# Resonant triad interactions of gravity waves
in cylindrical basins

Matthew Durey1 and Paul A. Milewski2

( 1School of Mathematics and Statistics, University of Glasgow, University
Place, Glasgow G12 8QQ, UK
2Department of Mathematical Sciences, University of Bath, Claverton Down, Bath
BA2 7AY, UK )

###### Abstract

We present the results of a theoretical investigation into the existence,
evolution and excitation of resonant triads of nonlinear free-surface gravity
waves confined to a cylinder of finite depth. It is well known that resonant
triads are impossible for gravity waves in laterally unbounded domains; we
demonstrate, however, that horizontal confinement of the fluid may induce
resonant triads for particular fluid depths. For any three correlated wave
modes arising in a cylinder of arbitrary cross-section, we prove necessary and
sufficient conditions for the existence of a depth at which nonlinear
resonance may arise, and show that the resultant critical depth is unique. We
enumerate the low-frequency triads for circular cylinders, including a new
class of resonances between standing and counter-propagating waves, and also
briefly discuss annular and rectangular cylinders. Upon deriving the triad
amplitude equations for a finite-depth cylinder of arbitrary cross-section, we
deduce that the triad evolution is always periodic, and determine parameters
controlling the efficiency of energy exchange. In order to excite a particular
triad, we explore the influence of external forcing; in this case, the triad
evolution may be periodic, quasi-periodic, or chaotic. Finally, our results
have potential implications on resonant water waves in man-made and natural
basins, such as industrial-scale fluid tanks, harbours and bays.

##  1 Introduction

Nonlinear resonance is a mechanism by which energy is continuously transferred
between a small number of linear wave modes. This phenomenon, first observed
in Wilton’s analysis of gravity-capillary wave trains [64], has been the
subject of frequent investigation over the past century [37, 38, 39, 55, 53,
20, 16]; indeed, nonlinear resonance has since been observed for wave trains
in a growing number of dispersive wave systems, including gravity waves [49,
22, 34, 3], acoustic-gravity waves [28, 27], flexural-gravity waves [59], two-
layer flows [2, 26, 54], and atmospheric flows [52, 51]. Whilst the
aforementioned studies typically consider nonlinear resonance for laterally
unbounded domains, the purpose of this study is to demonstrate that energy
exchange between free-surface gravity waves may be induced and accentuated by
horizontal confinement.

We focus our study on the collective resonance of three linear wave modes,
henceforth referred to as a _triad_ [6]. In laterally unbounded domains, the
monotonic and concave form of the dispersion curve precludes the existence of
resonant triads for gravity wave trains at finite depth [49, 22], with
resonant quartets instead being the smallest possible collective resonant
interaction [3, 34, 5]. However, confinement of the fluid to a vertical
cylinder results in linear wave modes that differ in form to sinusoidal plane
waves (except for a rectangular cylinder), so the preclusion of resonant
triads no longer applies. Indeed, our study demonstrates that, under certain
conditions, resonant triads may arise in cylinders of arbitrary cross-section
for specific values of the fluid depth. As resonant triads evolve over a much
faster time scale than that of resonant quartets, the exchange of energy in
gravity waves is thus more efficient under the influence of lateral
confinement [41], with potential implications on resonant sloshing in man-made
and natural basins [9].

Prior investigations of confined resonant free-surface gravity waves have
predominantly focused on the so-called 1:2 resonance, which arises when two of
the three linear wave modes comprising a triad coincide. For axisymmetric
standing waves in a circular cylinder, Mack [35] determined a condition for
the existence of critical depth-to-radius ratios at which a 1:2 resonance may
arise, a result later generalised to cylinders of arbitrary cross-section
[45]. Miles [43, 44] then characterised the weakly nonlinear evolution of such
internal resonances, demonstrating that a 1:2 resonance is impossible in a
rectangular cylinder [43]. Although Miles’ seminal results provide an
informative view of the weakly nonlinear dynamics, the influence of fully
nonlinear effects was later assessed by Bryant [9] and Yang _et al._ [65]. For
the case of a circular cylinder of finite depth, Bryant [9] and Yang _et al._
[65] characterised new steadily propagating nonlinear waves arising in the
vicinity of a 1:2 resonance, and Yang _et al._ [65] also computed nonlinear
near-resonant axisymmetric standing waves. Finally, broader mathematical
properties of water waves exhibiting O(2) symmetry (of which a circular
cylinder is one example) were analysed by Bridges & Dias [7] and Chossat &
Dias [13].

Given the restrictive set of critical depths at which a 1:2 resonance may
arise [9, 65], it is natural to explore the possibility of nonlinear resonance
in cylinders whose depth departs from the depths that trigger a 1:2 resonance.
To the best of our knowledge, the first and only such study was the seminal
experimental investigation performed by Michel [41], who focused on resonant
triads arising for free-surface gravity waves confined to a finite-depth
circular cylinder. Notably, the cylinder depth in Michel’s experiment was
judiciously chosen so as to isolate a specific triad. Michel utilised
bandlimited random horizontal vibration so as to excite two members of the
triad, whose nonlinear interaction led to the growth of the third mode.
Significantly, the energy of the third mode was, on average, the product of
the energies of the remaining two modes, thereby satisfying the quadratic
energy exchange typical of resonant triads.

In order to exemplify the mechanism of nonlinear resonance, Michel [41] also
calculated the response of a child mode due to the nonlinear interaction
between two parent modes (where all three wave modes comprise the triad).
Notably, Michel’s calculation is restricted to the early stages of growth and
to particular relative phases of the wave modes. In addition, Michel
considered a fluid of infinite depth for all but the resonance conditions, for
which finite-depth corrections were included. In contrast, we consider general
resonances in arbitrary cylinders of finite depth and derive equations for the
triad evolution over long time-scales. We also believe some nonlinear
contributions to the interactions were omitted from Michel’s calculation,
resulting in quantitative differences (see §4.3).

The goal of our study is to unify the existence and evolution of 1:2 and
triadic resonances into a single mathematical framework, effectively
characterising all triad interactions of this type. Based on existing theory,
it is unclear how the existence of resonant triads depends on the form of the
cylinder cross-section, and which combinations of wave modes are permissible
for judicious choice of the fluid depth. Furthermore, the range of depths that
may excite a particular triad is uncertain, with 1:2 resonances only excited
in a very narrow window about each critical depth [35, 45]. Once a particular
triad is excited, one anticipates that the triad evolution will be governed by
the canonical triad equations [6, 16]; however, quantifying the triad
evolution and relative energy exchange requires computation of the triad
coupling coefficients. Finally, it is unclear how best to excite triads in
arbitrary cylinders, both with and without external forcing.

We here present a relatively comprehensive characterisation of the existence,
evolution and excitation of resonant triads for gravity waves confined to a
cylinder of arbitrary cross-section and finite depth. In order to reduce the
problem to its key components, we first truncate the Euler equations,
recasting the fluid evolution in terms of a finite-depth Benney-Luke equation
(§2), incorporating only the nonlinear interactions necessary for resonant
triads. In §3, we prove necessary and sufficient conditions for there to exist
a finite depth at which three linear wave modes may form a resonant triad. In
particular, we prove that resonant triads are impossible for rectangular
cylinders, yet there is an abundance of resonant triads for circular
cylinders. We then use multiple-scales analysis to determine the long-time
evolution of a triad in a cylinder of arbitrary cross-section (§4), from which
we characterise the relative coupling of different triads. Finally, we explore
the excitation of resonant triads (§5), and discuss the potential extension of
our theoretical developments to the cases of applied forcing and two-layer
flows (§6).

##  2 Formulation

We consider the irrotational flow of an inviscid, incompressible liquid that
is bounded above by a free surface, confined laterally by the vertical walls
of a cylinder whose horizontal cross-section, $\mathcal{D}$, is enclosed by
the curve $\partial\mathcal{D}$, and bounded below by a rigid horizontal plane
lying a distance $H$ below the undisturbed free surface; see figure 1. We
consider the fluid evolution in dimensionless variables, taking the cylinder’s
typical horizontal extent, $a$, as the unit of length, and $\sqrt{ag^{-1}}$ as
the unit of time, where $g$ is the acceleration due to gravity. It follows
that the dimensionless free-surface elevation, $\eta(\bm{x},t)$, and velocity
potential, $\phi(\bm{x},z,t)$, evolve according to the equations

$\displaystyle\Delta\phi+\phi_{zz}$ $\displaystyle=0\qquad$
$\displaystyle\mathrm{for}\,\,\,\bm{x}\in\mathcal{D},\quad-h<z<\epsilon\eta,$
(1a)
$\displaystyle\phi_{t}+\eta+\frac{\epsilon}{2}\Big{(}|\nabla\phi|^{2}+\phi_{z}^{2}\Big{)}$
$\displaystyle=0\quad$
$\displaystyle\mathrm{for}\,\,\,\bm{x}\in\mathcal{D},\quad z=\epsilon\eta,$
(1b) $\displaystyle\eta_{t}+\epsilon\nabla\phi\cdot\nabla\eta$
$\displaystyle=\phi_{z}\quad$
$\displaystyle\mathrm{for}\,\,\,\bm{x}\in\mathcal{D},\quad z=\epsilon\eta,$
(1c) $\displaystyle\bm{n}\cdot\nabla\phi$ $\displaystyle=0$
$\displaystyle\mathrm{for}\,\,\,\bm{x}\in\partial\mathcal{D},\quad-h<z<\epsilon\eta,$
(1d) $\displaystyle\phi_{z}$ $\displaystyle=0$
$\displaystyle\mathrm{for}\,\,\,\bm{x}\in\mathcal{D},\quad z=-h,$ (1e)

corresponding to the continuity equation, dynamic and kinematic boundary
conditions, and no-flux through the vertical walls and horizontal base,
respectively. In equation (1), the dimensionless parameter $\epsilon$ is
proportional to the typical wave slope, $h=H/a$ is the ratio of the fluid
depth to the typical horizontal extent, $\bm{n}$ is a unit vector normal to
the boundary $\partial\mathcal{D}$, and the operators $\nabla$ and $\Delta$
denote the horizontal gradient and Laplacian, respectively. Moreover,
conservation of mass implies that the free surface satisfies
$\iint_{\mathcal{D}}\eta\,\mathrm{d}A=0$ for all time. Finally, in dimensional
variables, $a\bm{x}$ is the two-dimensional horizontal coordinate, $az$ is the
upward-pointing vertical coordinate, $\sqrt{ag^{-1}}t$ denotes time, $\epsilon
a\eta$ is the free-surface displacement, and $\epsilon a\sqrt{ag}\phi$ is the
velocity potential.

Figure 1:  Schematic diagram of the cylindrical tank (with cross-section
$\mathcal{D}$ and boundary $\partial\mathcal{D}$) partially filled with
liquid. The undisturbed free surface (dashed lines) lies on $z=0$, a distance
$H$ above the rigid bottom plane (grey). The disturbed free surface is
sketched in dash-dotted lines.

We aim to develop a broad framework for understanding resonant triads in a
cylinder of finite depth; however, care must be taken when modelling fluid-
boundary interactions and determining the class of permissible cylinder cross-
sections. From a modelling perspective, we employ an assumption generally
implicit to the water-wave problem in bounded domains; specifically, we
neglect the meniscus and dissipation arising near the vertical walls [42],
thus determining that the free surface intersects the boundary normally, i.e.
$\bm{n}\cdot\nabla\eta=0$ for $\bm{x}\in\partial\mathcal{D}$ [47]. In order to
maximise the generality of our investigation, we allow the cylinder cross-
section, $\mathcal{D}$, to be fairly arbitrary; however, the mathematical
developments presented herein require $\mathcal{D}$ to be bounded with a
piecewise-smooth boundary, thereby allowing us to utilise the spectral theorem
for compact self-adjoint operators [30] and the divergence theorem. As most
cylinders of practical interest consist of a piecewise-smooth boundary, this
mathematical restriction fails to limit the breadth of our study.

###  2.1 Derivation of the Benney-Luke equation

As our study is focused on the weakly nonlinear evolution of small-amplitude
waves, we proceed to simplify (1) in the case $0<\epsilon\ll 1$ and $h=O(1)$.
We begin by expanding the dynamic and kinematic boundary conditions (equations
(1b)–(1c)) about $z=0$ in powers of $\epsilon$, which, upon eliminating
$\eta$, gives rise to the equation [3, 48]

$\phi_{tt}+\phi_{z}=\epsilon\Big{(}\partial_{t}(\phi_{tz}\phi_{t})+\phi_{zz}\phi_{t}-\partial_{t}(|\nabla\phi|^{2})-\phi_{z}\phi_{zt}\Big{)}+O(\epsilon^{2})\quad\mathrm{for}\quad\bm{x}\in\mathcal{D},\quad
z=0.$ (2)

To reduce the fluid evolution to the dynamics arising on the linearised free
surface, $z=0$, we define the Dirichlet-to-Neumann operator, $\mathscr{L}$, so
that $\mathscr{L}\phi|_{z=0}=\phi_{z}|_{z=0}$. Here $\phi$ satisfies Laplace’s
equation (1a) over the linearised domain $-h<z<0$, with $\partial_{z}\phi=0$
on $z=-h$ (see equation (1e)) and $\bm{n}\cdot\nabla\phi=0$ for
$\bm{x}\in\partial\mathcal{D}$ (see equation (1d)). Notably, the Dirichlet-to-
Neumann operator may be defined in terms of its spectral representation, as
detailed in §2.2. By denoting $u(\bm{x},t)=\phi(\bm{x},0,t)$, we finally
obtain the finite-depth Benney-Luke equation [3, 4, 48]

$u_{tt}+\mathscr{L}u+\epsilon\bigg{(}u_{t}\big{(}\mathscr{L}^{2}+\Delta\big{)}u+\frac{\partial{}}{\partial{t}}\Big{[}(\mathscr{L}u)^{2}+|\nabla
u|^{2}\Big{]}\bigg{)}=O(\epsilon^{2})\quad\mathrm{for}\quad\bm{x}\in\mathcal{D},$
(3)

where we have simplified the nonlinear terms in equation (2) using
$\phi_{zz}=-\Delta\phi$ and $u_{tt}=-\mathscr{L}u+O(\epsilon)$.

The remainder of our investigation will be focused on the evolution of
resonant triads governed by the Benney-Luke equation (3). As resonant triads
arising in confined geometries are governed primarily by quadratic
nonlinearities, it is sufficient to neglect terms of size $O(\epsilon^{2})$ in
equation (3); however, higher-order corrections to the Benney-Luke equation
may be derived by following a similar expansion procedure [3, 48, 5]. Although
our investigation is mainly focused on the evolution of the velocity
potential, $u$, one may recover the leading-order free-surface elevation from
the dynamic boundary condition (1b), namely $\eta=-u_{t}+O(\epsilon)$.

###  2.2 Spectral representation of the Dirichlet-to-Neumann operator

The Dirichlet-to-Neumann operator, $\mathscr{L}$, may be understood in terms
of the discrete set of orthogonal eigenfunctions of the horizontal Laplacian
operator [30]. Specifically, we consider the set of real-valued
eigenfunctions, $\Phi_{n}(\bm{x})$, satisfying

$-\Delta\Phi_{n}=k_{n}^{2}\Phi_{n}\,\,\,\mathrm{for}\,\,\,n=0,1,\ldots,$

where the corresponding eigenvalues, $k_{n}^{2}$, are ordered so that
$0=k_{0}<k_{1}\leq k_{2}\leq\ldots$. Moreover, each eigenfunction satisfies
the boundary condition $\bm{n}\cdot\nabla\Phi_{n}=0$ on $\partial\mathcal{D}$,
as motivated by the no-flux condition (1d). Finally, the orthogonal
eigenfunctions are normalised so that
$\langle\Phi_{m},\Phi_{n}\rangle=\delta_{mn}$, where

$\langle f,g\rangle=\frac{1}{S}\iint_{\mathcal{D}}fg\,\mathrm{d}A$

defines an inner product for real functions $f$ and $g$, $S$ is the area of
$\mathcal{D}$, and $\delta_{mn}$ is the Kronecker delta. Notably,
$\Phi_{0}(\bm{x})=1$ is the constant eigenfunction, with corresponding
eigenvalue $k_{0}=0$.

To determine the Dirichlet-to-Neumann operator for sufficiently smooth $\phi$,
we first substitute the series expansion
$\phi(\bm{x},z)=\sum_{n=0}^{\infty}\phi_{n}(z)\Phi_{n}(\bm{x})$ into Laplace’s
equation (1a), where we have temporally omitted the time dependence. We then
solve the resulting equation for $\phi_{n}(z)$ over the linearised domain
$-h<z<0$, in conjunction with the no-flux condition on $z=-h$ (see equation
(1e)). It follows that
$\partial_{z}\phi_{n}(0)=\hat{\mathscr{L}}_{n}\phi_{n}(0)$, where

$\hat{\mathscr{L}}_{n}=k_{n}\tanh(k_{n}h)$ (4)

is the spectral multiplier of the Dirichlet-to-Neumann operator,
$\mathscr{L}$. By expressing the time-dependent free-surface velocity
potential, $u=\phi|_{z=0}$, in terms of the basis expansion
$u(\bm{x},t)=\sum_{n=0}^{\infty}u_{n}(t)\Phi_{n}(\bm{x})$, it follows that the
Dirichlet-to-Neumann map has the spectral representation
$\mathscr{L}u=\sum_{n=0}^{\infty}\hat{\mathscr{L}}_{n}u_{n}\Phi_{n}$.

##  3 The existence of resonant triads

Resonant triads arise due to the exchange of energy between linear wave modes,
an effect induced by nonlinear wave interactions. In order to define resonant
triads mathematically, it is necessary to first determine the angular
frequency associated with each linear wave mode. In the limit
$\epsilon\rightarrow 0$, the Benney-Luke equation (3) reduces to the linear
equation $u_{tt}+\mathscr{L}u=0$. By seeking a solution to the linearised
Benney-Luke equation of the form
$u(\bm{x},t)=\Phi_{n}(\bm{x})\mathrm{e}^{-\mathrm{i}\omega_{n}t}$, we conclude
that the angular frequency, $\omega_{n}$, satisfies
$\omega_{n}^{2}=\hat{\mathscr{L}}_{n}$, or the more familiar [33]

$\omega_{n}^{2}=k_{n}\tanh(k_{n}h).$ (5)

As we will see, a crucial aspect of the following analysis is that the angular
frequency depends on the fluid depth, i.e. $\omega_{n}(h)$. Finally, we note
that the angular frequency is larger for more oscillatory eigenfunctions (i.e.
for larger values of $k_{n}$); by analogy to the evolution of plane gravity
waves, we refer to $k_{n}$ as a ‘wavenumber’ henceforth.

We proceed by considering three linear wave modes, enumerated $n_{1}$, $n_{2}$
and $n_{3}$, where we denote

$\Omega_{j}=\omega_{n_{j}},\quad
K_{j}=k_{n_{j}},\quad\mathrm{and}\quad\Psi_{j}(\bm{x})=\Phi_{n_{j}}(\bm{x})\quad\mathrm{for}\,\,\,j=1,2,3.$

Notably, we exclude the wavenumber $k_{0}=0$ from consideration as the
corresponding eigenmode, $\Phi_{0}$, simply reflects the invariance of the
Benney-Luke equation (3) under the mapping $u\mapsto u+\mathrm{constant}$;
henceforth, we consider only wavenumbers $K_{j}>0$. The three linear wave
modes form a resonant triad if there is a critical fluid depth, $h_{c}$,
satisfying

$\Omega_{1}(h_{c})\pm\Omega_{2}(h_{c})\pm\Omega_{3}(h_{c})=0,$ (6)

where all four sign combinations are permissible (we consider $\Omega_{j}>0$
without loss of generality). To simplify notation in the following arguments,
we restrict our attention to the particular case

$\Omega_{1}(h_{c})+\Omega_{2}(h_{c})=\Omega_{3}(h_{c}),$ (7)

where the other three sign combinations in equation (6) may be recovered by
suitable re-indexing of the $\Omega_{j}$ terms. However, as we will see in §4,
an additional constraint necessary for triads to exist is the eigenmode
correlation condition,

$\iint_{\mathcal{D}}\Psi_{1}\Psi_{2}\Psi_{3}\,\mathrm{d}A\neq 0,$ (8)

which implies that the product of any two eigenmodes is non-orthogonal to the
remaining eigenmode.

###  3.1 The existence of a critical depth

We proceed to determine necessary and sufficient conditions on the
wavenumbers, $K_{j}$, for there to exist a depth, $h_{c}$, at which a resonant
triad forms, where such a critical depth is unique. We summarise our results
in terms of the following theorem.

######  Theorem 1.

There exists a positive and finite value of $h$ such that
$\Omega_{1}+\Omega_{2}=\Omega_{3}$ if and only if

$K_{1}+K_{2}<K_{3}<\big{(}\sqrt{K_{1}}+\sqrt{K_{2}}\big{)}^{2}.$ (9)

When this pair of inequalities is satisfied, the corresponding value of $h$ is
unique.

We briefly sketch the proof of Theorem 1, with full details presented in
appendix A. We first demonstrate that no solutions to
$\Omega_{1}+\Omega_{2}=\Omega_{3}$ are possible when the bounds in equation
(9) are violated, i.e. when $K_{1}+K_{2}\geq K_{3}$ or when
$\sqrt{K_{1}}+\sqrt{K_{2}}\leq\sqrt{K_{3}}$. We then consider the case where
the inequalities (9) are satisfied and determine the existence of positive
roots to the function $F(h)=(\Omega_{1}(h)+\Omega_{2}(h))/\Omega_{3}(h)-1$. In
this case, we demonstrate that $\lim_{h\rightarrow 0}F(h)<0$ and
$\lim_{h\rightarrow\infty}F(h)>0$, from which we conclude that $F(h)$ has at
least one root (by continuity of $F$). Finally, we deduce that this root is
unique by proving that $F(h)$ is a strictly monotonically increasing function
of $h$ when the inequalities (9) are satisfied.

Two important conclusions may be deduced from Theorem 1. First, it follows
from equation (9) that the wavenumber, $K_{3}$, corresponding to the largest
angular frequency, $\Omega_{3}$, is larger than both the other two wavenumbers
($K_{1}$ and $K_{2}$), but it cannot be arbitrarily large (as supplied by the
upper bound). For a given pair of eigenmodes (say $\Psi_{1}$ and $\Psi_{2}$),
we conclude that there are likely to be only finitely many eigenmodes that can
resonate with this pair (indeed, that number might fairly small, or even
zero). Second, when modes 1 and 2 coincide (a 1:2 resonance), one deduces that
$\Omega_{1}=\Omega_{2}$ and $K_{1}=K_{2}$; as such, the existence bounds (9)
simplify to $2K_{1}<K_{3}<4K_{1}$, or $2<K_{3}/K_{1}<4$ [35, 45].

###  3.2 Determining the critical depth

Although Theorem 1 determines necessary and sufficient conditions on the
wavenumbers, $K_{j}$, for there to be a critical depth, $h_{c}$, at which a
resonant triad exists, the critical depth remains to be determined. In
general, the critical depth must be computed numerically (being the unique
root of the nonlinear function $F(h)$); however, we demonstrate that useful
quantitative and qualitative information may be obtained via asymptotic
analysis. For the remainder of this section, we consider the rescaled
wavenumbers, $\xi_{1}=K_{1}/K_{3}$ and $\xi_{2}=K_{2}/K_{3}$, and the rescaled
depth, $\zeta=K_{3}h$; it remains to determine the root, $\zeta_{c}$, of

$F(\zeta)=\sqrt{\frac{\xi_{1}\tanh(\xi_{1}\zeta)}{\tanh(\zeta)}}+\sqrt{\frac{\xi_{2}\tanh(\xi_{2}\zeta)}{\tanh(\zeta)}}-1$
(10)

when $\xi_{1},\xi_{2}>0$ satisfy

$\xi_{1}+\xi_{2}<1<\sqrt{\xi_{1}}+\sqrt{\xi_{2}}.$ (11)

Figure 2:  Contours of the rescaled critical depth, $\zeta_{c}=h_{c}K_{3}$, as
a function of the rescaled wavenumbers, $\xi_{1}=K_{1}/K_{3}$ and
$\xi_{2}=K_{2}/K_{3}$. $(a)$ The contours computed numerically from equation
(10). The black lines indicate the limiting cases of $\zeta_{c}\rightarrow 0$
(at $\xi_{1}+\xi_{2}=1$) and $\zeta_{c}\rightarrow\infty$ (at
$\sqrt{\xi_{1}}+\sqrt{\xi_{2}}=1$). $(b)$ The contours are overlaid by the
leading-order approximation (equation (17); circles) and the higher-order
correction (equation (18); diamonds) for $\zeta_{c}$ equal to 0.5, 1, 1.5 and
2.

In figure 2$(a)$, we present contours of the critical rescaled depth,
$\zeta_{c}$, in the $(\xi_{1},\xi_{2})$-plane, restricted to the region
demarcated by equation (11). Consistent with the limits
$\lim_{\zeta\rightarrow 0}F(\zeta)=\xi_{1}+\xi_{2}-1$ and
$\lim_{\zeta\rightarrow\infty}F(\zeta)=\sqrt{\xi_{1}}+\sqrt{\xi_{2}}-1$, we
observe that the root, $\zeta_{c}$, tends to zero at the line
$\xi_{1}+\xi_{2}=1$, and approaches infinity at the curve
$\sqrt{\xi_{1}}+\sqrt{\xi_{2}}=1$. Furthermore, the uniqueness of the root of
$F$ for given $(\xi_{1},\xi_{2})$ is reflected in the observation that the
contours of $\zeta_{c}$ do not cross. Finally, we note that the contours are
symmetric about the line $\xi_{1}=\xi_{2}$, which is a direct consequence of
the invariance of $F(\zeta)$ under the mapping $\xi_{1}\leftrightarrow\xi_{2}$
(see equation (10)).

Although we are primarily interested in the physically relevant case for which
the cylinder’s depth-to-width ratio, $h$, is of size $O(1)$, an informative
analytic result may be obtained by considering $F(\zeta)$ in the limit
$\zeta\ll 1$ (or $K_{3}h\ll 1$). By utilising the Taylor expansion

$\sqrt{\tanh(x)}\sim\sqrt{x}\bigg{(}1-\frac{x^{2}}{6}+\frac{19}{360}x^{4}+O\big{(}x^{6}\big{)}\bigg{)},$

we obtain

$\sqrt{\xi_{1}\tanh(\xi_{1}\zeta)}+\sqrt{\xi_{2}\tanh(\xi_{2}\zeta)}-\sqrt{\tanh(\zeta)}\sim\sqrt{\zeta}\bigg{[}\Big{(}\xi_{1}+\xi_{2}-1\Big{)}-\frac{\zeta^{2}}{6}\Big{(}\xi_{1}^{3}+\xi_{2}^{3}-1\Big{)}+O(\zeta^{4})\bigg{]}$
(12)

for $0<\zeta\ll 1$. Whilst deriving equation (12), we have utilised the bound
$\xi_{1},\xi_{2}<1$ (see equation (11)), which additionally ensures that
$0<\xi_{j}\zeta\ll 1$ for $j=1,2$. We note that the left-hand side of equation
(12) is equal to $F(\zeta)\tanh(\zeta)$, so $\zeta_{c}$ satisfies

$\xi_{1}+\xi_{2}-1-\frac{\zeta_{c}^{2}}{6}\Big{(}\xi_{1}^{3}+\xi_{2}^{3}-1\Big{)}=O(\zeta_{c}^{4}),$
(13)

provided that $0<\zeta_{c}\ll 1$. By neglecting terms of size
$O(\zeta_{c}^{4})$ in equation (13), one may then easily solve for $\zeta_{c}$
in terms of $\xi_{1}$ and $\xi_{2}$.

Alternatively, a more succinct expression for $\zeta_{c}$ may be found by
first noting that

$\xi_{1}^{3}+\xi_{2}^{3}=(\xi_{1}+\xi_{2})^{3}-3\xi_{1}\xi_{2}(\xi_{1}+\xi_{2})=1-3\xi_{1}\xi_{2}+O(\zeta_{c}^{2}),$
(14)

where we have utilised the leading-order approximation
$\xi_{1}+\xi_{2}=1+O(\zeta_{c}^{2})$ (see equation (13)) to determine the
second equality. Upon substituting equation (14) into equation (13), we find
that $\xi_{1}$, $\xi_{2}$ and $\zeta_{c}$ are now related by the notably
simpler expression

$\xi_{1}+\xi_{2}-1+\frac{\zeta_{c}^{2}}{2}\xi_{1}\xi_{2}=O(\zeta_{c}^{4}).$
(15)

By neglecting terms of $O(\zeta_{c}^{4})$, the leading-order approximation for
the rescaled critical depth, $\zeta_{c}$, is given by

$\zeta_{c}\sim\sqrt{\frac{2(1-\xi_{1}-\xi_{2})}{\xi_{1}\xi_{2}}},$ (16)

an expression valid when $0<\zeta_{c}\ll 1$ and $\xi_{1}+\xi_{2}<1$ (see
equation (11)). Alternatively, one may deduce from equation (15) that the
contours of $\zeta_{c}$ satisfy the approximate form

$\xi_{2}\sim\frac{1-\xi_{1}}{1+\frac{1}{2}\zeta_{c}^{2}\xi_{1}},$ (17)

where the term in the denominator is responsible for the increased ‘bending’
of the contours as $\zeta_{c}$ becomes progressively larger (see figure 2). We
note that the additional simplification afforded by equation (14) allows for a
far more tractable representation of the contours relative to solving equation
(13) directly for $\xi_{2}$ given $\xi_{1}$ and $\zeta_{c}$.

Despite being derived under the assumption $0<\zeta_{c}\ll 1$, we see in
figure 2$(b)$ that the contours given by equation (17) agree favorably with
the numerical solution even up to $\zeta_{c}\approx 1$. However, it is readily
verified from equation (16) that the asymptotic approximation of each contour
crosses the boundary curve $\sqrt{\xi_{1}}+\sqrt{\xi_{2}}=1$ at $\zeta_{c}=4$
(for which $\xi_{1}=\xi_{2}=\frac{1}{4}$), thereby demonstrating that the
reduced asymptotic form has limited applicability (even in a qualitative
sense) for slightly larger values of $\zeta_{c}$. One may further improve the
quantitative (and, to an extent, qualitative) agreement between the asymptotic
analysis and numerical computation by including terms of size $O(\zeta^{4})$
in equation (12); indeed, an analogous calculation gives rise to the following
higher-order correction to equation (15):

$\xi_{1}+\xi_{2}-1+\frac{\zeta_{c}^{2}}{2}\xi_{1}\xi_{2}+\frac{\zeta_{c}^{4}}{72}\xi_{1}\xi_{2}\big{(}\xi_{1}\xi_{2}-1\big{)}=O(\zeta_{c}^{6}).$
(18)

Although one may then solve for $\zeta_{c}$ given $\xi_{1}$ and $\xi_{2}$ (or,
alternatively, determine the contours of $\zeta_{c}$) by truncating terms of
$O(\zeta_{c}^{6})$ in equation (18), the resulting algebraic expressions yield
little qualitative information. However, one may, in principle, use this
reduced form as a reasonable initial guess for a numerical root-finding
algorithm for determining the root of $F(\zeta)$, provided that $\zeta_{c}$ is
not too large.

###  3.3 Example cavities

Our investigation into the emergence of resonant triads has been focused, thus
far, on finite-depth cylinders with arbitrary horizontal cross-section.
However, it is convenient to understand how the results of Theorem 1 influence
the formation (or not) of resonant triads for some specific cross-sections,
namely rectangular, circular, and annular cylinders.

####  3.3.1 Rectangular cylinder

It is well known that resonant triads are impossible for plane gravity waves
evolving across an unbounded horizontal domain of finite depth [49, 22].
111Weak interactions are possible, however, in the shallow-water limit,
$K_{j}h\rightarrow 0$, for which $\tanh(K_{j}h)$ in the dispersion relation
(5) is replaced by its leading-order approximation, $K_{j}h$ [49, 8, 43]. We
now utilise Theorem 1 to demonstrate a similar result: resonant triads are
impossible for gravity waves evolving within a rectangular cylinder of finite
depth. Our result generalises the special case of a 1:2 resonance, for which
the impossibility of internal resonance in a rectangular cylinder was
demonstrated by Miles [43].

To proceed, we consider a rectangular cylinder with side lengths $L_{x}$ and
$L_{y}$. By orientating the Cartesian coordinate system, $\bm{x}=(x,y)$, so
that the cylinder cross-section is defined by the region $0<x<L_{x}$ and
$0<y<L_{y}$, the eigenmodes are of the form

$\Phi_{mn}(x,y)=\frac{1}{\mathscr{N}_{mn}}\cos(p_{m}x)\cos(q_{n}y),$

where $\mathscr{N}_{mn}>0$ is a normalisation constant. Notably, the
wavenumbers $p_{m}=m\pi/L_{x}$ and $q_{n}=n\pi/L_{y}$ are chosen so that the
no-flux condition is satisfied (see equation (1d)). For a triad determined by
the non-negative integers $m_{j}$ and $n_{j}$ (for $j=1,2,3$), the
corresponding wavenumbers, $P_{j}=p_{m_{j}}$ and $Q_{j}=q_{n_{j}}$, must
satisfy $P_{1}+P_{2}=P_{3}$ and $Q_{1}+Q_{2}=Q_{3}$ (under suitable reordering
of the subscripts) in order for the eigenmode correlation condition (8) to be
satisfied. By defining the wave vector $\bm{k}_{j}=(P_{j},Q_{j})$, the
conditions on $P_{j}$ and $Q_{j}$ simplify to the single requirement
$\bm{k}_{1}+\bm{k}_{2}=\bm{k}_{3}$, where the triangle inequality supplies
that $|\bm{k}_{3}|\leq|\bm{k}_{1}|+|\bm{k}_{2}|$. As the eigenvalues,
$K_{j}^{2}$, of the negative Laplacian operator are related to the wave
vectors via $K_{j}=|\bm{k}_{j}|$, we deduce that $K_{3}\leq K_{1}+K_{2}$.
Owing to the violation of the left-hand bound in equation (9), we conclude
that resonant triads _cannot_ exist in a rectangular cylinder of finite depth.

####  3.3.2 Circular cylinder

We consider a circular cylinder of unit radius in dimensionless variables
(i.e. the dimensional radius is equal to $a$; see §2). For polar coordinates
$\bm{x}=(r,\theta)$, it is well known that the corresponding (complex-valued)
eigenmodes may be expressed in the form

$\Phi_{mn}(r,\theta)=\frac{1}{\mathscr{N}_{mn}}\mathrm{J}_{m}(k_{mn}r)\mathrm{e}^{\mathrm{i}m\theta},\quad\mathrm{where}\quad\mathscr{N}_{mn}=\big{|}\mathrm{J}_{m}(k_{mn})\big{|}\sqrt{1-\frac{m^{2}}{k_{mn}^{2}}}$
(19)

is the normalisation factor and $m$ is the azimuthal wavenumber (an integer).
Furthermore, the no-flux condition (1d) determines that the radial
wavenumbers, denoted $k_{mn}$, satisfy $\mathrm{J}_{m}^{\prime}(k_{mn})=0$,
where $0<k_{m1}<k_{m2}<\ldots$ (we exclude $k_{00}=0$ from consideration; see
§3). Notably, the eigenvalues of the negative Laplacian operator are
_precisely_ the squared wavenumbers, $k_{mn}^{2}$; consequently, the antinodes
of each Bessel function play a pivotal role in determining the existence of
resonant triads.

Akin to the rectangular cylinder, we find that the eigenmode correlation
condition imparts an important restriction on the combination of eigenmodes
that may resonate. For given $m_{j}$ and $n_{j}$ (for $j=1,2,3$), we denote
$K_{j}=k_{m_{j}n_{j}}$, $\Psi_{j}=\Phi_{m_{j}n_{j}}$ and
$N_{j}=\mathscr{N}_{m_{j}n_{j}}$. Although the correlation condition given in
equation (8) is defined for real eigenmodes, a similar condition holds for
complex-valued eigenmodes, namely
$\iint_{\mathcal{D}}\Psi_{1}\Psi_{2}\Psi_{3}^{*}\,\mathrm{d}A\neq 0$. By
considering the quantity

$\iint_{\mathcal{D}}\Psi_{1}\Psi_{2}\Psi_{3}^{*}\,\mathrm{d}A=\frac{1}{N_{1}N_{2}N_{3}}\bigg{(}\int_{0}^{1}r\mathrm{J}_{m_{1}}(K_{1}r)\mathrm{J}_{m_{2}}(K_{2}r)\mathrm{J}_{m_{3}}(K_{3}r)\,\mathrm{d}r\bigg{)}\bigg{(}\int_{0}^{2\pi}\mathrm{e}^{\mathrm{i}(m_{1}+m_{2}-m_{3})\theta}\,\mathrm{d}\theta\bigg{)},$

we deduce from the azimuthal integral that a necessary condition for the
correlation integral to be nonzero is $m_{1}+m_{2}=m_{3}$ [41]. This condition
thus restricts the permissible combinations of azimuthal wavenumbers in a
manner similar to the restriction on the permissible planar wavenumbers for
the case of a rectangular cylinder. Unlike rectangular cylinders, however, we
demonstrate that resonant triads _are_ possible in a circular cylinder.

No. | $m_{1}$ | $m_{2}$ | $m_{3}$ | $n_{1}$ | $n_{2}$ | $n_{3}$ | $K_{1}$ | $K_{2}$ | $K_{3}$ | $\overline{K}$ | $h_{c}$ | $\iint_{\mathcal{D}}\Psi_{1}\Psi_{2}\Psi_{3}^{*}\,\mathrm{d}A$
---|---|---|---|---|---|---|---|---|---|---|---|---
1 | -1 | 1 | 0 | 1 | 1 | 2 | 1.841 | 1.841 | 7.016 | 3.566 | 1.00970 | -0.02032
2 | 1 | 1 | 2 | 1 | 1 | 2 | 1.841 | 1.841 | 6.706 | 3.463 | 0.83138 | 0.02801
3 | -1 | 2 | 1 | 1 | 1 | 3 | 1.841 | 3.054 | 8.536 | 4.477 | 0.60375 | -0.02595
4 | 1 | 2 | 3 | 1 | 1 | 2 | 1.841 | 3.054 | 8.015 | 4.304 | 0.50595 | 0.03712
5 | -1 | 3 | 2 | 1 | 1 | 3 | 1.841 | 4.201 | 9.969 | 5.337 | 0.48152 | -0.02717
6 | -2 | 2 | 0 | 1 | 1 | 3 | 3.054 | 3.054 | 10.173 | 5.427 | 0.39129 | -0.03050
7 | 0 | 1 | 1 | 1 | 1 | 3 | 3.832 | 1.841 | 8.536 | 4.736 | 0.38516 | 0.00542
8 | -1 | 1 | 0 | 1 | 2 | 3 | 1.841 | 5.331 | 10.173 | 5.782 | 0.30197 | -0.00603
9 | 1 | 1 | 2 | 1 | 2 | 3 | 1.841 | 5.331 | 9.969 | 5.714 | 0.28691 | 0.01818
10 | 0 | 2 | 2 | 1 | 1 | 3 | 3.832 | 3.054 | 9.969 | 5.619 | 0.26387 | -0.00087
11 | 1 | 2 | 3 | 1 | 2 | 3 | 1.841 | 6.706 | 11.346 | 6.631 | 0.23678 | 0.02590
12 | 0 | 3 | 3 | 1 | 1 | 3 | 3.832 | 4.201 | 11.346 | 6.460 | 0.21395 | -0.00640
13 | 1 | 2 | 3 | 2 | 1 | 3 | 5.331 | 3.054 | 11.346 | 6.577 | 0.19839 | 0.01522
14 | 0 | 0 | 0 | 1 | 1 | 3 | 3.832 | 3.832 | 10.173 | 5.946 | 0.19814 | 0.03327
15 | -1 | 2 | 1 | 1 | 1 | 2 | 1.841 | 3.054 | 5.331 | 3.409 | 0.17266 | 0.85581
16 | -2 | 2 | 0 | 1 | 1 | 2 | 3.054 | 3.054 | 7.016 | 4.375 | 0.17030 | 0.46429
17 | -1 | 3 | 2 | 1 | 1 | 2 | 1.841 | 4.201 | 6.706 | 4.250 | 0.16313 | 0.64211
18 | -2 | 3 | 1 | 1 | 1 | 3 | 3.054 | 4.201 | 8.536 | 5.264 | 0.15767 | 0.30704
19 | -1 | 1 | 0 | 1 | 1 | 1 | 1.841 | 1.841 | 3.832 | 2.505 | 0.15227 | 1.28795
20 | -3 | 3 | 0 | 1 | 1 | 3 | 4.201 | 4.201 | 10.173 | 6.192 | 0.14591 | 0.19061
21 | -2 | 2 | 0 | 1 | 2 | 3 | 3.054 | 6.706 | 10.173 | 6.645 | 0.06331 | 0.68257
22 | -1 | 3 | 2 | 2 | 1 | 3 | 5.331 | 4.201 | 9.969 | 6.501 | 0.06286 | 0.66930
23 | -1 | 2 | 1 | 2 | 1 | 3 | 5.331 | 3.054 | 8.536 | 5.641 | 0.04664 | 0.99088
24 | -1 | 3 | 2 | 1 | 2 | 3 | 1.841 | 8.015 | 9.969 | 6.609 | 0.03928 | 1.08903
25 | 0 | 3 | 3 | 2 | 1 | 3 | 7.016 | 4.201 | 11.346 | 7.521 | 0.02782 | 1.00669
Table 1:  Combinations of the azimuthal wavenumbers, $m_{j}$, and radial mode
indices, $n_{j}$, that form a resonant triad ($m_{1}+m_{2}=m_{3}$ and
$\Omega_{1}+\Omega_{2}=\Omega_{3}$) at critical depth, $h_{c}$, in a circular
cylinder of unit radius. For each triad, the corresponding wavenumbers,
$K_{j}=k_{m_{j}n_{j}}$, satisfy (9), and the correlation condition,
$\iint_{\mathcal{D}}\Psi_{1}\Psi_{2}\Psi_{3}^{*}\,\mathrm{d}A\neq 0$, is met.
The list is restricted to resonant triads arising for $|m_{j}|,n_{j}\leq 3$,
and we consider $m_{1}\leq m_{2}$ and $m_{3}\geq 0$ without loss of
generality. We have omitted resonances that give rise to the same critical
depth, but with the roles of modes 1 and 2 swapped. The triad numbers (left
column) and shaded rows are referenced in the text.

Despite the apparent restriction of the Bessel antinodes, $K_{j}$, and
summation condition on the azimuthal wavenumbers, $m_{j}$, Theorem 1
determines that a vast array of resonant triads may be excited for judicious
choices of the fluid depth. In table 1, we list a small number of resonant
triads and each corresponding critical depth, $h_{c}$, subject to the
restrictions $|m_{j}|\leq 3$ and $n_{j}\leq 3$; for larger values of $|m_{j}|$
and $n_{j}$, the corresponding wave field becomes increasingly oscillatory, to
the extent that the effects of surface tension and dissipation might become
appreciable. Moreover, even marginally relaxing the upper bounds on $|m_{j}|$
and $n_{j}$ vastly increases the number of resonant triads; indeed, the
restriction $|m_{j}|\leq 4$ and $n_{j}\leq 4$ introduces 70 additional
resonant triads relative to table 1. As the upper bounds for $|m_{j}|$ and
$n_{j}$ are further increased, the typical difference between the various
critical depths decreases and an increasingly large number of triads form at
small values of the critical depth. Triads forming in shallow fluids (e.g.
triads 21 to 25 in table 1) have physical relevance only at larger length
scales (e.g. lakes) as dissipation could become a dominant factor at smaller
scales.

Although the list of triads in table 1 is restricted to the lowest radial and
azimuthal modes, we observe some general trends. In particular, we observe
that the correlation integral,
$\iint_{\mathcal{D}}\Psi_{1}\Psi_{2}\Psi_{3}^{*}\,\mathrm{d}A$, generally
decreases in magnitude as the fluid depth increases. Although the correlation
integral remains non-zero (as is necessary to satisfy the correlation
condition), its small value in some cases (e.g. triad 10) potentially
corresponds to an elongation of the triad evolution time-scale (see §4.1).
Moreover, we observe that the average wavenumber involved in the triad,
$\overline{K}=\frac{1}{3}(K_{1}+K_{2}+K_{3})$, is appreciably larger when the
critical depth is very small. This correlation is consistent with the form of
the corresponding angular frequency,
$\Omega_{j}=\sqrt{K_{j}\tanh(K_{j}h_{c})}$, for which a small critical depth,
$h_{c}$, is necessary for finite-depth effects to be appreciable when the
typical wavenumber is large. To enumerate the myriad resonant triads arising
in a circular cylinder when the upper bounds on $|m_{j}|$ and $n_{j}$ are
relaxed, we provide MATLAB code in the supplementary material.

At this juncture, it is informative to assess how the triads listed in table 1
relate to the resonances explored in prior investigations. First, triad 8 in
table 1 (dark grey row) was explored by Michel [41] for a circular cylinder of
radius 9.45 cm and an approximate fluid depth of 3 cm; it follows that the
depth-to-radius ratio in Michel’s experiment was approximately 0.317, close to
the value of 0.30197 reported in table 1. Furthermore, table 1 (grey rows)
incorporates two well-known examples of a 1:2 resonance, for which modes 1 and
2 coincide: (i) the critical depth $h_{c}=0.83138$ (triad 2) corresponds to
the second-harmonic resonance with the fundamental mode [43, 44, 9, 65]; (ii)
the critical depth $h_{c}=0.19814$ (triad 14) corresponds to a standing wave
composed of two resonant axisymmetric modes [35, 65]. Finally, triads 1, 6,
16, 19 and 20 (table 1, light grey rows) form an interesting class of resonant
triad, for which an axisymmetric mode ($m_{3}=0$) interacts with two identical
counter-propagating non-axisymmetric modes ($m_{1}=-m_{2}\neq 0$ and
$n_{1}=n_{2}$). In fact, our investigation in §5.1 demonstrates that the
axisymmetric mode is the so-called _pump_ mode, and may thus excite the non-
axisymmetric modes, even when the initial energy in each non-axisymmetric mode
is negligible. We draw an analogy between this novel class of resonant triad
and the excitation of beach edge waves [19] in §6.

We conclude our exploration of resonant triads arising in a circular cylinder
by remarking that the fluid depth may, in some cases, be judiciously chosen so
as to excite multiple triads. In general, the condition on the angular
frequencies, $\Omega_{1}+\Omega_{2}=\Omega_{3}$ (see equation (7)), cannot be
satisfied for two distinct triads at the same fluid depth; however, nonlinear
resonance may persist for both triads provided that each condition on the
angular frequencies is _approximately_ satisfied [6, 40, 16], at the cost of
weak detuning (see §4.3.1 for further details). Specifically, if triads 1 and
2 have critical depths $h_{c,1}$ and $h_{c,2}$, respectively, then there is
potential excitement of both triads when the fluid depth, $h$, satisfies
$|h-h_{c,j}|=O(\epsilon)$ for $j=1,2$ (where $0<\epsilon\ll 1$ is the typical
wave slope; see §2), giving rise to the approximation
$\Omega_{1}+\Omega_{2}-\Omega_{3}=O(\epsilon)$ for each triad. For example, if
$0<h_{c,2}-h_{c,1}\ll 1$, then it may be sufficient to excite both triads at
an intermediate depth, $h_{c,1}\leq h\leq h_{c,2}$. We note, however, that the
excitation of multiple triads at a single fluid depth is not possible when the
depth discrepancy, $|h-h_{c,j}|$, becomes too large (relative to the typical
wave slope) for any of the triads under consideration.

To demonstrate the potential for the simultaneous excitation of two triads
within a circular cylinder of finite depth, we consider two scenarios: (i) the
excitation of two triads that share a common wave mode; and (ii) the
excitation of two triads that do not share any common wave modes.
Heuristically, case (ii) is more common than case (i) owing to the number of
similar fluid depths in table 1; however, case (i) will likely generate a far
richer set of dynamics owing to the nonlinear interaction between the two
triads [36, 16, 14, 12]. As an example of case (i), we consider triads 11 and
12 in table 1, with nearby critical depths $h_{c,1}=0.23678$ and
$h_{c,2}=0.21395$, respectively. As mode $(m_{3},n_{3})=(3,3)$ is common to
both triads, inter-triad resonance may arise at an intermediate depth, e.g.
$h=0.225$. Furthermore, an example of case (ii) arises for triads 13 and 14 in
table 1, with nearby critical depths $h_{c,1}=0.19839$ and $h_{c,2}=0.19814$.
Neither of these triads share a common wave mode, so one would _not_ expect
the inter-triad energy exchange discussed in case (i). Nevertheless, one might
anticipate a signature of these two triads to be visible in the surface
evolution for an intermediate depth, e.g. $h=0.19825$. The theoretical and
numerical exploration of coupled triads in a circular cylinder will be the
focus of future investigation.

####  3.3.3 Annular cylinder

A natural variation upon a circular cylinder is an annulus of inner radius
$r_{0}\in(0,1)$ and outer radius 1. By varying $r_{0}$, the annulus approaches
a circular cylinder as $r_{0}\rightarrow 0^{+}$, and a quasi-one-dimensional
periodic ring as $r_{0}\rightarrow 1^{-}$. Notably, resonant triads are
impossible for a one-dimensional periodic ring, as can be shown by modifying
the arguments presented for the case of a rectangular cylinder (see §3.3.1).
Thus, one might anticipate that the existence of triads in an annular cylinder
depends critically on the inner radius, $r_{0}$. Rather than enumerating some
possible triads for given values of $r_{0}$, we instead track the
corresponding critical depth, $h_{c}$, for the triads identified for a
circular cylinder (see table 1) as $r_{0}$ is progressively increased from
zero. Of particular interest is determining whether a given triad exists for
all $r_{0}<1$, or whether there is some critical inner radius, $r_{c}$, beyond
which the triad ceases to exist, with either $h_{c}\rightarrow 0$ or
$h_{c}\rightarrow\infty$ as $r_{0}\rightarrow r_{c}^{-}$.

The (complex-valued) eigenmodes in an annular domain are cylinder functions of
the form

$\Phi_{mn}(r,\theta)=\frac{1}{\mathscr{N}_{mn}}\bigg{[}\mathrm{J}_{m}(k_{mn}r)\cos(\gamma_{mn}\pi)+\mathrm{Y}_{m}(k_{mn}r)\sin(\gamma_{mn}\pi)\bigg{]}\mathrm{e}^{\mathrm{i}m\theta},$
(20)

where $\mathscr{N}_{mn}>0$ is a normalisation constant, $\mathrm{Y}_{m}$ is
the Bessel function of the second kind with order $m$ (an integer), and
$\gamma_{mn}\in[0,1]$ determines the weighting between the two Bessel
functions. As shown in appendix B, the no-flux condition (see equation (1d))
on the inner and outer walls determines that the wavenumbers, $k_{mn}(r_{0})$,
satisfy the equation

$\mathrm{J}_{m}^{\prime}(k_{mn}r_{0})\mathrm{Y}_{m}^{\prime}(k_{mn})-\mathrm{J}_{m}^{\prime}(k_{mn})\mathrm{Y}_{m}^{\prime}(k_{mn}r_{0})=0.$
(21)

A formula for the corresponding value of $\gamma_{mn}$ is determined in
appendix B. Once again, the wavenumbers, $k_{mn}$, are ordered so that
$0<k_{m1}<k_{m2}<\ldots$ (excluding $k_{00}=0$) and satisfy
$-\Delta\Phi_{mn}=k_{mn}^{2}\Phi_{mn}$. Three correlated wave modes may form a
resonant triad (for a judicious choice of the fluid depth) provided that the
corresponding wavenumbers, $K_{j}$, which depend on the channel width,
$1-r_{0}$, satisfy the bounds given in Theorem 1.

Figure 3:  The existence and predominant characteristics of a triad in an
annular cylinder with inner radius $r_{0}$ and outer radius 1. The triad
bifurcates from the critical depth $h_{c}=0.17266$ as $r_{0}\rightarrow 0$
(the limiting case of a circular cylinder), with corresponding wavenumbers
presented in table 1 (see triad 15). $(a)$ The critical depth, $h_{c}$ (blue
curve), with $h_{c}\rightarrow\infty$ as $r_{0}\rightarrow r_{c}^{-}$, where
$r_{c}\approx 0.57$ (black line). $(b)$ The corresponding wavenumbers,
$K_{j}$, all of which remain finite for $r_{0}<r_{c}$ (black line). $(c)$ The
normalised wavenumbers, $K_{1}/K_{3}$ and $K_{2}/K_{3}$, parametrised by
increasing $r_{0}$ (blue arrow), with the limiting case $r_{0}\rightarrow 0$
denoted by the white dot. The wavenumbers leave the triad existence region
(see Theorem 1) via the left-hand boundary (black curve) as $r_{0}\rightarrow
r_{c}^{-}$.

Bifurcating from the limiting case of a circular cylinder, we track the
critical depth (when such a depth exists) of different triads as $r_{0}$ is
progressively increased. The predominant behaviour is characterised by the
example presented in figure 3, for which we consider the triad whose critical
depth is $h_{c}=0.17266$ as $r_{0}\rightarrow 0^{+}$ (see triad 15 in table
1). Given that $h_{c}$ is fairly small in this limit, one might anticipate
that the triad ceases to exist with $h_{c}\rightarrow 0$; somewhat
surprisingly, however, the opposite scenario arises, with
$h_{c}\rightarrow\infty$ as $r_{0}\rightarrow r_{c}^{-}$ ($r_{c}\approx 0.57$
in this example). It follows, therefore, that the triad may persist for narrow
channels only when the fluid is sufficiently deep. We note, however, that
there exist (at least) two relatively rare transitions for increasing $r_{0}$,
which we briefly describe as follows: (i) the triad ceases to exist when
$h_{c}\rightarrow 0$ as $r_{0}\rightarrow r_{c}^{-}$, which may arise when
bifurcating from a sufficiently shallow circular cylinder (e.g. triad 25 in
table 1); and (ii) the triad continues to exist for all $r_{0}<1$, with
$h_{c}\rightarrow 0$ and $K_{j}\rightarrow\infty$ as $r_{0}\rightarrow 1$, yet
the normalised depth, $h_{c}K_{3}$, remains finite, and the normalised
wavenumbers, $K_{1}/K_{3}$ and $K_{2}/K_{3}$, remain within the triad
existence region (e.g. triad 14 in table 1). Owing to the appreciable
influence of viscous effects for relatively shallow fluids, the physical
relevance of these latter two scenarios is somewhat nebulous, however.

##  4 The evolution of resonant triads

Having established the existence of resonant triads, we now determine the
long-time triad evolution, utilising the method of multiple scales.
Ostensibly, the calculations necessary for determining the triad equations are
a variation upon the pioneering work of McGoldrick [37, 38, 39] in the absence
of surface tension. However, the confinement of the fluid to a cylinder
imposes some additional considerations, the salient details of which we
outline below. Finally, we note that an alternative approach to multiple
scales is Whitham’s technique of averaging the system’s Lagrangian [60, 61,
62, 63], which has the advantage of streamlining some algebraic calculations
[55, 43, 44]; nevertheless, multiple-scales analysis is sufficient for our
purposes and allows for the possible inclusion of higher-order corrections in
the asymptotic expansion [39].

In a manner similar to §3, we consider three linear wave modes (with real-
valued eigenfunctions), enumerated $n_{1}$, $n_{2}$ and $n_{3}$, where we
denote

$\Omega_{j}=\omega_{n_{j}},\quad K_{j}=k_{n_{j}},\quad
L_{j}=\hat{\mathscr{L}}_{n_{j}},\quad\mathrm{and}\quad\Psi_{j}(\bm{x})=\Phi_{n_{j}}(\bm{x})\quad\mathrm{for}\,\,\,j=1,2,3.$

In contrast to §3, however, we now allow each (nonzero) angular frequency to
be either negative or positive: the resonance condition on the angular
frequencies is henceforth defined

$\Omega_{1}+\Omega_{2}+\Omega_{3}=0.$ (22)

The modified requirement on the angular frequencies (equation (22)) is not
restrictive on the possible triad combinations; one may recover equation (7)
by mapping $\Omega_{3}\mapsto-\Omega_{3}$, for example. The decision behind
the summation condition on the angular frequencies is motivated by the
cyclical symmetry of equation (22), a property that will be inherited by the
resultant amplitude equations [55]. As a consequence, one need only derive the
amplitude equation for one of the wave modes; the amplitude equations for the
remaining two wave modes follow by cyclic permutation of the subscripts
$(1,2,3)$.

Before embarking on the multiple-scales analysis presented in §4.1, we remark
upon two caveats. First, we note that equation (22) corresponds to an exact
resonance, for which the fluid depth, $h$, is chosen to be precisely equal to
the critical depth, $h_{c}$. In practice, however, there may be a small
discrepancy between $h$ and $h_{c}$, resulting in a the sum of the angular
frequencies being slightly offset from zero. When the frequency detuning is
sufficiently weak, e.g. $\Omega_{1}+\Omega_{2}+\Omega_{3}=O(\epsilon)$, one
may modify the following asymptotic analysis to derive a similar set of
amplitude equations (see §4.3.1). Second, our analysis in §4.1 is not valid
when two of the wave modes coincide. This case corresponds to a 1:2 resonance,
for which the corresponding evolution equations were derived by Miles [43]
using Whitham modulation theory (as summarised in §4.3.2).

###  4.1 Multiple-scales analysis

In order to determine the evolution of each of the three dominant wave modes
involved in an exact resonance, we utilise the method of multiple scales [29,
56]. Specifically, we seek a perturbation solution to the Benney-Luke equation
(3) of the form $u\sim u_{0}+\epsilon u_{1}+O(\epsilon^{2})$. The leading-
order terms in equation (3) determine that $u_{0}$ satisfies
$\partial_{tt}u_{0}+\mathscr{L}u_{0}=0$; we _choose_ to consider a leading-
order solution comprised only of the three triad modes (all other modes are
assumed to be smaller in magnitude and appear at higher order), giving rise to
the leading-order form

$u_{0}(\bm{x},t,\tau)=\sum_{j=1}^{3}\Big{[}A_{j}(\tau)\Psi_{j}(\bm{x})\mathrm{e}^{-\mathrm{i}\Omega_{j}t}+\mathrm{c.c.}\Big{]}.$
(23)

In equation (23), we have introduced the slow time-scale $\tau=\epsilon t$,
which governs the evolution of each complex amplitude, $A_{j}$. As $\epsilon$
and $t$ are both independent variables, we treat $\tau$ and $t$ as independent
time-scales, giving rise to the transformation of derivatives
$\partial_{t}\mapsto\partial_{t}+\epsilon\partial_{\tau}$. Finally,
$\mathrm{c.c.}$ denotes the complex conjugate of the preceding term, a
contribution necessary for real $u_{0}$.

So as to determine coupled evolution equations for each complex amplitude,
$A_{j}$, we consider terms of $O(\epsilon)$ in the Benney-Luke equation (3).
By substituting the leading-order solution, $u_{0}$, into the nonlinear terms
and applying the triad condition for the angular frequencies (equation (22)),
we obtain the following problem for $u_{1}$:

$\partial_{tt}u_{1}+\mathscr{L}u_{1}=-\bigg{[}\sum_{j=1}^{3}f_{j}(\bm{x},\tau)\mathrm{e}^{-\mathrm{i}\Omega_{j}t}+\mathrm{c.c.}\bigg{]}+\mbox{nonresonant
terms}.$ (24)

As we will see below, each of the functions $f_{j}(\bm{x},\tau)$ appearing on
the right-hand side of equation (24) will play a fundamental role when
determining the amplitude equations; specifically,

$f_{1}=-2\mathrm{i}\Omega_{1}\frac{\mathrm{d}{A_{1}}}{\mathrm{d}{\tau}}\Psi_{1}+\mathrm{i}A_{2}^{*}A_{3}^{*}\bigg{(}\Big{[}\Omega_{2}\big{(}L_{3}^{2}-K_{3}^{2}\big{)}+\Omega_{3}\big{(}L_{2}^{2}-K_{2}^{2}\big{)}-2\Omega_{1}L_{2}L_{3}\Big{]}\Psi_{2}\Psi_{3}-2\Omega_{1}\nabla\Psi_{2}\cdot\nabla\Psi_{3}\bigg{)},$

where $f_{2}$ and $f_{3}$ follow upon cyclic permutation of the subscripts
$(1,2,3)$. Finally, we note that the ‘nonresonant terms’ in equation (24) are
of the general form $p(\bm{x},\tau)\mathrm{e}^{\mathrm{i}\varsigma t}$, where
we assume that the angular frequency, $\varsigma$, is not equal (or close) to
any of the angular frequencies, $\pm\omega_{n}$, associated with linear wave
modes (see §3).

We proceed by projecting equation (24) onto each of the three wave modes,
giving rise to differential equations of the form (for $j=1,2,3$)

$\partial_{tt}\hat{u}_{1,j}+L_{j}\hat{u}_{1,j}=-\Big{[}\langle\Psi_{j},f_{j}\rangle\mathrm{e}^{-\mathrm{i}\Omega_{j}t}+\mathrm{c.c.}\Big{]}+\mbox{nonresonant
terms},$ (25)

where $\hat{u}_{1,j}=\langle\Psi_{j},u_{1}\rangle$ is the projection of
$u_{1}$ onto the mode $\Psi_{j}$. By recalling that $L_{j}=\Omega_{j}^{2}$, we
immediately see that the term in square brackets in equation (25) is itself a
solution to the linear operator $\partial_{tt}+\Omega_{j}^{2}$. It follows
that the solution of equation (25) comprises of particular solutions that have
temporal dependence $t\mathrm{e}^{\pm\mathrm{i}\Omega_{j}t}$, leading to an
ill-posed asymptotic expansion when $\epsilon t=O(1)$. The resolution to this
problem is achieved via the solubility condition
$\langle\Psi_{j},f_{j}\rangle=0,$ which suppresses the secular growth.

By applying the solubility condition $\langle\Psi_{j},f_{j}\rangle=0$ for
$j=1,2,3$, we conclude that the complex amplitude, $A_{j}(\tau)$, of each wave
mode, $\Psi_{j}(\bm{x})\mathrm{e}^{-\mathrm{i}\Omega_{j}t}$, evolves according
to the triad system of canonical form [6, 16]

$\frac{\mathrm{d}{A_{1}}}{\mathrm{d}{\tau}}=\alpha_{1}A_{2}^{*}A_{3}^{*},\qquad\frac{\mathrm{d}{A_{2}}}{\mathrm{d}{\tau}}=\alpha_{2}A_{1}^{*}A_{3}^{*},\qquad\frac{\mathrm{d}{A_{3}}}{\mathrm{d}{\tau}}=\alpha_{3}A_{1}^{*}A_{2}^{*},$
(26)

where

$\alpha_{1}=\frac{1}{2\Omega_{1}}\bigg{(}\Big{[}\Omega_{2}\big{(}L_{3}^{2}-K_{3}^{2}\big{)}+\Omega_{3}\big{(}L_{2}^{2}-K_{2}^{2}\big{)}-2\Omega_{1}L_{2}L_{3}\Big{]}\mathscr{C}-2\Omega_{1}\big{\langle}\Psi_{1},\nabla\Psi_{2}\cdot\nabla\Psi_{3}\big{\rangle}\bigg{)},$
(27)

while $\alpha_{2}$ and $\alpha_{3}$ follow by cyclic coefficient of the
subscripts $(1,2,3)$. Furthermore, the correlation integral, $\mathscr{C}$, is
defined

$\mathscr{C}=\frac{1}{S}\iint_{\mathcal{D}}\Psi_{1}\Psi_{2}\Psi_{3}\,\mathrm{d}A,$
(28)

where we recall that $S$ is the area of the cylinder cross-section (see §2.2).
As the triad equations (26) are valid for $\tau=O(1)$ (or $t=O(1/\epsilon)$),
their dynamics yield an informative view of the long-time evolution of the
resonant triad.

In order to assess the influence of the triad coefficients on the triad
evolution (see §4.2), we first simplify the algebraic form given in equation
(27). As shown by Miles [43], one may simplify the inner product
$\langle\Psi_{1},\nabla\Psi_{2}\cdot\nabla\Psi_{3}\rangle$ by repeated
application of the divergence theorem and utilisation of the relationship
$-\Delta\Psi_{j}=K_{j}^{2}\Psi_{j}$; it follows that

$\big{\langle}\Psi_{1},\nabla\Psi_{2}\cdot\nabla\Psi_{3}\big{\rangle}=\frac{1}{2}\Big{(}K_{2}^{2}+K_{3}^{2}-K_{1}^{2}\Big{)}\mathscr{C},$
(29)

where $\mathscr{C}$ is the correlation integral defined in equation (28). We
then substitute equation (29) into equation (27) and simplify using the
relation $\Omega_{1}+\Omega_{2}+\Omega_{3}=0$. After some algebra, we derive
the reduced expression

$\alpha_{1}=\frac{\mathscr{C}}{2\Omega_{1}}\bigg{(}\Omega_{2}L_{3}^{2}+\Omega_{3}L_{2}^{2}-2\Omega_{1}L_{2}L_{3}+\sum_{l=1}^{3}\Omega_{l}K_{l}^{2}\bigg{)},$
(30)

where $\alpha_{2}$ and $\alpha_{3}$ follow similarly. Finally, we demonstrate
in appendix C that the algebraic form of the triad coefficients may be further
reduced to

$\alpha_{j}=\frac{\mathscr{C}\beta}{2\Omega_{j}}\quad\mathrm{for}\quad
j=1,2,3,$ (31)

where

$\beta=\sum_{l=1}^{3}\Omega_{l}K_{l}^{2}-\frac{1}{2}\Omega_{1}\Omega_{2}\Omega_{3}\big{(}\Omega_{1}^{2}+\Omega_{2}^{2}+\Omega_{3}^{2}\big{)}.$
(32)

Equations (26), (28), (31) and (32) constitute the triad equations for
resonant gravity waves confined to a cylinder of finite depth. Although the
triad equations are of canonical form [6], the novelty of our investigation is
the computation of the coefficients, $\alpha_{j}$, whose algebraic form is
specific to our system.

###  4.2 Properties of the triad coefficients

The simplified form of the coefficients, $\alpha_{j}$ (equation (31)), allows
for some important theoretical observations that were obfuscated by the more
complicated expressions for $\alpha_{j}$ given in equations (27) and (30). In
particular, as exactly two of the angular frequencies, $\Omega_{j}$, have the
same sign, we deduce from equation (31) that the two corresponding
coefficients, $\alpha_{j}$, also have the same sign, with the third
coefficient having the opposite sign. By utilising well-known results
pertaining to the canonical triad equations, we conclude that all solutions to
the triad equations (26) are periodic in time, with solutions expressible in
terms of elliptic functions [2, 6, 55, 16]. Typically, these solutions result
in an exchange of energy between the comprising modes, although there is a
class of periodic solution that, perhaps counter-intuitively, results in zero
energy exchange for all time [10, 11]. Moreover, it is readily verified that
the leading-order energy density,
$\mathscr{E}_{1}+\mathscr{E}_{2}+\mathscr{E}_{3}$, is conserved, where
$\mathscr{E}_{j}=\Omega_{j}^{2}|A_{j}|^{2}$, consistent with the Hamiltonian
structure of the Euler equations [6, 16]. The reader is directed to the work
of Craik [16] for a more detailed account of the various properties of the
canonical triad equations.

Figure 4:  Contours of $\beta/K_{3}^{2}$ (see equation (32)) for the case
$\Omega_{1},\Omega_{2}>0$ and $\Omega_{3}<0$ (with
$\Omega_{1}+\Omega_{2}+\Omega_{3}=0$), for which $\beta<0$ (see equation
(33)).

Of particular relevance to the evolution of the triad is the quantity $\beta$
(see equation (32)), which, together with $\mathscr{C}$, determines the time
scale over which energy exchange arises. In particular, we present the form of
$\beta$ in figure 4 for the case $\Omega_{1},\Omega_{2}>0$ and $\Omega_{3}<0$.
As we will demonstrate below, $\beta<0$ in this case; in general, the sign of
$\beta$ is the same as the sign of the largest (in magnitude) angular
frequency, $\Omega_{j}$. Notably, $|\beta|$ decreases sharply towards zero as
$K_{1}+K_{2}\rightarrow K_{3}$, corresponding to the limit $h_{c}\rightarrow
0$. Similarly, $|\beta|$ approaches zero in the limiting cases $K_{1}\ll
K_{3}$ or $K_{2}\ll K_{3}$, corresponding to one low-oscillatory wave mode
interacting with two highly-oscillatory wave modes. Away from these limiting
cases, however, $|\beta|$ depends only weakly on the wavenumbers, $K_{j}$,
suggesting that the correlation integral, $\mathscr{C}$, predominantly
controls the time-scale of the triad evolution. Finally, we observe that
$\beta$ is symmetric about the line $K_{1}=K_{2}$, consistent with the
invariance of equation (32) under the mapping $K_{1}\leftrightarrow K_{2}$
(and hence, $\Omega_{1}\leftrightarrow\Omega_{2}$).

We conclude this section by proving that $\beta<0$ in the case
$\Omega_{1},\Omega_{2}>0$ and $\Omega_{3}<0$. By comparing the forms of
equations (32) and (30), and then permuting the subscripts
$(1,2,3)\mapsto(3,1,2)$, we first note that $\beta$ may be equivalently
expressed as

$\beta=\Omega_{1}L_{2}^{2}+\Omega_{2}L_{1}^{2}-2\Omega_{3}L_{1}L_{2}+\sum_{l=1}^{3}\Omega_{l}K_{l}^{2},$

or

$\beta=\Omega_{1}(L_{2}^{2}+K_{1}^{2})+\Omega_{2}(L_{1}^{2}+K_{2}^{2})+|\Omega_{3}|(2L_{1}L_{2}-K_{3}^{2}).$

By bounding $L_{j}=K_{j}\tanh(K_{j}h_{c})<K_{j}$ for $0<h_{c}<\infty$ and
utilising the relation $\Omega_{1}+\Omega_{2}=|\Omega_{3}|$, we obtain

$\beta<|\Omega_{3}|\big{(}K_{1}^{2}+K_{2}^{2}+2K_{1}K_{2}-K_{3}^{2}\big{)}=|\Omega_{3}|\big{(}(K_{1}+K_{2})^{2}-K_{3}^{2}\big{)}.$
(33)

As resonant triads exist only when $K_{1}+K_{2}<K_{3}$ (see Theorem 1), we
conclude that $\beta<0$ in this case.

###  4.3 Summary

To summarise our theoretical developments, the velocity potential, $u$, at the
fluid rest level ($z=0$) evolves according to

$u(\bm{x},t)\sim\sum_{j=1}^{3}\Big{[}A_{j}(\tau)\Psi_{j}(\bm{x})\mathrm{e}^{-\mathrm{i}\Omega_{j}t}+\mathrm{c.c.}\Big{]}+O(\epsilon),$
(34)

where the complex amplitudes, $A_{j}(\tau)$, evolve over the slow time-scale,
$\tau=\epsilon t$, according to the triad equations (26). In particular, the
triad coefficients, $\alpha_{j}$ (see equation (31)), are defined in terms of
the correlation integral, $\mathscr{C}$ (equation (28)), and the coefficient
$\beta$ (equation (32)). Notably, we assume that $\mathscr{C}$ is nonzero; if
this condition were violated then all three of the triad coefficients,
$\alpha_{j}$, would be equal to zero, giving rise to non-interacting wave
modes at leading order (contradicting the notion of a triad). Indeed, the
condition $\mathscr{C}\neq 0$ is identical to the correlation condition
detailed in equation (8), the origins of which we have now justified. Finally,
the evolution of the free surface, $\eta$, may be recovered by recalling that
$\eta=-u_{t}+O(\epsilon)$: we conclude that $\eta(\bm{x},t)$ has a similar
leading-order form to $u(\bm{x},t)$, but each complex amplitude,
$A_{j}(\tau)$, in (34) is replaced by $\mathrm{i}\Omega_{j}A_{j}(\tau)$ (see
equation (37) below).

We briefly contrast our investigation of triad interaction with the early-time
calculation of Michel [41], who characterised the initial linear growth of a
child mode induced by the nonlinear interaction of two parent modes (where all
three modes comprise the triad). If modes 1 and 2 are the parent modes and
mode 3 is the child mode, then the initial linear growth may be deduced
directly from triad equations (26) in the limit
$|A_{3}|\ll|A_{1}|\sim|A_{2}|$. Specifically, the initial variation of $A_{1}$
and $A_{2}$ is slow relative to that of $A_{3}$, which has the approximate
early-time form $A_{3}(\tau)\approx\alpha_{3}C_{1}^{*}C_{2}^{*}\tau+C_{3}$,
where $C_{j}=A_{j}(0)$. Notably, the linear growth rate of the child mode
depends on the corresponding triad coefficient, $\alpha_{3}$, and the product
of the initial amplitudes of the two parent modes. However, our result for
circular cylinders differs to that of Michel; we believe that the author
neglected some important nonlinear contributions (compare Michel’s equation
(A2) to equations (2.4) and (2.4a) of Longuet-Higgins [34]). As Michel’s
experiment verified the scaling of the interaction only up to a
proportionality constant, this discrepancy was not captured.

####  4.3.1 The influence of weak detuning

As discussed earlier in §4, the analysis in §§4.1 and 4.2 does not account for
weak detuning of the angular frequencies, as might arise when the fluid depth,
$h$, differs slightly from the critical depth, $h_{c}$. We now briefly
consider the case of weak detuning, for which equation (22) is replaced by the
condition $\Omega_{1}+\Omega_{2}+\Omega_{3}=\epsilon\sigma$ (see §3.3.2); here
$\epsilon$ is the small parameter representative of the typical wave slope
(see §2) and $\sigma=O(1)$ determines the extent of the detuning [6, 40]. By
following a very similar multiple-scales procedure to the case $\sigma=0$, we
obtain amplitude equations that are now augmented by a time-dependent
modulation. Specifically, each complex amplitude now evolves according to

$\frac{\mathrm{d}{A_{1}}}{\mathrm{d}{\tau}}=\alpha_{1}A_{2}^{*}A_{3}^{*}\mathrm{e}^{\mathrm{i}\sigma\tau},\quad\frac{\mathrm{d}{A_{2}}}{\mathrm{d}{\tau}}=\alpha_{2}A_{1}^{*}A_{3}^{*}\mathrm{e}^{\mathrm{i}\sigma\tau},\quad\frac{\mathrm{d}{A_{3}}}{\mathrm{d}{\tau}}=\alpha_{3}A_{1}^{*}A_{2}^{*}\mathrm{e}^{\mathrm{i}\sigma\tau},$

where each coefficient, $\alpha_{j}$, is defined in equation (31). Although
detuning yields non-autonomous amplitude equations, autonomous equations may
be derived by mapping $A_{j}(\tau)\mapsto
A_{j}(\tau)\mathrm{e}^{\mathrm{i}\sigma\tau/3}$ for all $j=1,2,3$ [16].
Finally, we note that the energy,
$\mathscr{E}_{1}+\mathscr{E}_{2}+\mathscr{E}_{3}$, is not exactly conserved
when considering the effects of detuning; instead, the energy slowly
oscillates about a constant value [16].

####  4.3.2 The case of a 1:2 resonance

A 1:2 resonance is a resonant triad for which two modes comprising the triad
coincide. For this case, we define two angular frequencies, $\Omega_{1}$ and
$\Omega_{2}$, so that $\Omega_{2}=2\Omega_{1}$ [43], where the connection to
resonant triads is clear when writing $\Omega_{1}+\Omega_{1}=\Omega_{2}$. By
following a very similar multiple-scales procedure to that outlined in §4.1,
we obtain

$u(\bm{x},t)\sim\sum_{j=1}^{2}\Big{[}A_{j}(\tau)\Psi_{j}(\bm{x})\mathrm{e}^{-\mathrm{i}\Omega_{j}t}+\mathrm{c.c.}\Big{]}+O(\epsilon),$

where

$\frac{\mathrm{d}{A_{1}}}{\mathrm{d}{\tau}}=-\gamma
A_{1}^{*}A_{2}\quad\mathrm{and}\quad\frac{\mathrm{d}{A_{2}}}{\mathrm{d}{\tau}}=\frac{\gamma}{4}A_{1}^{2}.$
(35)

In particular, the evolution of the amplitude equations (35) depends on the
coefficient
$\gamma=\mathscr{C}\big{(}K_{2}^{2}-K_{1}^{2}-3\Omega_{1}^{4}\big{)}$, where
$\mathscr{C}=\frac{1}{S}\iint_{\mathcal{D}}\Psi_{1}^{2}\Psi_{2}\,\mathrm{d}A$
is the correlation integral. Indeed, the amplitude equations (35) and
coefficient, $\gamma$, are consistent with the results of Miles [43] when
expressing the evolution of each complex amplitude, $A_{j}$, in polar form
(with appropriate rescaling). Finally, we note that a weak detuning (see
§4.3.1) may also be incorporated within the amplitude equations (35), thereby
accounting for a slight mismatch between the fluid depth, $h$, and the
corresponding critical depth, $h_{c}$ [43].

Of particular interest is the evolution of weakly nonlinear waves steadily
propagating around a circular cylinder of unit radius, focusing on the case
where the fluid depth is precisely equal to the critical depth of a 1:2
resonance [65]. For the complex-valued eigenmodes defined in equation (19),
the correlation condition,
$\iint_{\mathcal{D}}\Psi_{1}^{2}\Psi_{2}^{*}\,\mathrm{d}A\neq 0$, determines
that the angular wavenumbers satisfy $m_{2}=2m_{1}$ [13, 65]. By expressing
the complex wave amplitudes in polar form,
$A_{j}(\tau)=a_{j}(\tau)\mathrm{e}^{\mathrm{i}\theta_{j}(\tau)}$ (for
$j=1,2$), equation (35) may be recast as [43]

$\frac{\mathrm{d}{a_{1}}}{\mathrm{d}{\tau}}=-\gamma
a_{1}a_{2}\cos\Theta,\quad\frac{\mathrm{d}{a_{2}}}{\mathrm{d}{\tau}}=\frac{\gamma}{4}a_{1}^{2}\cos\Theta,\quad\frac{\mathrm{d}{\Theta}}{\mathrm{d}{\tau}}=2\gamma
a_{2}\bigg{[}1-\frac{a_{1}^{2}}{8a_{2}^{2}}\bigg{]}\sin\Theta,$

where $\Theta(\tau)=\theta_{2}(\tau)-2\theta_{1}(\tau)$ is the time-dependent
phase shift. Steadily propagating waves correspond to time-independent
solutions for $a_{1}$, $a_{2}$ (both nonzero) and $\Theta$, from which we
deduce that $\cos\Theta=0$ and $a_{1}/a_{2}=2\sqrt{2}$. Indeed, it is
remarkable that the amplitude ratio of the two dominant (normalised) wave
modes is independent of the angular wavenumbers, $m_{j}$, the radial
wavenumbers, $K_{j}$, and the corresponding angular frequencies, $\Omega_{j}$
(see §3.3.2 for details). Furthermore, one may readily determine the
relationship between the angular velocity of the steady wave rotation and the
corresponding wave amplitude, which may then be compared to the numerical
solution of the full Euler equations [65]. This comparison, as well as a
comparison to steadily propagating waves computed from various truncations of
the Euler equations, will be the subject of future investigation.

##  5 The excitation of resonant triads

Having established the existence and evolution of resonant triads, we now
focus on the excitation of a particular triad via external forcing. So as to
motivate the method of excitation, we first recall (§5.1) the well-known
result that one mode in the triad may, or may not, excite the other two modes
[17, 23, 55]; in the case of excitation, the initial mode is referred to as
the _pump_ mode [16]. We will then utilise the criterion of the pump mode to
excite all three modes in the triad via a pulsating pressure source (§5.2).
Throughout this section, we continue with the convention that the triad
angular frequencies satisfy $\Omega_{1}+\Omega_{2}+\Omega_{3}=0$, as set forth
in §4.

###  5.1 Excitation via the triad pump mode

To first identify the triad pump mode and then characterise the resultant
excitation, we consider the case for which $A_{3}$, say, is much larger in
magnitude than the other two mode amplitudes, so $|A_{1}|,|A_{2}|\ll|A_{3}|$
[17, 23, 55]. By linearising the triad equations (26), we obtain

$\frac{\mathrm{d}{A_{1}}}{\mathrm{d}{\tau}}=\alpha_{1}A_{2}^{*}A_{3}^{*},\qquad\frac{\mathrm{d}{A_{2}}}{\mathrm{d}{\tau}}=\alpha_{2}A_{1}^{*}A_{3}^{*},\qquad\frac{\mathrm{d}{A_{3}}}{\mathrm{d}{\tau}}=0,$
(36)

from which we immediately conclude that $A_{3}$ is constant (whilst the
linearisation assumption holds); we denote $A_{3}(\tau)=C$ for some given
complex number $C$. By considering second derivatives of $A_{1}$ and $A_{2}$,
we deduce the linearised evolution equations [16]

$\frac{\mathrm{d}{{}^{2}A_{1}}}{\mathrm{d}{\tau^{2}}}=\alpha_{1}\alpha_{2}|C|^{2}A_{1}\quad\mathrm{and}\quad\frac{\mathrm{d}{{}^{2}A_{2}}}{\mathrm{d}{\tau^{2}}}=\alpha_{1}\alpha_{2}|C|^{2}A_{2},$

where $\alpha_{1}\alpha_{2}=\mathscr{C}^{2}\beta^{2}/(4\Omega_{1}\Omega_{2})$
(see equation (31)). We conclude that $A_{1}(\tau)$ and $A_{2}(\tau)$ grow
exponentially in time (whilst the linearisation approximation holds) when
$\Omega_{1}\Omega_{2}>0$, and exhibit sinusoidal oscillations when
$\Omega_{1}\Omega_{2}<0$ [17, 23, 16]. Thus, mode 3 may excite modes 1 and 2
when $\Omega_{1}$ and $\Omega_{2}$ have the same sign (and likewise for other
mode permutations). As one angular frequency must have a different sign from
the other two (so as to satisfy $\Omega_{1}+\Omega_{2}+\Omega_{3}=0$), we
conclude that the mode whose angular frequency is largest in magnitude (i.e.
differs in sign) is the triad pump mode [16]. Equivalently, the pump mode is
the mode with largest wavenumber, $K_{j}$, providing a robust mechanism for an
inverse energy cascade to lower wavenumbers [1].

Figure 5:  Excitation of a triad via its pump mode for the case of a circular
cylinder. We consider triad 11 in table 1, but with $m_{3}\mapsto-m_{3}$. We
choose $\Omega_{1},\Omega_{2}>0$ and $\Omega_{3}<0$, so that mode 3 is the
pump mode. $(a)$ Evolution of the free-surface, $\eta\sim-u_{t}$, over the
slow time-scale, $\tau=\epsilon t$, with $\epsilon=10^{-3}$. $(b)$ The
evolution of the wave amplitudes, $|A_{j}|$, according to the triad equations
(equation (26), solid curves) and the pump-mode approximation (equation (36),
dashed-dotted curves). Insets: modes 1 (blue), 2 (red) and 3 (gold) at
$\tau=0$; all three modes rotate counter-clockwise. The simulations were
initialised from $A_{1}(0)=0.01$ and $A_{2}(0)=0.01\mathrm{i}$, where
$A_{3}(0)$ was chosen to be the positive real number satisfying
$\mathscr{E}_{1}+\mathscr{E}_{2}+\mathscr{E}_{3}=1$, with
$\mathscr{E}_{j}=\Omega_{j}^{2}|A_{j}|^{2}$ (see §4.2).

To visualise the influence of the pump mode on the resultant free-surface
pattern, we present the solution of the triad equations (26) and the
corresponding pump-mode approximation (equation (36)) in figure 5. By
recalling that the free surface satisfies $\eta=-u_{t}+O(\epsilon)$, we first
deduce that

$\eta(\bm{x},t)\sim\sum_{j=1}^{3}\Big{[}\mathrm{i}\Omega_{j}A_{j}(\tau)\Psi_{j}(\bm{x})\mathrm{e}^{-\mathrm{i}\Omega_{j}t}+\mathrm{c.c.}\Big{]}+O(\epsilon).$
(37)

For the case of a circular cylinder, we utilise the complex-valued eigenmodes
defined in equation (19), corresponding to the superposition of steadily
propagating waves for $m_{j}\neq 0$ (the rotation direction depends on the
sign of $\Omega_{j}/m_{j}$). Upon initialising the system so that the energy
is primarily within the pump mode (mode 3), modes 1 and 2 are gradually
excited due to nonlinear interaction, with exponential growth evident for
$\tau\lesssim 10$. As time further increases, the dynamics depart from the
pump-mode approximation: the energy in the pump mode appreciably decreases,
whilst the energy in modes 1 and 2 saturates. The free surface varies
qualitatively during this evolution, with an appreciable change in pattern
structure visible by $\tau=24$ (primarily a superposition of modes 1 and 2).
Notably, the system evolution is periodic, which becomes apparent over longer
time scales.

###  5.2 Excitation via an applied pressure source

Based on the ideas of the previous section, we consider a methodology for
exciting the pump mode of a triad, which will subsequently excite the
remaining two modes (provided that the initial disturbance of each of the
remaining modes is nonzero). Notably, several methods for exciting internal
resonances have been considered in prior investigations, primarily focusing on
imposed motion of the fluid vessel via horizontal [43, 46] or vertical
vibration [43, 45, 47, 25]. Furthermore, one may, in principle, utilise
sinusoidal paddles or plungers to excite a particular triad’s pump mode for a
given geometry (similar wave makers are used in rectangular wave tanks [38,
24]). However, for large-scale fluid tanks, imposed motion of the vessel may
be impractical (if the tank were set in a concrete base, for example), and it
may be challenging to determine the correct paddle motion necessary to excite
a chosen pump mode for geometrically complex cylinders. We choose, therefore,
to consider a slightly different approach: we instead excite the pump mode via
a pulsating pressure source located just above the free surface (e.g. an air
blower).

In order to incorporate a pressure source within our mathematical framework,
we first reformulate the dimensionless dynamic boundary condition (equation
(1b)) as

$\phi_{t}+\eta+\frac{\epsilon}{2}\Big{(}|\nabla\phi|^{2}+\phi_{z}^{2}\Big{)}+\epsilon
P(\bm{x},t)=0\quad\mbox{for}\quad\bm{x}\in\mathcal{D},\quad z=\epsilon\eta,$

where the dimensional pressure is $\epsilon^{2}a\rho gP$ for fluid density
$\rho$ ($P=0$ corresponds to atmospheric pressure). The pressure source is
chosen to be small in magnitude so that the resultant wave excitation arises
over the slow time-scale, $\tau=\epsilon t$, and may thus be saturated by
weakly nonlinear effects. By modifying the developments outlined in §2.1, we
derive the forced Benney-Luke equation

$u_{tt}+\mathscr{L}u+\epsilon\bigg{(}u_{t}\big{(}\mathscr{L}^{2}+\Delta\big{)}u+\frac{\partial{}}{\partial{t}}\Big{[}(\mathscr{L}u)^{2}+|\nabla
u|^{2}\Big{]}+\partial_{t}P\bigg{)}=O(\epsilon^{2})\quad\mathrm{for}\quad\bm{x}\in\mathcal{D},$
(38)

which will be the starting point for the asymptotic analysis.

Before proceeding further, we first describe two forms of the pressure source
relevant to our investigation. For a stationary pressure source oscillating
periodically over the fast time-scale, $t$, we express
$P(\bm{x},t)=f(\tau)s(\bm{x})\mathrm{e}^{-\mathrm{i}\Omega_{p}t}+\mathrm{c.c.}$,
where $s(\bm{x})$ is a fixed spatial profile (generally spanning the cavity),
$f(\tau)$ accounts for a slow modulation in the magnitude of the pressure, and
$\Omega_{p}$ is the pulsation angular frequency. We choose $\Omega_{p}$ to be
close to the angular frequency of the pump mode, which, without loss of
generality, we assume to be mode 3 (i.e. $\Omega_{3}$ has the opposite sign
from $\Omega_{1}$ and $\Omega_{2}$). We denote, therefore,
$\Omega_{p}=\Omega_{3}+\epsilon\mu$, where $\mu=O(1)$ determines the extent of
the frequency mismatch. For a pressure source orbiting the centre of a
circular cylinder at a constant angular velocity, we instead posit that $P$
has the form $P(r,\theta,t)=f(\tau)s(r,\theta-\Omega_{p}t)$, where
$\Omega_{p}=(\Omega_{3}+\epsilon\mu)/m_{3}$ is the angular velocity of the
pressure source (assuming that the pump mode is non-axisymmetric, i.e.
$m_{3}\neq 0$).

For both standing and orbiting pressure sources, we now follow a similar
multiple-scales procedure to that outlined in §4.1, starting from the forced
Benney-Luke equation (38). So as to discount the possibility that the pressure
source excites more than one mode in the triad, we assume that neither
$|\Omega_{1}|$ or $|\Omega_{2}|$ are close to $|\Omega_{3}|$. Furthermore, we
incorporate a weak detuning in the triad angular frequencies, denoting
$\Omega_{1}+\Omega_{2}+\Omega_{3}=\epsilon\sigma$ (see §4.3.1). It follows
that each complex amplitude, $A_{j}(\tau)$, evolves according to

$\frac{\mathrm{d}{A_{1}}}{\mathrm{d}{\tau}}=\alpha_{1}A_{2}^{*}A_{3}^{*}\mathrm{e}^{\mathrm{i}\sigma\tau},\quad\frac{\mathrm{d}{A_{2}}}{\mathrm{d}{\tau}}=\alpha_{2}A_{1}^{*}A_{3}^{*}\mathrm{e}^{\mathrm{i}\sigma\tau},\quad\frac{\mathrm{d}{A_{3}}}{\mathrm{d}{\tau}}=\alpha_{3}A_{1}^{*}A_{2}^{*}\mathrm{e}^{\mathrm{i}\sigma\tau}-\Omega_{3}s_{3}f(\tau)\mathrm{e}^{-\mathrm{i}\mu\tau},$
(39)

where the coefficients, $\alpha_{j}$, are defined in equation (31). Notably,
the pump mode may only be excited provided that the corresponding eigenmode is
non-orthogonal to the pressure source, corresponding to a nonzero projection,
i.e. $s_{3}\neq 0$, where $s_{3}=\langle\Psi_{3},s\rangle$. Similar equations
describing the evolution of forced resonant triads have been explored by
McEwan _et al._ [36] (with the inclusion of linear damping) and Raupp & Silva
Dias [51].

Figure 6:  Evolution of the forced triad equations (39) for $\sigma=\mu=0$ and
constant $f$. We consider the same triad as figure 5, with $s_{3}f=0.1$. In
all three panels, $A_{1}(0)=0.02\mathrm{i}$ and $A_{2}(0)=0.01$. For
$A_{3}(0)=0.01$, we observe $(a)$ the initial excitation of the triad and
$(b)$ the resultant periodic dynamics (the initial growth is highlighted
within the grey box). $(c)$ For $A_{3}(0)=0.01\mathrm{i}$, the triad evolution
appears chaotic. $(d)$ The separation distance, $\delta(\tau)$, of a
trajectory randomly perturbed at time $\tau_{\mathrm{pert}}$, with
$\delta(\tau_{\mathrm{pert}})=10^{-10}$ and the same initialisation (at
$\tau=0$) as $(c)$. The black curves bound $\delta(\tau)$ for 30 equally
spaced values of $\tau_{\mathrm{pert}}$ in the interval
$100\leq\tau_{\mathrm{pert}}\leq 115$; the blue curve corresponds to
$\tau_{\mathrm{pert}}=100$.

In the special case of time-independent forcing ($f$ constant) and no
frequency detuning ($\sigma=\mu=0$), the dynamics of the forced triad
equations has been analysed by Harris _et al._ [21], with both periodic and
quasi-periodic dynamics reported. We also consider this case, leaving the
effects of detuning and variable forcing for future investigation. In this
setting, when $|A_{1}|$, $|A_{2}|$ and $|A_{3}|$ are initially small relative
to the magnitude of the forcing, $|\Omega_{3}s_{3}f|$, the initial growth in
$A_{3}$ is approximately linear (see figure 6$(a)$). As mode 3 is the pump
mode, the growth in $A_{3}$ excites $A_{1}$ and $A_{2}$, thus activating the
triad. The conservation laws of the forced triad equations [21] result in a
temporary diminution of mode 3, which is later augmented by the external
forcing; whence the process repeats. In some parameter regimes, the resulting
evolution of the forced triad is periodic in time (see figure 6$(b)$ and Raupp
& Silva Dias [51]); in contrast to the findings of Harris _et al._ [21],
however, we also identify initial conditions (with all other parameters
unchanged) that result in hitherto unidentified chaotic dynamics (see figure
6$(c)$).

To verify the chaotic nature of this latter example, we consider the
separation distance of two initially adjacent trajectories in phase space, for
which we observe exponential divergence in time (see figure 6$(d)$). This
exponential divergence is indicative of a positive maximal Lyapunov exponent
[56], which characterises the sensitivity to initial conditions exhibited by
chaotic systems. Specifically, for a solution, $A_{j}(\tau)$, and its
perturbation, $A_{j,\mathrm{pert}}(\tau)$, we consider the evolution of the
separation distance, defined

$\delta(\tau)=\sqrt{\sum_{j=1}^{3}\big{|}A_{j}(\tau)-A_{j,\mathrm{pert}}(\tau)\big{|}^{2}}.$

To compute $\delta(\tau)$, we first simulate the forced triad equations (39)
on the interval $0\leq\tau\leq\tau_{\mathrm{pert}}$, with
$A_{j}(0)=A_{j,\mathrm{pert}}(0)$ and $\tau_{\mathrm{pert}}$ chosen to be
sufficiently large so as to ensure that the chaotic attractor (should one
exist) be approached. At $\tau=\tau_{\mathrm{pert}}$, the real and imaginary
parts of $A_{j,\mathrm{pert}}$ are both randomly perturbed according to a
uniform distribution on the interval $(-1,1)$, with the resulting complex
perturbation scaled so that $\delta(\tau_{\mathrm{pert}})=10^{-10}$. We then
evolve $A_{j}(\tau)$ and $A_{j,\mathrm{pert}}(\tau)$ for
$\tau\geq\tau_{\mathrm{pert}}$, giving rise to initial exponential growth of
$\delta(\tau)$, with saturation when the perturbation distance is comparable
to the ‘diameter’ of the chaotic attractor (see figure 6$(d)$).

To confirm that the exponential growth was not specific to a perturbation
about a particular point on the chaotic attractor [56], we considered 30
equally spaced values of $\tau_{\mathrm{pert}}$ on the interval
$100\leq\tau_{\mathrm{pert}}\leq 115$, roughly corresponding to the time taken
for one ‘loop’ of the chaotic attractor to take place (see figure 6$(c)$).
Each simulation was computed with a fourth-order Runge-Kutta method and a time
step of 0.005. For each value of $\tau_{\mathrm{pert}}$, we observed similar
exponential divergence of trajectories (see figure 6$(d)$); moreover, the
evolution of $\delta(t)$ during the growth phase remained unchanged when the
time step was decreased to 0.001, with numerical errors only accumulating over
longer time scales. Our results thus provide strong evidence that there is a
positive maximal Lyapunov exponent in this particular portion of parameter
space, indicative of chaotic dynamics.

##  6 Discussion

We have performed a systematic investigation into nonlinear resonant triads of
free-surface gravity waves confined to a cylinder of finite depth; previously
studied 1:2 resonances are obtained as special cases. A key result of our
study is Theorem 1, which determines whether there exists a fluid depth at
which three given wave modes resonate due to the nonlinear evolution of the
fluid. Equipped with this result, we determined the long-time fluid evolution
using multiple-scales analysis, from which we deduced that all solutions to
the triad equations are periodic in time. Finally, we determined that a given
triad may be excited via external forcing of the triad’s pump mode, thereby
providing a mechanism for exciting a given triad in a wave tank. All our
results are derived for cylinders of arbitrary cross-section (barring some
technical assumptions; see §2), thus forming a broad framework for
characterising nonlinear resonance of confined free-surface gravity waves. In
particular, our theoretical developments buttress experimental observations
[41] and demonstrate the potential generality of confinement as a mechanism
for promoting nonlinear resonance.

A second fundamental component of our study is the influence of the cylinder
cross-section on the existence of resonant triads; for example, resonant
triads are impossible in rectangular cylinders, yet abundant within circular
and annular cylinders (for particular fluid depths). Of the vast array of
resonances arising in a circular cylinder, triads consisting of an
axisymmetric pump mode and two identical counter-propagating waves are of
notable interest. This combination of axisymmetric and non-axisymmetric modes
possesses an interesting analogy to the excitation of counter-propagating
subharmonic beach edge waves due to a normally incident standing wave [19].
Specifically, the wave crests of the standing axisymmetric mode are always
parallel to the bounding wall of the circular cylinder, and may excite
steadily propagating waves that are periodic in the azimuthal direction. For
the special case for which the amplitudes of the two counter-propagating modes
coincide, one observes the resonant interaction of standing axisymmetric and
non-axisymmetric waves.

So as to gain a deeper insight into the influence of nonlinearity on resonant
triads, a primary focus for future investigations will be the simulation of
the Euler equations within a cylindrical domain, with consideration of various
truncated systems [15, 48, 5, 58]. From a computational perspective, the most
natural geometry to consider is a circular cylinder [50]; this geometry has
been previously explored in the context of steadily propagating nonlinear
waves in the vicinity of a 1:2 resonance [9, 65], but it remains to assess the
efficacy of the amplitude equations (26) for predicting the evolution of
nonlinear triads. Indeed, exploration of the nonlinear dynamics may reveal
additional resonant triads arising beyond the small-wave-amplitude limit
explored herein. Of similar interest is the fluid evolution when multiple
triads are excited at a single depth, with the potential for energy exchange
via triad-triad interactions [36, 16, 14, 12]. The simulation of free-surface
gravity waves in non-circular cylinders presents a more formidable challenge,
however, except for cylinder cross-sections that possess a tractable eigenmode
decomposition.

A second natural avenue for future investigation is to characterise the
influence of applied forcing on resonant triads. For example, when the fluid
bath is subjected to sufficiently vigorous vertical vibration, Faraday waves
[18, 31] may appear on the free surface; although this scenario has been
studied in the case of a 1:2 internal resonance [45, 47, 25], resonant triads
may give rise to the formation of more exotic free-surface patterns,
particularly at fluid depths that differ from that of a 1:2 resonance. In a
similar vein, horizontal vibration [43, 46] or a pulsating pressure source at
the frequency of the triad’s pump mode may lead to a wealth of periodic and
quasi-periodic dynamics, as predicted by the forced triad equations [21]. Our
study has indicated, however, that chaotic dynamics are also possible in some
parameter regimes, and might thus be excited in numerical simulation or
experiments. Lastly, our study has focused on flat-bottomed cylinders; it
seems plausible, however, that submerged topography may enhance or mitigate
certain resonances, which may be an important consideration in the design of
industrial-scale fluid tanks.

Finally, our study has focused on the special case of a liquid-air interface,
for which the dynamics of the air are neglected within the Euler equations. It
is natural, however, to extend our formulation to the case of two-layer flows
(in the absence of surface tension), with two immiscible fluids (e.g. air and
water) confined within a cylinder whose lid and base are both rigid. In this
setting, the density difference across the fluid-fluid interface has a strong
influence of the system dynamics; it seems plausible, therefore, that
additional resonances may be excited in this configuration, relative to the
liquid-air interface considered herein. Notably, the anticipated resonances
would arise across a single interface, in contrast to the cross-interface
resonances explored in previous investigations [2, 55, 26, 54, 57, 12].
Finally, exploring the influence of parametric forcing [32] on resonant triads
arising for two-layer flows opens up exciting new vistas in nonlinear
resonance induced by confinement.

##  Appendix A Proof of Theorem 1

###### Proof.

To prove Theorem 1, we first show that there are no values of $h\in(0,\infty)$
satisfying $\Omega_{1}+\Omega_{2}=\Omega_{3}$ when $K_{1}+K_{2}\geq K_{3}$ or
when $\sqrt{K_{1}}+\sqrt{K_{2}}\leq\sqrt{K_{3}}$, where we recall that
$\Omega_{j}(h)=\sqrt{K_{j}\tanh(K_{j}h)}$ and $K_{j}>0$ for $j=1,2,3$. We then
prove that there exists a solution to $\Omega_{1}+\Omega_{2}=\Omega_{3}$ when
$K_{1}+K_{2}<K_{3}<(\sqrt{K_{1}}+\sqrt{K_{2}})^{2}$, and that this solution is
unique.

In the case $K_{1}+K_{2}\geq K_{3}$, we first define
$\chi(K;h)=\sqrt{K\tanh(Kh)}$. For fixed $h>0$, we observe that

$\chi(K_{3};h)\leq\chi(K_{1}+K_{2};h)<\chi(K_{1};h)+\chi(K_{2};h),$

where we have utilised that $\chi(K;h)$ is a positive, monotonically
increasing, concave function of $K>0$. We conclude that
$\Omega_{3}<\Omega_{1}+\Omega_{2}$ for any $h>0$, so there are no values of
$h$ for which $\Omega_{1}+\Omega_{2}=\Omega_{3}$.

In the case $\sqrt{K_{1}}+\sqrt{K_{2}}\leq\sqrt{K_{3}}$, we first note that
the lower bound $K_{j}>0$ (for $j=1,2,3$) implies that $K_{1}<K_{3}$ and
$K_{2}<K_{3}$. Furthermore, as $\tanh(x)$ is a monotonically increasing
function for $x>0$, we conclude that $\tanh(K_{j}h)<\tanh(K_{3}h)$ for $j=1,2$
and all $h>0$. We now utilise this property to deduce that

$\sqrt{K_{1}\tanh(K_{1}h)}+\sqrt{K_{2}\tanh(K_{2}h)}<\big{(}\sqrt{K_{1}}+\sqrt{K_{2}}\big{)}\sqrt{\tanh(K_{3}h)}\leq\sqrt{K_{3}\tanh(K_{3}h)}.$

We conclude that $\Omega_{3}>\Omega_{1}+\Omega_{2}$ for any $h>0$, so there
are no values of $h$ for which $\Omega_{1}+\Omega_{2}=\Omega_{3}$.

For the remainder of the proof, we consider the case

$K_{1}+K_{2}<K_{3}\quad\mathrm{and}\quad\sqrt{K_{3}}<\sqrt{K_{1}}+\sqrt{K_{2}},$
(40)

which is equivalent to the pair of inequalities given by equation (9). Indeed,
we will show that there exists a unique value of $h>0$ satisfying
$\Omega_{1}+\Omega_{2}=\Omega_{3}$ in this case. Equivalently, we demonstrate
that $F(h)=\big{(}\Omega_{1}(h)+\Omega_{2}(h)\big{)}/\Omega_{3}(h)-1$ has a
unique positive root, where we express

$F(h)=\sqrt{\psi_{1}(h)}+\sqrt{\psi_{2}(h)}-1,$

with the positive functions $\psi_{1}$ and $\psi_{2}$ defined

$\psi_{j}(h)=\frac{K_{j}\tanh(K_{j}h)}{K_{3}\tanh(K_{3}h)}\quad\mathrm{for}\,\,\,j=1,2.$

In order to show the existence of a root of $F(h)$, we first note that

$\lim_{h\rightarrow
0}F(h)=\frac{K_{1}+K_{2}}{K_{3}}-1<0\quad\mathrm{and}\quad\lim_{h\rightarrow\infty}F(h)=\frac{\sqrt{K_{1}}+\sqrt{K_{2}}}{\sqrt{K_{3}}}-1>0,$

where we have used the limits $\lim_{x\rightarrow 0}(\tanh(x)/x)=1$ and
$\lim_{x\rightarrow\infty}\tanh(x)=1$, respectively, and implemented the
inequalities given in equation (40). As $F(h)$ is a continuous function, the
intermediate-value theorem determines that $F(h)$ has at least one positive
root.

To prove that such a root is unique, we demonstrate that $F(h)$ is a strictly
monotonically increasing function for $h>0$. Specifically, we note that (for
$j=1,2$)

$\frac{\mathrm{d}{\psi_{j}}}{\mathrm{d}{h}}=2K_{3}\psi_{j}(h)\bigg{(}\frac{K_{j}}{K_{3}}\mathrm{cosech}(2K_{j}h)-\mathrm{cosech}(2K_{3}h)\bigg{)}>0\quad\mathrm{for}\,\,\,0<K_{j}<K_{3},$

where the inequality follows from the convexity of $\mathrm{cosech}(x)$ for
$x>0$, i.e. $b\,\mathrm{cosech}(bx)>\mathrm{cosech}(x)$ for $0<b<1$ and all
$x>0$ (associating $x=2K_{3}h$ and $b=K_{j}/K_{3}$). As the bounds
$K_{1}<K_{3}$ and $K_{2}<K_{3}$ incorporate the region determined by equation
(40), we deduce that $F(h)$ is strictly monotonically increasing. We conclude,
therefore, that the root of $F(h)$ must be unique, thereby completing the
proof.

∎

##  Appendix B Wavenumbers in an annulus

The no-flux condition (equation (1d)) on the inner and outer radii of an
annulus requires that $\partial_{r}\Phi_{mn}(r_{0},\theta)=0$ and
$\partial_{r}\Phi_{mn}(1,\theta)=0$ for all $\theta$, where
$\Phi_{mn}(r,\theta)$ is the cylinder function defined in equation (20). It
follows, therefore, that the corresponding wavenumber, $k_{mn}$, and weighting
factor, $\gamma_{mn}$, satisfy the equations

$\displaystyle\mathrm{J}_{m}^{\prime}(k_{mn}r_{0})\cos(\gamma_{mn}\pi)+\mathrm{Y}_{m}^{\prime}(k_{mn}r_{0})\sin(\gamma_{mn}\pi)$
$\displaystyle=0,$ (41a)
$\displaystyle\mathrm{J}_{m}^{\prime}(k_{mn})\cos(\gamma_{mn}\pi)+\mathrm{Y}_{m}^{\prime}(k_{mn})\sin(\gamma_{mn}\pi)$
$\displaystyle=0.$ (41b)

By rearranging equation (41), we determine the following expressions for
$\tan(\gamma_{mn}\pi)$:

$\tan(\gamma_{mn}\pi)=-\frac{\mathrm{J}^{\prime}_{m}(k_{mn}r_{0})}{\mathrm{Y}^{\prime}_{m}(k_{mn}r_{0})}\quad\mathrm{and}\quad\tan(\gamma_{mn}\pi)=-\frac{\mathrm{J}^{\prime}_{m}(k_{mn})}{\mathrm{Y}^{\prime}_{m}(k_{mn})}.$
(42)

By eliminating $\tan(\gamma_{mn}\pi)$ and rearranging, we find that $k_{mn}>0$
satisfies equation (21). Upon computing $k_{mn}$, one may then determine
$\gamma_{mn}\in[0,1]$ using either of the equivalent expressions for
$\tan(\gamma_{mn}\pi)$ given in equation (42).

##  Appendix C Reduction of the triad coefficients

As motivated by the form of $\alpha_{1}$ given in equation (30), we
demonstrate that

$\Omega_{2}L_{3}^{2}+\Omega_{3}L_{2}^{2}-2\Omega_{1}L_{2}L_{3}=-\frac{1}{2}\Omega_{1}\Omega_{2}\Omega_{3}\big{(}\Omega_{1}^{2}+\Omega_{2}^{2}+\Omega_{3}^{2}\big{)},$
(43)

where we recall that $\Omega_{1}+\Omega_{2}+\Omega_{3}=0$ and
$L_{j}=\Omega_{j}^{2}$. In fact, the equality given in equation (43) holds
under cyclic permutation of the indices $(1,2,3)$ (as is necessary when
defining $\alpha_{2}$ and $\alpha_{3}$), where we note that the right-hand
side is unchanged under such permutations. We conclude that $\alpha_{2}$ and
$\alpha_{3}$ may be simplified in a similar manner, with the right-hand side
of equation (43) appearing as a constant term in all three coefficients (see
§4.2).

We now detail the algebraic manipulations necessary to transform the left-hand
side of equation (43) into the right-hand side. By substituting
$L_{j}=\Omega_{j}^{2}$ into the left-hand side of equation (43) and
factorising, we obtain

$\Omega_{2}L_{3}^{2}+\Omega_{3}L_{2}^{2}-2\Omega_{1}L_{2}L_{3}=\Omega_{2}^{4}\Omega_{3}+\Omega_{2}\Omega_{3}^{2}\big{(}\Omega_{3}^{2}-2\Omega_{1}\Omega_{2}\big{)}.$
(44)

Next, we substitute

$\Omega_{3}^{2}=(\Omega_{1}^{2}+\Omega_{2}^{2})=\Omega_{1}^{2}+2\Omega_{1}\Omega_{2}+\Omega_{2}^{2}$
(45)

into equation (44), yielding

$\Omega_{2}L_{3}^{2}+\Omega_{3}L_{2}^{2}-2\Omega_{1}L_{2}L_{3}=\Omega_{2}\Omega_{3}\Big{[}\Omega_{2}^{3}+\Omega_{3}\big{(}\Omega_{1}^{2}+\Omega_{2}^{2}\big{)}\Big{]}.$
(46)

We proceed by substituting $\Omega_{3}=-(\Omega_{1}+\Omega_{2})$ within the
square brackets in equation (46); by distributing and cancelling common terms,
we obtain

$\Omega_{2}L_{3}^{2}+\Omega_{3}L_{2}^{2}-2\Omega_{1}L_{2}L_{3}=-\Omega_{1}\Omega_{2}\Omega_{3}\Big{[}\Omega_{1}^{2}+\Omega_{1}\Omega_{2}+\Omega_{2}^{2}\Big{]}.$
(47)

Finally, we rearrange equation (45) to give

$\Omega_{1}\Omega_{2}=\frac{1}{2}\Big{(}\Omega_{3}^{2}-\Omega_{1}^{2}-\Omega_{2}^{2}\Big{)},$

which, upon substitution into equation (47), supplies the required result
(equation (43)).

## References

  * [1] S. Y. Annenkov and V. I. Shrira.  Direct numerical simulation of downshift and inverse cascade for water wave turbulence.  Phys. Rev. Lett., 96(20):204501, 2006.
  * [2] F. K. Ball.  Energy transfer between external and internal gravity waves.  J. Fluid Mech., 19(3):465–478, 1964.
  * [3] D. J. Benney.  Non-linear gravity wave interactions.  J. Fluid Mech., 14(4):577–584, 1962.
  * [4] D. J. Benney and J. C. Luke.  On the Interactions of Permanent Waves of Finite Amplitude.  J. Math. Phys., 43:309–313, 1964.
  * [5] K. M. Berger and P. A. Milewski.  Simulation of Wave Interactions and Turbulence in One-Dimensional Water Waves.  SIAM J. Appl. Math., 63(4):1121–1140, 2003.
  * [6] F. Bretherton.  Resonant interactions between waves. The case of discrete oscillations.  J. Fluid Mech., 20(3):457–479, 1964.
  * [7] T. J. Bridges and F. Dias.  An analysis of two-dimensional water waves based on O(2) symmetry.  Nonlinear Anal. Theory Methods Appl., 14(9):733–764, 1990.
  * [8] P. J. Bryant.  Periodic waves in shallow water.  J. Fluid Mech., 59(4):625–644, 1973.
  * [9] P. J. Bryant.  Nonlinear progressive free waves in a circular basin.  J. Fluid Mech., 205:453–467, 1989.
  * [10] K. M. Case and S. C. Chiu.  Three-wave resonant interactions of gravity-capillary waves.  Phys. Fluids, 20:742, 1977.
  * [11] M. Chabane and W. Choi.  On resonant interactions of gravity-capillary waves without energy exchange.  Stud. Appl. Math., 142:528–550, 2019.
  * [12] W. Choi, M. Chabane, and T. M. A. Taklo.  Two-dimensional resonant triad interactions in a two-layer system.  J. Fluid Mech., 907:A5, 2021.
  * [13] P. Chossat and F. Dias.  The 1:2 Resonance with O(2) Symmetry and Its Applications in Hydrodynamics.  J. Nonlinear Sci., 5:105–129, 1995.
  * [14] C. Chow, D. Henderson, and H. Segur.  A generalized stability criterion for resonant triad interactions.  J. Fluid Mech., 319:67–76, 1996.
  * [15] W. Craig and C. Sulem.  Numerical Simulation of Gravity Waves.  J. Comp. Phys., 108(1):73–83, 1993.
  * [16] A. D. D. Craik.  Wave interactions and fluid flows.  Cambridge Monographs on Mechanics. Cambridge University Press, 1986.
  * [17] R. E. Davis and A. Acrivos.  The stability of oscillatory internal waves.  J. Fluid Mech., 30(4):723–736, 1967.
  * [18] M. Faraday.  On a Peculiar Class of Acoustical Figures; and on Certain Forms Assumed by Groups of Particles upon Vibrating Elastic Surfaces.  Philos. Trans. R. Soc., 121:299–340, 1831.
  * [19] R. T. Guza and R. E. Davis.  Excitation of Edge Waves by Waves Incident on a Beach.  J. Geophys. Res., 79(9):1285– 1291, 1974.
  * [20] J. L. Hammack and D. M. Henderson.  Resonant Interactions Among Surface Water Waves.  Annu. Rev. Fluid Mech., 25(1):55–97, 1993.
  * [21] J. Harris, M. D. Bustamante, and C. Connaughton.  Externally forced triads of resonantly interacting waves: Boundedness and integrability properties.  Commun. Nonlinear Sci. Numer. Simulat., 17:4988–5006, 2012.
  * [22] K. Hasselmann.  On the non-linear energy transfer in a gravity-wave spectrum Part 1\. General theory.  J. Fluid Mech., 12(4):481–500, 1961.
  * [23] K. Hasselmann.  A criterion for nonlinear wave stability.  J. Fluid Mech., 30(4):737–739, 1967.
  * [24] D. M. Henderson and J. L. Hammack.  Experiments on ripple instabilities. Part 1. Resonant triads.  J. Fluid Mech., 184:15–41, 1987.
  * [25] D. M. Henderson and J. W. Miles.  Faraday waves in 2:1 internal resonance.  J. Fluid Mech., 222:449–470, 1991.
  * [26] T. M. Joyce.  Nonlinear interactions among standing surface and internal gravity waves.  J. Fluid Mech., 63(4):801–825, 1974.
  * [27] U. Kadri and T. R. Akylas.  On resonant triad interactions of acoustic-gravity waves.  J. Fluid Mech., 788:R1, 2016.
  * [28] U. Kadri and M. Stiassnie.  Generation of an acoustic-gravity wave by two gravity waves, and their subsequent mutual interaction.  J. Fluid Mech., 735:R6, 2013.
  * [29] J. K. Kevorkian and J. D. Cole.  Multiple Scale and Singular Perturbation Methods, volume 114 of Appl. Math. Sci. Springer, New York, 1996.
  * [30] E. Kreyszig.  Introductory Functional Analysis with Applications.  John Wiley & Sons, 1989.
  * [31] K. Kumar.  Linear theory of Faraday instability in viscous fluids.  Proc. R. Soc. Lond. A, 452:1113–1126, 1996.
  * [32] K. Kumar and L. S. Tuckerman.  Parametric instability of the interface between two fluids.  J. Fluid Mech., 279:49–68, 1994.
  * [33] H. Lamb.  Hydrodynamics (6th ed.).  Dover Publications, 1932.
  * [34] M. S. Longuet-Higgins.  Resonant interactions between two trains of gravity waves.  J. Fluid Mech., 12(3):321–332, 1962.
  * [35] L. R. Mack.  Periodic, finite-amplitude, axisymmetric gravity waves.  J. Geophys. Res., 67(2):829–843, 1962.
  * [36] A. McEwan, D. Mander, and R. Smith.  Forced resonant second-order interaction between damped internal waves.  J. Fluid Mech., 55(4):589–608, 1972.
  * [37] L. F. McGoldrick.  Resonant interactions among capillary-gravity waves.  J. Fluid Mech., 21(2):305–331, 1965.
  * [38] L. F. McGoldrick.  An experiment on second-order capillary gravity resonant wave interactions.  J. Fluid Mech., 40(2):251–271, 1970.
  * [39] L. F. McGoldrick.  On Wilton’s ripples: a special case of resonant interactions.  J. Fluid Mech., 42(1):193–200, 1970.
  * [40] L. F. McGoldrick.  On the rippling of small waves: a harmonic nonlinear nearly resonant interaction.  J. Fluid Mech., 52(4):725–751, 1972.
  * [41] G. Michel.  Three-wave interactions among surface gravity waves in a cylindrical container.  Phys. Rev. Fluids, 4:012801(R), 2019.
  * [42] J. W. Miles.  Surface-wave damping in closed basins.  Proc. Roy. Soc. A, 297(1451):459–475, 1967.
  * [43] J. W. Miles.  Nonlinear surface waves in closed basins.  J. Fluid Mech., 75(3):419–448, 1976.
  * [44] J. W. Miles.  Internally resonant surface waves in a circular cylinder.  J. Fluid Mech., 149:1–14, 1984.
  * [45] J. W. Miles.  Nonlinear Faraday resonance.  J. Fluid Mech., 146:285–302, 1984.
  * [46] J. W. Miles.  Resonantly forced surface waves in a circular cylinder.  J. Fluid Mech., 149:15–31, 1984.
  * [47] J. W. Miles and D. M. Henderson.  Parametrically forced surface waves.  Annu. Rev. Fluid Mech., 22:143–165, 1990.
  * [48] P. A. Milewski and J. B. Keller.  Three-Dimensional Water Waves.  Stud. Appl. Math., 97:149–166, 1996.
  * [49] O. M. Phillips.  On the dynamics of unsteady gravity waves of finite amplitude Part 1. The elementary interactions.  J. Fluid Mech., 9(2):193–217, 1960.
  * [50] S. Qadeer and J. A. Wilkening.  Computing the Dirichlet–Neumann Operator on a Cylinder.  SIAM J. Numer. Anal., 57(3):1183–1204, 2019.
  * [51] C. F. M. Raupp and P. L. Silva Dias.  Resonant Wave Interactions in the Presence of a Diurnally Varying Heat Source.  J. Atmos. Sci., 66(10):3165–3183, 2009.
  * [52] C. F. M. Raupp, P. L. Silva Dias, E. G. Tabak, and P. A. Milewski.  Resonant wave interactions in the equatorial waveguide.  J. Atmos. Sci., 65(11):3398–3418, 2008.
  * [53] L. Schwartz and J.-M. Vanden-Broeck.  Numerical solution of the exact equations for capillary–gravity waves.  J. Fluid Mech., 95(1):119–139, 1979.
  * [54] H. Segur.  Resonant interactions of surface and internal gravity waves.  Phys. Fluids, 23:2556, 1980.
  * [55] W. F. Simmons.  A variational method for weak resonant wave interactions.  Proc. Roy. Soc. Lond. A, 309(1499):551–577, 1969.
  * [56] S. H. Strogatz.  Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry and Engineering (2nd ed.).  CRC Press, 2015.
  * [57] T. M. A. Taklo and W. Choi.  Group resonant interactions between surface and internal gravity waves in a two-layer system.  J. Fluid Mech., 892:A14, 2020.
  * [58] Z. Wang and P. A. Milewski.  Dynamics of gravity-capillary solitary waves in deep water.  J. Fluid Mech., 708:480–501, 2012.
  * [59] Z. Wang, J.-M. Vanden-Broeck, and P. A. Milewski.  Two-dimensional flexural–gravity waves of finite amplitude in deep water.  IMA J. App. Math., 78(4):750–761, 2013.
  * [60] G. B. Whitham.  A general approach to linear and non-linear dispersive waves using a Lagrangian.  J. Fluid Mech., 22(2):273–283, 1965.
  * [61] G. B. Whitham.  Non-Linear Dispersive Waves.  Proc. Roy. Soc. Lond. A, 283(1393):238–261, 1965.
  * [62] G. B. Whitham.  Non-linear dispersion of water waves.  J. Fluid Mech., 27(2):399–412, 1967.
  * [63] G. B. Whitham.  Variational methods and applications to water waves.  Proc. Roy. Soc. Lond. A, 299:6–25, 1967.
  * [64] J. R. Wilton.  On ripples.  Phil. Mag. Ser. 6, 29(173):688–700, 1915.
  * [65] X. Yang, F. Dias, Z. Liu, and S. Liao.  Finite-amplitude steady-state resonant waves in a circular basin.  J. Fluid Mech., 915:A136, 2021.