter-propagating waves. The reason is that, when $\Lambda = \text{const.}$, the wave equation (8) may be written as

$$ \left[ (\partial_t + \partial_x u) \frac{1}{c} (\partial_t + u \partial_x) - \partial_x c \partial_x \right] \phi = 0, \qquad (63) $$

and it is easily shown that this can be factorized, i.e., it can be written in the form

$$ [\partial_t + \partial_x (u \mp c)] \frac{1}{c} [\partial_t + (u \pm c) \partial_x] \phi = 0. \qquad (64) $$

The two orderings of the plus and minus signs are equivalent. Taken together, they immediately imply that the general solution is a sum of two solutions to two distinct first-order PDEs:

$$ \phi = \phi_{+} + \phi_{-} \qquad (65) $$

where

$$ [\partial_t + (u + c) \partial_x] \phi_+ = 0, \quad [\partial_t + (u - c) \partial_x] \phi_- = 0, \qquad (66) $$

and where the $+(-)$ sign corresponds precisely to the co- (counter-) propagating mode. This factorization is no longer possible when $\Lambda$ is $x$-dependent, which is why it engenders mixing between the co-propagating and counter-propagating solutions.

## B. Mode mixing and scattering amplitudes

Let us introduce the following “unscattered” modes:

$$ W_{\pm}(x) = \frac{1}{\sqrt{\Lambda(x)}} \exp \left( i \int^{x} k_{\pm}(x') dx' \right) = \frac{1}{\sqrt{\Lambda(x)}} \exp \left( i\omega \int^{x} \frac{dx'}{u(x') \pm c(x')} \right). \qquad (67) $$

These are exact solutions of the wave equation when $\Lambda$ is constant, and they are good approximate solutions when $\Lambda$ varies slowly. The general solution to the wave equation can be written as a sum of the two with position-dependent coefficients:

$$ \varphi(x) = \mathcal{A}_{+}(x) W_{+}(x) + \mathcal{A}_{-}(x) W_{-}(x). \qquad (68) $$

The amplitudes $\mathcal{A}_+$ and $\mathcal{A}_-$ are precisely the “normalized” amplitudes introduced in Sec. IV D, since the $1/\sqrt{\Lambda(x)}$ factor is precisely what is needed to conserve the norm flux of the modes. The wave equation is then transformed into an evolution equation for $(\mathcal{A}_+, \mathcal{A}_-)$, in the spirit of [40–42]. The detailed derivation is left to Appendix B; here we simply write the result:

$$ \partial_x \begin{bmatrix} \mathcal{A}_+ \\ \mathcal{A}_- \end{bmatrix} = \frac{\Lambda'(x)}{2\Lambda(x)} \begin{bmatrix} 0 & e^{2i\omega\theta(x)} \\ e^{-2i\omega\theta(x)} & 0 \end{bmatrix} \begin{bmatrix} \mathcal{A}_+ \\ \mathcal{A}_- \end{bmatrix}, \quad \theta(x) = \int^x \frac{c(x')}{u^2(x') - c^2(x')} dx'. \qquad (69) $$

This writing gives us a more direct route to the scattering coefficients. We see that the mode amplitudes $\mathcal{A}_\pm$ becomes constant asymptotically if $\Lambda$ becomes constant asymptotically. We simply need to specify appropriate initial conditions, which are typically determined via the identification of which mode plays the role of an incident mode.

It is straightforward to show that $|\mathcal{A}_+|^2 - |\mathcal{A}_-|^2$ is a conserved quantity. This is a necessary consequence of norm conservation and the fact that the two modes have opposite norm flux (see Sec. IV D).
An important complication is that the phase $\theta$ entering the evolution matrix in Eq. (69) is singular whenever $u^2 - c^2 = 0$. We have already seen how that engenders the Hawking process in the previous section. Here, it means that we cannot integrate across the horizon, but only up to points very close to it at $x = x_{\text{hor}} \pm \epsilon$. For the co-propagating mode, which encounters no phase singularity, we may simply impose continuity across the horizon by setting its value equal at $x = x_{\text{hor}} \pm \epsilon$. For the counter-propagating modes, these two points represent their “in” state, since these modes begin at the horizon and propagate outward.

### C. Scattering of incident co-propagating waves

For concreteness, we focus on the scenario illustrated in Figure 1(b): an incident co-propagating wave falling onto the horizon from the subcritical side, which is mainly transmitted across the horizon but is partially scattered into counter-propagating waves on either side. The initial conditions required for application of Eq. (69) are that the counter-propagating waves are products of the scattering and have no support on the horizon itself, so we may set their amplitudes to zero at the points $x = x_{\text{hor}} \pm \epsilon$ while that of the co-propagating mode is the same at those two points. We then integrate outward from the horizon on both sides, and the two pieces are finally recombined and normalized with respect to the amplitude of the incident wave.

We have performed the numerical integration of Eqs. (69) in precisely the manner just described. For simplicity, we assume a fixed obstacle with a Gaussian profile (see Appendix C for a treatment of asymmetrical obstacles):

$$b(x) = b_{\text{max}} \exp \left( -\frac{x^2}{2\sigma^2} \right), \qquad (70)$$

with $b_{\text{max}} = 10$ cm and $\sigma = 30$ cm. We consider four different values of the vorticity: $\Omega = 0, 5, 10$ and $15 \text{ s}^{-1}$. In one run, we fix the flow rate $q$ at $0.2 \text{ m}^2/\text{s}$, and this fixes the background flow in the manner described in Sec. V A. The results are plotted in Fig. 2. The coefficients plotted are the squared amplitudes of the waves in the asymptotic regions:

$$ \begin{aligned} R &= A_{-}(x \to -\infty) & (\text{reflected wave}), \\ T &= A_{+}(x \to +\infty) & (\text{transmitted wave}), \\ N &= A_{-}(x \to +\infty) & (\text{negative-energy wave}). \end{aligned} \qquad (71) $$

We have checked that the scattering coefficients satisfy the unitarity relation

$$|T|^2 + |R|^2 - |N|^2 = 1. \qquad (72)$$

In particular, when $|N|^2$ is larger than $|R|^2$, the transmission coefficient must be larger than 1 in order to compensate for the production of negative-energy waves, and we clearly see this occurring for some frequencies in Fig. 2. We can think of this as a form of energy extraction from the flow, with the negative-energy wave encoding the energy deficit. This is very similar to how, in the case of gravitational black holes, the Penrose process [43] allows energy extraction via the transfer of matter into a negative-energy state.

An important point concerns the effect of the vorticity: as $\Omega$ is increased, there is a noticeable trend for the waves to be less affected by the flow. Specifically, the reflection and negative-energy scattering coefficients decrease, while the transmission coefficient approaches unity. This observation is consistent with the remarks made at the end of Sec. III, where we noticed that if $Br \gg 1$, then $\Lambda^2(x)$ is almost independent of $x$. Equation (69) shows that, in this limit, we do indeed expect the scattering to be suppressed.

## VII. CONCLUSION AND DISCUSSION

We have reviewed the application of Analogue Gravity to the theory of shallow water waves, showing how the wave equation has the same form as that for a scalar field in an effective (2+1)-dimensional spacetime. In doing so, we have generalized the standard treatment of an irrotational flow to one with constant shear, deriving the change to the effective metric and reproducing the long-wavelength limit of the dispersion relation derived in previous works. This is, to our knowledge, the first demonstration that a flow with constant vorticity remains perfectly compatible with the existence of an effective spacetime metric. Importantly, it alters the form of the conformal factor, which is the key quantity governing the mixing of co- and counter-propagating waves on an inhomogeneous background. The presence of shear vorticity is found to reduce the inhomogeneity of the conformal factor, and we have confirmed this
<figure><img src="image_3.png" /><figcaption>Figure 2. Scattering coefficients over a fixed obstacle with identical values of the flow rate $q = 0.2 \text{ m}^2/\text{s}$, across several values of the vorticity. The top row shows the background flow: the water depth (left), the Froude number (center), and the profiles of $\bar{u}$ and $c$ (right). On the bottom are shown the scattering coefficients for the reflected mode (left), the transmitted mode (center), and the negative-energy mode (right).</figcaption></figure>

by numerically solving the scattering equations to show that the scattering coefficients are suppressed as vorticity is increased. We also considered the analogue Hawking effect, a peculiar scattering process induced by the phase singularity encountered by counter-propagating modes at the horizon (where the flow speed is equal to the wave speed). Application of the Hawking-Unruh prediction shows that the magnitude of this effect is only mildly affected by the presence of vorticity, the associated “surface gravity” (related to black hole temperature) experiencing, at most, an increase of about 15% with respect to the irrotational case.

The compatibility of a constant vorticity with the effective metric formalism is not a priori obvious, and is likely particular to this simplified model. While we have introduced it in order to begin to account for complications that are likely to arise experimentally, the assumption of a strictly constant vorticity is clearly an idealisation. A natural extension of this work, therefore, would be to allow the vorticity to vary with depth, as is typically the case in real flows where vorticity is induced by boundary friction and recirculation. Work on a layered flow, with non-zero vorticity restricted to the lower layer, will be presented in a follow-up paper. In addition, a ubiquitous feature that all analogue systems must contend with is dispersion, which (as mentioned in Sec. VB) regularises the phase singularity at the horizon. We thus also envisage the inclusion of dispersive effects by relaxing the long-wavelength limit, as has been studied quite extensively in the irrotational case [44, 45].

ACKNOWLEDGMENTS

AB and SR are funded by the CNRS Chair in Physical Hydrodynamics (reference ANR-22-CPJ2-0039-01). This work pertains (namely, is not funded but enters in the scientific perimeter) to the French government programs “Investissement d’Avenir” EUR INTREE (reference ANR-18-EURE-0010) and LABEX INTERACTIFS (reference ANR-11-LABX-0017-01).

Appendix A FIELD THEORY FORMULATION

In this appendix, we briefly introduce the effective field theory description of the system that would typically be used in the context of curved spacetime. The point is simply to show that such a mapping can be made. We also emphasize that the formalism used here brings with it a powerful mathematical structure.
A Lagrangian and canonical fields

The action associated with a massless scalar field has the form [46, 47]

$$
S = \iint dt \, dx \sqrt{-g} \mathcal{L}, \quad \mathcal{L} = -\frac{1}{2} g^{\mu\nu} \partial_{\mu} \phi \partial_{\nu} \phi. \tag{73}
$$

$\mathcal{L}$ is itself a scalar, while $\sqrt{-g} \, dx \, dt$ is the invariant volume element of the spacetime (that is, both are invariant under general coordinate transformations). The canonical field $\phi$ comes with an associated canonical momentum $\pi$:

$$
\pi = \frac{\partial(\sqrt{-g}\mathcal{L})}{\partial(\partial_t \phi)} = -\sqrt{-g} g^{t\mu} \partial_{\mu} \phi = -\sqrt{-g} (g^{tt} \partial_t + g^{tx} \partial_x) \phi. \quad (74)
$$

In our case, we have

$$
g_{\mu\nu} = \Lambda^2 \begin{bmatrix} -(c^2 - u^2) & -u & 0 \\ -u & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}, \quad g^{\mu\nu} = \frac{1}{\Lambda^2 c^2} \begin{bmatrix} -1 & -u & 0 \\ -u & c^2 - u^2 & 0 \\ 0 & 0 & c^2 \end{bmatrix}, \quad \sqrt{-g} = \Lambda^3 c. \tag{75}
$$

Our canonical field $\phi$ is just the perturbation of the velocity potential $\delta\varphi$, while the canonical momentum is

$$
\pi = \frac{\Lambda}{c} (\partial_t + u \partial_x) \phi = \frac{(\partial_t + \bar{u}_0 \partial_x) \delta\varphi}{1 + \frac{\Omega^2 h_0}{4g}} = g \delta h. \quad (76)
$$

B Hamiltonian

The energy density is identified with the Hamiltonian density, which is given by

$$
\begin{align}
\mathcal{H} &= \pi \partial_t \phi - \sqrt{-g} \mathcal{L} \nonumber \\
&= \frac{\Lambda}{c} (\partial_t + u \partial_x) \phi \partial_t \phi - \frac{\Lambda}{c} \left\{ \frac{1}{2} \left[ (\partial_t + u \partial_x) \phi \right]^2 - \frac{c^2}{2} (\partial_x \phi)^2 \right\} \nonumber \\
&= \frac{\Lambda}{c} \left\{ \frac{1}{2} \left[ (\partial_t + u \partial_x) \phi \right]^2 + \frac{c^2}{2} (\partial_x \phi)^2 - u \partial_x \phi (\partial_t + u \partial_x) \phi \right\}. \tag{77}
\end{align}
$$

This agrees (up to a constant prefactor) with the energy density of the previous subsection, and we clearly see the
appearance of the cross-term that allows the energy to be negative. In fact this can be rewritten in an even simpler
form:

$$
\mathcal{H} = \frac{\Lambda}{c} \left\{ \frac{1}{2} (\partial_t \phi)^2 + \frac{c^2 - u^2}{2} (\partial_x \phi)^2 \right\}, \quad (78)
$$

where we see clearly that the energy is allowed to be negative only when $u^2 > c^2$ (i.e., when $Fr > 1$).

C Scalar product and norm

We finish this section by noting an important symmetry and its associated conservation law. Let us complexify the
field $\phi$, whereupon the Lagrangian becomes

$$
\mathcal{L} = -g^{\mu\nu} \partial_{\mu} \phi^{*} \partial_{\nu} \phi. \tag{79}
$$

Although the actual fields we consider are real, there is nothing to prevent us from generalizing the field to a complex
one when the wave equation we work with is linear. The key advantage is that it allows us to exploit the symmetry
of the Lagrangian under phase rotations of the field, $\phi \to \phi e^{i\theta}$. This engenders, via Noether’s theorem, an associated
conservation law, which is:

$$
\partial_t (\phi_1, \phi_2) = 0, \quad (\phi_1, \phi_2) = i \int dx (\phi_1^* \pi_2 - \pi_1 \phi_2). \tag{80}
$$

The structure $(\phi_1, \phi_2)$ defines a scalar product on the space of all wave solutions. The scalar product of a solution with itself is often called its *norm*, and one can easily show that it coincides (up to a constant factor) with the norm defined in Sec. IV D. We see clearly that, although it is a real number, it is not necessarily positive: indeed, if a solution $\phi$ has positive norm, then $\phi^*$ (which is also a solution because the wave equation has real coefficients) has negative norm. We also see that real solutions have zero norm.
Appendix B DERIVATION OF MODE MIXING

We introduce two plane waves:

$$w_{\pm}(x) = \exp \left[ i \int^{x} k_{\pm}(x') dx' \right], \quad k_{\pm}(x') = \frac{\omega}{u(x') \pm c(x')} \qquad (81)$$

These are chosen because they almost solve the wave equation, but not quite. In fact, they *would* be exact solutions if the conformal factor $\Lambda$ were constant. To see the extent to which they do not solve the wave equation, we may plug them into the left-hand side of Eq. (8) and see how it differs from zero. Firstly, note that

$$\partial_x w_{\pm} = ik_{\pm} w_{\pm}, \quad \partial_x^2 w_{\pm} = (-k_{\pm}^2 + ik'_{\pm}) w_{\pm}. \qquad (82)$$

Also, by setting $\partial_t \to -i\omega$ and expanding, the d'Alembertian on the left-hand side of Eq. (8) becomes

$$\Box = -\frac{\Lambda}{c}\omega^2 - i\omega \left(u\frac{\Lambda}{c}\right)' - 2i\omega u \frac{\Lambda}{c} \partial_x + \left[\frac{\Lambda}{c}(u^2 - c^2)\right]' \partial_x + \frac{\Lambda}{c}(u^2 - c^2) \partial_x^2. \qquad (83)$$

Applying this to the plane waves $w_{\pm}$, we have

$$\begin{aligned}
\Box w_{\pm} &= \left[ -\frac{\Lambda}{c}\omega^2 - i\omega \left(u\frac{\Lambda}{c}\right)' + 2\omega u \frac{\Lambda}{c} k + i \left[\frac{\Lambda}{c}(u^2-c^2)\right]' k - \frac{\Lambda}{c}(u^2-c^2)k^2 + i\frac{\Lambda}{c}(u^2-c^2)k' \right] w_{\pm} \\
&= \left[ -\frac{\Lambda}{c} \left\{ (\omega-uk)^2 - c^2k^2 \right\} + i \left[ \frac{\Lambda}{c} \left\{ (u^2-c^2)k - \omega u \right\} \right]' \right] w_{\pm} \\
&= i \left[ \frac{\Lambda}{c} \left\{ \mp \omega c \right\} \right]' w_{\pm} \\
&= \mp i\omega \Lambda' w_{\pm}.
\end{aligned} \qquad (84)$$

This agrees with our expectation that $w_{\pm}$ are exact solutions of the wave equation whenever $\Lambda$ is constant.

To combine the $w_{\pm}(x)$ into an exact solution of the wave equation, we introduce amplitudes $a_{\pm}(x)$ (note that these are also $x$-dependent), and we write

$$\phi = a_+ w_+ + a_- w_-, \quad \phi' = a_+ w'_+ + a_- w'_-. \qquad (85)$$

The first of these equations is simply a decomposition of the exact solution into the plane waves, but it is not sufficient to uniquely determine how the amplitudes $a_{\pm}$ behave. So the second equation acts as a constraint by imposing that the derivative satisfies exactly the same decomposition. This is a choice that happens to simplify the calculations. Note that it is equivalent to stating the following in terms of the derivatives of the amplitudes:

$$a'_{+} w_{+} + a'_{-} w_{-} = 0. \qquad (86)$$

Furthermore, it yields the following expression for the second derivative of $\phi$:

$$\phi'' = a_+ w''_+ + a_- w''_- + a'_+ w'_+ + a'_- w'_-. \qquad (87)$$

Examining Eqs. (85) and (87), we see that $\phi^{(n)} = a_+ w_+^{(n)} + a_- w_-^{(n)}$, except for an additional contribution to the second derivative $\phi''$. This yields a very simple expression when we take the d'Alembertian of $\phi$: applying (83) to $\phi$, we get

$$\Box \phi = a_+ \Box w_+ + a_- \Box w_- + \frac{\Lambda}{c} (u^2 - c^2) (a'_+ w'_+ + a'_- w'_-) = 0. \qquad (88)$$

We may substitute the results (84) to get

$$\begin{aligned}
\Box \phi &= -i\omega \Lambda' (a_+ w_+ - a_- w_-) + \frac{\Lambda}{c} (u^2 - c^2) (a'_+ w'_+ + a'_- w'_-) \\
&= -i\omega \left[ \Lambda' (a_+ w_+ - a_- w_-) - \frac{\Lambda}{c} (u^2 - c^2) \left( \frac{1}{u+c} a'_+ w_+ + \frac{1}{u-c} a'_- w_- \right) \right] \\
&= -i\omega \left[ \Lambda' (a_+ w_+ - a_- w_-) - \frac{\Lambda}{c} \left( (u-c) a'_+ w_+ + (u+c) a'_- w_- \right) \right] \\
&= -i\omega \left[ \Lambda' (a_+ w_+ - a_- w_-) + \Lambda (a'_+ w_+ - a'_- w_-) \right] \\
&= 0.
\end{aligned} \qquad (89)$$
Recalling Eq. (86), we may isolate $a'_\pm$ in terms of $a_\pm$:

$$a'_+ = -\frac{\Lambda'}{2\Lambda} \left( a_+ - a_- \frac{w_-}{w_+} \right), \quad a'_- = -\frac{\Lambda'}{2\Lambda} \left( a_- - a_+ \frac{w_+}{w_-} \right). \qquad (90)$$

Finally, we may remove the diagonal contribution to the evolution of $a_\pm$ by defining normalized amplitudes $A_\pm$:

$$a_\pm = \frac{A_\pm}{\sqrt{\Lambda}}, \qquad (91)$$

upon which the above equations become

$$A'_+ = \frac{\Lambda'}{2\Lambda} \frac{w_-}{w_+} A_-, \quad A'_- = \frac{\Lambda'}{2\Lambda} \frac{w_+}{w_-} A_+, \qquad (92)$$

or, in matrix form,

$$\partial_x \begin{bmatrix} A_+ \\ A_- \end{bmatrix} = \frac{\Lambda'}{2\Lambda} \begin{bmatrix} 0 & \exp \left( 2i\omega \int^x \frac{c}{u^2-c^2} dx' \right) \\ \exp \left( -2i\omega \int^x \frac{c}{u^2-c^2} dx' \right) & 0 \end{bmatrix} \begin{bmatrix} A_+ \\ A_- \end{bmatrix}. \qquad (93)$$

The form of this result, as a coupling between two mode amplitudes, is also found when considering pair creation in cosmology [41, 42].

## Appendix C WAVE SCATTERING OVER AN ASYMMETRICAL OBSTACLE

Figure 2 was produced assuming an obstacle with a Gaussian profile (70). In reality there are advantages to considering an asymmetrical profile, particularly one with a mild slope on the downstream side so as to minimize boundary separation and recirculation [14, 15, 17, 48–50].

Here, we show the scattering coefficients pertaining to an asymmetrical obstacle, of similar shape to obstacles used in experiments at Institut Pprime. The shape of this obstacle is simply a trapezoid:

$$\frac{b(x)}{b_{\max}} = \begin{cases} \frac{x-x_1}{x_2-x_1} & x_1 \le x \le x_2 \\ 1 & x_2 \le x \le x_3, \\ \frac{x_4-x}{x_4-x_3} & x_3 \le x \le x_4 \end{cases}, \qquad (94)$$

where $b_{\max} = 10$ cm, $x_1 = -0.53$ m, $x_2 = -0.23$ m, $x_3 = 0.23$ m, and $x_4 = 0.98$ m.

Note that this obstacle has sharp corners, which is not strictly compatible with the long-wavelength limit in which the effective metric description is valid. This is particularly problematic for the analogue Hawking effect, which is concerned with the behavior of the flow near the horizon and is governed mainly by the second derivative of the obstacle profile at the top of the obstacle (see Eq. (62)). For the piecewise obstacle used here, with a flat top of finite extension, this second derivative is not well-defined. However, as far as mixing between the co- and counter-propagating modes is concerned – and especially for the scattering of an incident co-current mode, where the amplitudes of the counter-propagating waves is zero at the horizon and the phase singularity is thereby not encountered – then the evolution equations derived in Appendix B remain sensible and well-defined. The initial conditions need simply be specified a short distance $\epsilon$ on either side of the flat top (i.e., at $x = x_2 - \epsilon$ and $x = x_3 + \epsilon$) instead of at $x = x_{\text{hor}} \pm \epsilon$ as in Sec. VI.

The results are shown in Fig. 3, where again the flow rate $q$ has been fixed at $0.2 \text{ m}^2/\text{s}$, exactly as in Fig. 2. The sharp corners seen in the flow profile are inherited from the sharp corners of the obstacle as a result of our idealized treatment in the long-wavelength limit, and of course will be smoothed out in an actual experiment. The results for the scattering coefficients are very similar to what we saw in Sec. VI, particularly the fact that the scattering tends to be suppressed with increasing vorticity. The main visible difference here is that the transmission coefficient does not exceed 1 as clearly as it did in Fig. 2. This indicates a shift in the ratio between the scattering into the negative-energy mode on the downstream side of the obstacle, compared with the scattering into the positive-energy reflected mode on the upstream side. Indeed, comparing Fig. 3 with Fig. 2, we see that $|R|^2$ falls off quickly at higher frequencies in Fig. 2 while $|N|^2$ has an extended tail, whereas in Fig. 3 it is the other way round. This makes sense when we consider that, in Fig. 3, the downstream part of the flow varies more slowly by construction of the obstacle, and as the scattering amplitude is related to the derivative of $\Lambda(x)$ then we can expect a suppression of $|N|^2$.
<figure><img src="image_4.png" /><figcaption>Figure 3. Scattering coefficients over a fixed asymmetric obstacle (described in Eq. (94)) with identical values of the flow rate $q = 0.2 \text{ m}^2/\text{s}$, across several values of the vorticity. The top row shows the background flow: the water depth (left), the Froude number (center), and the profiles of $\bar{u}$ and $c$ (right). On the bottom are shown the scattering coefficients for the reflected mode (left), the transmitted mode (center), and the negative-energy mode (right).</figcaption></figure>

As a final check, we also consider the scattering over an obstacle which is essentially a smoothed-out version of that of Eq. (94). This obstacle has the following form:

$$b(x) = b_0 [\tanh(\kappa_A (x - x_A)) - \tanh(\kappa_B (x - x_B))], \quad (95)$$

where $b_0 = 6 \text{ cm}$, $\kappa_A = 4.74 \text{ m}^{-1}$, $x_A = -0.375 \text{ m}$, $\kappa_B = 1.87 \text{ m}^{-1}$, and $x_B = 0.602 \text{ m}$. Note that $b_{\text{max}}$ here is very slightly larger than before, at 10.5 cm. The associated scattering coefficients are presented in Fig. 4. They behave in a qualitatively similar way, the main difference being that they are suppressed more quickly at larger frequencies.

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<figure><img src="image_5.png" /><figcaption>Figure 4. Scattering coefficients over a fixed asymmetric obstacle (described in Eq. (95), which is a smoothed version of that in Eq. (94)) with identical values of the flow rate $q = 0.2 \text{ m}^2/\text{s}$, across several values of the vorticity. The top row shows the background flow: the water depth (left), the Froude number (center), and the profiles of $\bar{u}$ and $c$ (right). On the bottom are shown the scattering coefficients for the reflected mode (left), the transmitted mode (center), and the negative-energy mode (right).</figcaption></figure>

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