eurm10 msam10 1–11

# Stability of Fully Nonlinear Stokes Waves on Deep Water: Part 1.
Perturbation Theory

SHAHRDADG.SAJJADI  DAVIDL.ROSS  Division of Geophysics and Planetary Physics,
CAMAC, Department of Applied Mathematics,
Embry-Riddle Aeronautical University,
Florida 32114, USA.

(2009; 22 May 2009)

###### Abstract

We consider a full set of harmonics for the Stokes wave in deep water in the
absence of viscosity, and examine the role that higher harmonics play in
modifying the classical Benjamin-Feir instability. Using a representation of
the wave coefficients due to Wilton, a perturbation analysis shows that the
Stokes wave may become unbounded due to interactions between the $N^{th}$
harmonic of the primary wave train and a set of harmonics of a disturbance. If
the frequency of the $n^{th}$ harmonic is denoted
$\omega_{n}=\omega\left({1\pm\delta}\right)$ then instability will occur if

$\displaystyle 0<\delta<\frac{\sqrt{2}\ k\,n^{n}s_{n}}{\left({n-1}\right)!}$
(1)

subject to the disturbance initially having sufficiently large amplitude. We
show that, subject to initial conditions, all lower harmonics will contribute
to instability as well, and we identify the frequency of the disturbance
corresponding to maximum growth rate.

††volume: 538

##  1 Introduction

In Benjamin-Feir’s seminal work identifying criteria for the instability of
gravity waves on deep water [1], a bound was established on the frequency of
the disturbance as a function of wave steepness. This result was obtained
through a perturbation analysis, with the wave train represented by a second
order Stokes wave of the form

$\eta\left({x,t}\right)=a\cos\left({kx-\omega
t}\right)+\textstyle\frac{1}{2}a^{2}k\cos\left[{2(kx-\omega t)}\right]$

A disturbance was introduced in the form of two sidebands with slightly
different wave numbers and amplitude varying in time,

$\displaystyle\tilde{\eta}_{1,2}\left({x,t}\right)$ $\displaystyle=$
$\displaystyle\varepsilon_{1,2}\cos\left[{k(1\pm\kappa)x-\omega\left({1+\delta}\right)t+\gamma_{1,2}(t)}\right]$
$\displaystyle+$
$\displaystyle\varepsilon_{1,2}ak\cos\left[{kx+k(1\pm\kappa)x-\omega
t-\omega\left({1+\delta}\right)t+\gamma_{1,2}\left(t\right)}\right]$

with the positive sign corresponding to the index $i=1$ and the negative sign
to $i=2$. The phase functions $\gamma_{1}\left(t\right)$ and
$\gamma_{2}\left(t\right)$ were necessary to insure that the dispersion
relation holds to second order for the sidebands.

Benjamin and Feir demonstrated that interactions between the disturbance and
the first harmonic of the primary wave can lead to resonance effects so long
as $0<\delta<\sqrt{2}\,ak$, (i.e., the frequency of the disturbance lies
sufficiently close to the fundamental frequency of the wave train). This
result followed by obtaining a set of three coupled ordinary differential
equations in $\varepsilon_{1}\left(t\right)$, $\varepsilon_{2}\left(t\right)$,
and $\theta\left(t\right)=\gamma_{1}\left(t\right)+\gamma_{2}\left(t\right)$
and demonstrating that the amplitudes $\varepsilon_{1}\left(t\right)$ and
$\varepsilon_{2}\left(t\right)$ may become unbounded if $\delta$ is in the
stated range.

In this work, we consider the fully nonlinear Stokes wave

$\eta\left({x,t}\right)=H=\sum\limits_{n=1}^{\infty}{a_{n}\cos\left({k_{n}x-\omega_{n}t}\right)=}\sum\limits_{n=1}^{\infty}{a_{n}\cos\zeta_{n}}$

and a disturbance represented by the infinite sum

$\tilde{\eta}_{i}=\sum\limits_{n=1}^{\infty}{\varepsilon_{n,i}\cos\zeta_{n,i}+\sum\limits_{n=1}^{\infty}{\varepsilon_{n,i}a_{n}\cos\left({\zeta_{n}+\zeta_{n,i}}\right)\,\,\,}}.$

Out of all possible interactions between the disturbance modes and the primary
wavetrain, we show that critical mode interactions with the primary wavetrain
may be identified for every harmonic of the disturbance. Characterizing each
of these interactions leads to a coupled system of equations quite similar to
those considered by Benjamin and Feir. This leads to a range of frequencies
for the $n^{th}$harmonic of the sidebands where contributions to instability
may occur if $0<\delta<\sqrt{2}kn^{n}s_{n}/(n-1)!$, where $s_{n}$ is a
parameter related to wave steepness ak. The value of $\delta$ corresponding to
marginal stability decreases with $n$, so that if instability corresponding to
a particular harmonic is identified, all lower harmonics will contribute to
instability as well, providing that certain initial conditions are satisfied.
The initial conditions require
$\left|{\varepsilon_{n,i}\left({t_{0}}\right)}\right|$ to be sufficiently
large at some fixed time $t_{0}$.

##  2 Formulation of the problem

We consider a two-dimensional Stokes wave on water of finite depth, modeled as
an inviscid fluid, where $z=0$ represents the mean surface level, and
$z=\eta\left({x,t}\right)$ the free surface. Then Laplace’s equation governs
the motion, with nonlinearities captured through the kinematic and dynamic
conditions at the surface. We seek the velocity potential
$\phi\left({x,z,t}\right)$ and $\eta\left({x,t}\right)$ satisfying

$\displaystyle\begin{array}[]{l}\phi_{xx}+\phi_{zz}=0\\\
\left.{\begin{array}[]{l}\phi_{z}=\eta_{t}+\varepsilon\phi_{x}\eta_{x}\\\
\phi_{t}+\eta+\frac{1}{2}\varepsilon\left({\phi_{x}^{2}+\phi_{z}^{2}}\right)=0\\\
\end{array}}\right\\}{\rm on}\,\,\,z=1+\varepsilon\eta\\\ \nabla\phi\to
0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{\rm
as}\,\,\,\,z\to-\infty\\\ \end{array}$ (7)

We represent the free surface as

$\eta\left({x,t}\right)=H=\sum\limits_{n=1}^{\infty}{a_{n}\cos\left({k_{n}x-\omega_{n}t}\right)=}\sum\limits_{n=1}^{\infty}{a_{n}\cos\zeta_{n}}$

And the potential as

$\phi\left({x,y,t}\right)=\Phi=\sum\limits_{n=1}^{\infty}{\omega_{n}k_{n}^{-1}a_{n}e^{k_{n}z}\sin\zeta_{n}}$

where $\omega_{n}^{2}=gk_{n}$.

We introduce perturbations,

$\eta=H+\varepsilon\tilde{\eta}\quad\phi=\Phi+\varepsilon\tilde{\phi}$ (8)

and seek conditions that allow the disturbance $\tilde{\eta}$ to exhibit
unbounded growth. By linearity of Laplace’s equation, we have

$\nabla^{2}\tilde{\phi}=\tilde{\phi}_{xx}+\tilde{\phi}_{yy}=0$ (9)

$\nabla\tilde{\phi}\to 0$ as $z\to-\infty$.

The kinematic and dynamic boundary conditions may be written as

$\eta_{t}+\eta_{x}\left({\phi_{x}}\right)_{z=\eta}-\left({\phi_{z}}\right)_{z=\eta}=0$
(10)
$g\eta+\left({\phi_{t}}\right)_{z=\eta}+\textstyle\frac{1}{2}\left({\phi_{x}^{2}+\phi_{z}^{2}}\right)_{z=\eta}=0$
(11)

We conduct a standard perturbation analysis by substituting (8) into (10) and
(11), setting $O\left(\varepsilon\right)$ terms to zero, and expanding the
resulting expressions about $z=H$. Then,

$\tilde{\eta}_{t}+\tilde{\eta}_{x}\left({\Phi_{x}}\right)_{z=H}+\tilde{\eta}\left({-\Phi_{zz}+H_{x}\Phi_{xz}}\right)_{z=H}+\left({-\tilde{\phi}_{z}+H_{x}\phi_{x}}\right)_{z=H}=0$
(12)
$g\tilde{\eta}+\tilde{\eta}\left({\Phi_{x}\Phi_{xz}+\Phi_{z}\Phi_{zz}+\Phi_{tz}}\right)_{z=H}+\left({\tilde{\phi}_{t}+\Phi_{x}\tilde{\phi}_{x}+\Phi_{z}\tilde{\phi}_{z}}\right)_{z=H}=0$
(13)

Consider the analytic continuation of $\tilde{\phi}$ over the neighborhood of
the free surface, neglect terms greater than $O\left({a_{n}^{2}}\right)$, and
Taylor expand all expressions about the mean surface level $z=0$. Upon
substituting for $\Phi$ and $H,$ the kinematic condition (12) may be written

$\begin{array}[]{l}\tilde{\eta}_{t}+\tilde{\eta}_{x}\left({\sum\limits_{n=1}^{\infty}{\omega_{n}a_{n}\cos\zeta_{n}}+\sum\limits_{n=1}^{\infty}{\sum\limits_{m=1}^{\infty}{a_{n}a_{m}\omega_{n}k_{n}\cos\zeta_{n}\cos\zeta_{m}}}}\right)\\\
\,\,\,\,\,\,\,+\tilde{\eta}\left({-\sum\limits_{n=1}^{\infty}{\omega_{n}k_{n}a_{n}\sin\zeta_{n}}-\sum\limits_{n=1}^{\infty}{\sum\limits_{m=1}^{\infty}{a_{n}a_{m}k_{n}k_{m}\omega_{m}\sin\zeta_{n}\cos\zeta_{m}}}}\right)\\\
\,\,\,\,\,\,\,-\sum\limits_{n=1}^{\infty}{\sum\limits_{m=1}^{\infty}{a_{n}a_{m}\omega_{m}k_{m}^{2}\cos\zeta_{n}\sin\zeta_{m}}}\\\
\,\,\,\,\,\,\,-\sum\limits_{n=1}^{\infty}{\sum\limits_{m=1}^{\infty}{\sum\limits_{q=1}^{\infty}{a_{n}a_{m}a_{q}k_{m}k_{q}^{2}\omega_{q}\cos\zeta_{n}\sin\zeta_{m}\cos\zeta_{q}}}}\\\
\,\,\,\,\,\,\,-\left({\tilde{\phi}_{z}+\sum\limits_{n=1}^{\infty}{a_{n}k_{n}\sin\zeta_{n}\tilde{\phi}_{x}}}\right)_{z=0}\\\
\,\,\,\,\,\,\,-\left({\sum\limits_{n=1}^{\infty}{a_{n}\cos\zeta_{n}}\tilde{\phi}_{zz}+\sum\limits_{n=1}^{\infty}{\sum\limits_{m=1}^{\infty}{a_{n}a_{m}k_{m}\omega_{m}\cos\zeta_{n}\sin\zeta_{m}}}\tilde{\phi}_{xz}}\right)_{z=0}\\\
\,\,\,\,\,\,\,-\frac{1}{2}\left(\sum\limits_{n=1}^{\infty}{\sum\limits_{m=1}^{\infty}{a_{n}a_{m}\cos\zeta_{n}\cos\zeta_{m}}\tilde{\phi}_{zzz}}\right)_{z=0}\\\
\,\,\,\,\,\,\,-\frac{1}{2}\left(\sum\limits_{n=1}^{\infty}{\sum\limits_{m=1}^{\infty}{\sum\limits_{q=1}^{\infty}{a_{n}a_{m}a_{q}k_{q}\cos\zeta_{n}\cos\zeta_{m}\sin\zeta_{q}}}}\tilde{\phi}_{xzz}\right)_{z=0}=0\\\
\end{array}$ (14)

The dynamic condition (13) may be written

$\begin{array}[]{l}g\tilde{\eta}+\tilde{\eta}\left(\sum\limits_{n=1}^{\infty}\sum\limits_{m=1}^{\infty}{a_{n}a_{m}\omega_{n}\omega_{n}k_{m}\cos\zeta_{n}\cos\zeta_{m}}\right.\\\
\,\,\,\,\,\,\,\left.+\sum\limits_{n=1}^{\infty}{\sum\limits_{m=1}^{\infty}{a_{n}a_{m}\omega_{n}\omega_{n}k_{m}\sin\zeta_{n}\sin\zeta_{m}}-}\sum\limits_{n=1}^{\infty}{\omega_{n}^{2}a_{n}\cos\zeta_{n}}\right)\\\
\,\,\,\,\,\,\,+\sum\limits_{n=1}^{\infty}{\sum\limits_{m=1}^{\infty}{\sum\limits_{q=1}^{\infty}{a_{n}a_{m}a_{q}k_{q}\omega_{m}\omega_{q}k_{m}k_{q}\cos\zeta_{n}\cos\zeta_{m}\cos\zeta_{q}}}}\\\
\,\,\,\,\,\,\,+\sum\limits_{n=1}^{\infty}{\sum\limits_{m=1}^{\infty}{\sum\limits_{q=1}^{\infty}{a_{n}a_{m}a_{q}\omega_{m}\omega_{q}k_{q}^{2}\cos\zeta_{n}\cos\zeta_{m}\cos\zeta_{q}}}}\\\
\,\,\,\,\,\,\,+\sum\limits_{n=1}^{\infty}{\sum\limits_{m=1}^{\infty}{\sum\limits_{q=1}^{\infty}{a_{n}a_{m}a_{q}\omega_{m}\omega_{q}k_{m}k_{q}\cos\zeta_{n}\sin\zeta_{m}\sin\zeta_{q}}}}\\\
\,\,\,\,\,\,\,+\sum\limits_{n=1}^{\infty}{\sum\limits_{m=1}^{\infty}{\sum\limits_{q=1}^{\infty}{a_{n}a_{m}a_{q}k_{q}\omega_{m}\omega_{q}k_{q}^{2}\cos\zeta_{n}\cos\zeta_{m}\sin\zeta_{q}}}}\\\
\,\,\,\,\,\,\,+\left\\{{\tilde{\phi}_{t}+\sum\limits_{n=1}^{\infty}{\sum\limits_{m=1}^{\infty}{a_{n}a_{m}k_{m}\omega_{m}^{2}k_{m}\cos\zeta_{n}\cos\zeta_{m}}}+\sum\limits_{n=1}^{\infty}{a_{n}\omega_{n}\cos\zeta_{n}\tilde{\phi}_{x}}}\right.\\\
\,\,\,\,\,\,\,+\sum\limits_{n=1}^{\infty}{a_{n}\omega_{n}sn\zeta_{n}\tilde{\phi}_{z}}+\sum\limits_{n=1}^{\infty}{a_{n}\cos\zeta_{n}\left({\tilde{\phi}_{tz}+O\left({a_{n}}\right)}\right)}\\\
\,\,\,\,\,\,\,+\left.{\frac{1}{2}\sum\limits_{n=1}^{\infty}{\sum\limits_{m=1}^{\infty}{a_{n}a_{m}\cos\zeta_{n}\cos\zeta_{m}}\left({\tilde{\phi}_{tzz}+O\left({a_{n}}\right)}\right)}}\right\\}_{z=0}=0\\\
\end{array}$ (15)

Neglecting cross terms in (14) and (15) we obtain for the kinematic condition,

$\begin{array}[]{l}\tilde{\eta}_{t}+\tilde{\eta}_{x}\sum\limits_{n=1}^{\infty}{\omega_{n}a_{n}\cos\zeta_{n}}-\tilde{\eta}\sum\limits_{n=1}^{\infty}{\omega_{n}k_{n}a_{n}\sin\zeta_{n}}\\\
\,\,\,\,\,\,\,-\left({\tilde{\phi}_{z}+\tilde{\phi}_{x}\sum\limits_{n=1}^{\infty}{a_{n}k_{n}\sin\zeta_{n}}+\tilde{\phi}_{zz}\sum\limits_{n=1}^{\infty}{a_{n}\cos\zeta_{n}}}\right)_{z=0}=0\\\
\\\ \end{array}$ (16)

and for the dynamic condition,

$\begin{array}[]{l}g\tilde{\eta}-\tilde{\eta}\sum\limits_{n=1}^{\infty}{\omega_{n}^{2}a_{n}\cos\zeta_{n}}\\\
\,\,\,\,\,\,\,+\left\\{{\tilde{\phi}_{t}+\tilde{\phi}_{x}\sum\limits_{n=1}^{\infty}{a_{n}\omega_{n}\cos\zeta_{n}}\,+\tilde{\phi}_{z}\sum\limits_{n=1}^{\infty}{a_{n}\omega_{n}\sin\zeta_{n}}+\tilde{\phi}_{tz}\sum\limits_{n=1}^{\infty}{a_{n}\cos\zeta_{n}}}\right\\}_{z=0}=0\\\
\end{array}$ (17)

Equations (16) and (17) generalize the boundary conditions obtained by
Benjamin and Feir [1] insofar as terms corresponding to $n=1$ produce the
$O\left(a\right)$ terms they obtained by a similar perturbation analysis. In
that work, the $O\left(a\right)$ terms were used to specify the form of the
disturbance, while $O\left({a^{2}}\right)$ terms were used to produce coupled
equations for the amplitude of the sideband modes. Neglecting cross terms in
(14) and (15) eliminated terms which would correspond to these
$O\left({a^{2}}\right)$ terms in Benjamin and Feir’s work.

##  3 Conditions for instability due to a particular harmonic

While equations (16) and (17) were obtained by ignoring certain cross terms,
the form of the equations generalized the $O\left(a\right)$ terms obtained by
Benjamin and Feir [1]. We now retain precisely the collection of cross-terms
in (14) and (15) to produce $O\left({a_{n}^{2}}\right)$ terms that will
generalize the $O\left({a^{2}}\right)$ in that work. In this way, the
kinematic condition may be represented as

$\begin{array}[]{l}\tilde{\eta}_{t}-\left({\tilde{\phi}}\right)_{z=0}=\sum\limits_{n=1}^{\infty}{a_{n}\left\\{{k_{n}\omega_{n}\sin\zeta_{n}\,\tilde{\eta}-\omega_{n}\cos\zeta_{n}\,\tilde{\eta}_{x}+\left({k_{n}\sin\zeta_{n}\tilde{\phi}_{x}+\cos\zeta_{n}\tilde{\phi}_{zz}}\right)_{z=0}}\right\\}}\\\
\,\,\,\,\,\,\,+\frac{1}{2}\sum\limits_{n=1}^{\infty}{a_{n}^{2}\left({2k_{n}^{2}\omega_{n}\sin
2\zeta_{n}\,\tilde{\eta}-k_{n}\omega_{n}\left({1+2\cos
2\zeta_{n}}\right)\tilde{\eta}_{x}}\right)}\\\
\,\,\,\,\,\,\,+\frac{1}{2}\sum\limits_{n=1}^{\infty}{a_{n}^{2}}\left\\{{k_{n}\sin
2\zeta_{n}\left({2k_{n}\tilde{\phi}_{x}+\tilde{\phi}_{xz}}\right)+k_{n}\cos
2\zeta_{n}\tilde{\phi}_{zz}+\frac{1}{2}\left({1+\cos
2\zeta_{n}}\right)\tilde{\phi}_{zzz}}\right\\}_{z=0}\\\ \end{array}$ (18)

and the dynamic condition represented as

$\begin{array}[]{l}g\tilde{\eta}+\left({\tilde{\phi}_{t}}\right)_{z=0}=\sum\limits_{n=1}^{\infty}{a_{n}\left({\omega_{n}^{2}\cos\zeta_{n}\,\tilde{\eta}-\left({\omega_{n}\cos\zeta_{n}\,\tilde{\phi}_{x}+\omega_{n}\sin\zeta_{n}\,\tilde{\phi}_{z}+\cos\zeta_{n}\,\tilde{\phi}_{tz}}\right)_{z=0}}\right)}\\\
\,\,\,\,\,\,\,+\frac{1}{2}\sum\limits_{n=1}^{\infty}{a_{n}^{2}\left({k_{n}\omega_{n}^{2}\left({1-\cos
2\zeta_{n}}\right)\,\tilde{\eta}+\left({\omega_{n}\sin
2\zeta_{n}\left({k_{n}\tilde{\phi}_{z}+\tilde{\phi}_{zz}}\right)_{z=0}}\right)}\right)}\\\
\,\,\,\,\,\,\,+\frac{1}{2}\sum\limits_{n=1}^{\infty}{a_{n}^{2}\left({\left({1+\cos
2\zeta_{n}}\right)\left({k_{n}\omega_{n}\tilde{\phi}_{x}+\omega_{n}\tilde{\phi}_{xz}+\frac{1}{2}\tilde{\phi}_{zzt}}\right)+k_{n}\cos
2\zeta_{n}\,\tilde{\phi}_{zt}}\right)_{z=0}}\\\ \end{array}$ (19)

Now, assume that the disturbance consists of two sideband modes together with
the products of their interaction with the basic wave train,

$\tilde{\eta}=\tilde{\eta}_{1}+\tilde{\eta}_{2}\quad,\quad\tilde{\phi}=\tilde{\phi}_{1}+\tilde{\phi}_{2}.$

We define the arguments

$\zeta_{n,1}=k_{n}\left({1+\kappa}\right)x-\omega_{n}\left({1+\delta}\right)t-\gamma_{n,1}\left(t\right)$
(20)
$\zeta_{n,2}=k_{n}\left({1-\kappa}\right)x-\omega_{n}\left({1-\delta}\right)t-\gamma_{n,2}\left(t\right),$
(21)

where $\gamma_{n,1}\left(t\right)$ and $\gamma_{n,2}\left(t\right)$ are
required so that each harmonic satisfies the dispersion relation to
$O\left({\delta^{2}}\right)$. We denote frequencies and wave numbers for the
sidebands by

$\displaystyle\omega_{n,1}=\omega\left({1+\delta}\right)$ (22)
$\displaystyle\omega_{n,2}=\omega\left({1-\delta}\right),\quad$ (23)

and

$k_{n,1}=k_{n}\left({1+\kappa}\right)$ (24)
$k_{n,2}=k_{n}\left({1-\kappa}\right)$ (25)

where $\kappa$ and $\delta$ are small constants.

Then, for $i=1,2$, we define

$\tilde{\eta}_{i}=\sum\limits_{n=1}^{\infty}{\varepsilon_{n,i}\cos\zeta_{n,i}}+\sum\limits_{n=1}^{\infty}{\varepsilon_{n,i}k_{n}a_{n}\cos\left({\zeta_{n}+\zeta_{n,i}}\right)+O\left({k_{n}^{2}a_{n}^{2}\varepsilon_{n,i}}\right)}$
(26)

and

$\displaystyle\tilde{\phi}_{i}$ $\displaystyle=$
$\displaystyle\sum\limits_{n=1}^{\infty}{k_{n,i}^{-1}e^{k_{n,i}z}\left\\{{\varepsilon_{n,i}\left({\omega_{n,i}L_{n,i}+\dot{\gamma}_{n,i}M_{n,i}}\right)\sin\zeta_{n,i}+\dot{\varepsilon}_{n,i}N_{n,i}\cos\zeta_{n,i}}\right\\}}$
(27) $\displaystyle+$
$\displaystyle\sum\limits_{n=1}^{\infty}{\omega_{n}a_{n}\varepsilon_{n,i}D_{n,i}e^{\left|{k_{n}-k_{n,i}}\right|z}\sin\left({\zeta_{n}-\zeta_{n,i}}\right)}$

We further assume that $\varepsilon_{n,i}$and $\gamma_{n,i}$ are slowly
varying functions of time, such that their derivatives have the properties

$\dot{\varepsilon}_{n,i}=O\left({w_{n}k_{n}^{2}a_{n}^{2}\varepsilon_{n,i}}\right),\quad\quad\dot{\gamma}_{n,i}=O\left({w_{n}k_{n}^{2}a_{n}^{2}}\right).$
(28)

Now, among the infinitely many products arising from the nonlinear interaction
between these disturbance modes and the basic wave train, there will be
components with arguments

$\left.\begin{array}[]{l}2\zeta_{n}-\zeta_{n,1}=\zeta_{n,2}+\left({\gamma_{n,1}+\gamma_{n,2}}\right)\\\
2\zeta_{n}-\zeta_{n,2}=\zeta_{n,1}+\left({\gamma_{n,1}+\gamma_{n,2}}\right)\\\
\end{array}\right\\}$ (29)

suggesting that resonance may be induced between the sidebands by the
interaction of particular harmonics. This may contribute to instability if the
sum of the time-dependent phase functions approaches a constant,

$\displaystyle\theta_{n}=\gamma_{n,1}+\gamma_{n,2}\to\mbox{const. as}\quad
t\to\infty$ (30)

Indeed, upon substitution of (26) and (27) into (16) and (17), a variety of
cross terms involving interaction between the primary wave and the disturbance
may be identified so that (29) applies. For example,

$\sin
2\zeta_{n}\cos\zeta_{n,1}=\textstyle\frac{1}{2}\left[{\sin\left({2\zeta_{n}-\zeta_{n,1}}\right)+\sin\left({2\zeta_{n}+\zeta_{n,1}}\right)}\right]\sim\textstyle\frac{1}{2}\sin\left({\zeta_{n,2}+\theta_{n}}\right)$

and similarly

$\sin
2\zeta_{n}\cos\zeta_{n,2}=\textstyle\frac{1}{2}\left[{\sin\left({2\zeta_{n}-\zeta_{n,2}}\right)+\sin\left({2\zeta_{n}+\zeta_{n,2}}\right)}\right]\sim\textstyle\frac{1}{2}\sin\left({\zeta_{n,1}+\theta_{n}}\right).$

Upon substitution, there exists a set of interactions so that, after equating
coefficients, the kinematic boundary condition reduces to the pair of
equations

$\begin{array}[]{l}\varepsilon_{n,1}\left\\{{\omega_{n,1}\left({1-L_{n,1}}\right)+\dot{\gamma}_{n,1}\left({1-M_{n,1}}\right)}\right\\}\sin\zeta_{n,1}+\dot{\varepsilon}_{n,1}\left({1-N_{n,1}}\right)\cos\zeta_{n,1}\\\
\,\,\,\,\,\,\,=\omega_{n}k_{n}^{2}a_{n}^{2}\left\\{{\frac{5}{4}\varepsilon_{n,1}\sin\zeta_{n,1}+\frac{5}{8}\varepsilon_{n,2}\sin\left({\zeta_{n,1}+\theta_{n}}\right)}\right\\}\\\
\end{array}$ (31)
$\begin{array}[]{l}\varepsilon_{n,2}\left\\{{\omega_{n,2}\left({1-L_{n,2}}\right)+\dot{\gamma}_{n,2}\left({1-M_{n,2}}\right)}\right\\}\sin\zeta_{n,2}+\dot{\varepsilon}_{n,2}\left({1-N_{n,2}}\right)\cos\zeta_{n,2}\\\
\,\,\,\,\,\,\,=\omega_{n}k_{n}^{2}a_{n}^{2}\left\\{{\frac{5}{4}\varepsilon_{n,2}\sin\zeta_{n,2}+\frac{5}{8}\varepsilon_{n,1}\sin\left({\zeta_{n,2}+\theta_{n}}\right)}\right\\}\\\
\end{array}.$ (32)

for $n=1,2,3,...$. Furthermore, there exists a set of interactions so that the
dynamic boundary condition reduces to the pair of equations

$\begin{array}[]{l}\varepsilon_{n,1}\left\\{{\omega_{n,1}^{-1}\left({gk_{n,1}-\omega_{n,1}^{2}L_{n,1}}\right)-\dot{\gamma}_{n,1}\left({1+M_{n,1}}\right)}\right\\}\cos\zeta_{n,1}+\dot{\varepsilon}_{n,1}\left({1+N_{n,1}}\right)\sin\zeta_{n,1}\\\
\,\,\,\,\,\,\,=-\omega
k_{n}^{2}a_{n}^{2}\left\\{{\frac{3}{4}\varepsilon_{n,1}\cos\zeta_{n,1}+\frac{3}{8}\varepsilon_{n,2}\cos\left({\zeta_{n,1}+\theta_{n}}\right)}\right\\}\\\
\end{array}$ (33)

and

$\begin{array}[]{l}\varepsilon_{n,2}\left\\{{\omega_{n,2}^{-1}\left({gk_{n,2}-\omega_{n,2}^{2}L_{n,2}}\right)-\dot{\gamma}_{n,2}\left({1+M_{n,2}}\right)}\right\\}\cos\zeta_{n,2}+\dot{\varepsilon}_{n,2}\left({1+N_{n,2}}\right)\sin\zeta_{n,2}\\\
\,\,\,\,\,\,\,=-\omega
k_{n}^{2}a_{n}^{2}\left\\{{\frac{3}{4}\varepsilon_{n,2}\cos\zeta_{n,2}+\frac{3}{8}\varepsilon_{n,1}\cos\left({\zeta_{n,2}+\theta_{n}}\right)}\right\\}\\\
\end{array}$ (34)

for $n=1,2,3,...$. By adding the coefficients of $\cos\zeta_{n,i}$ in (22) and
coefficients of $\sin\zeta_{n,i}$in (23); then, adding coefficients of
$\sin\zeta_{n,i}$ in (22) and $\cos\zeta_{n,i}$ in (23), we obtain a system of
four equations,

$\left.{\begin{array}[]{l}\displaystyle\frac{d\varepsilon_{n,1}}{dt}=\left({\frac{1}{2}\omega
k_{n}^{2}a_{n}^{2}\sin\theta_{n}}\right)\varepsilon_{n,2}\\\ \\\
\displaystyle\frac{d\varepsilon_{n,2}}{dt}=\left({\frac{1}{2}\omega
k_{n}^{2}a_{n}^{2}\sin\theta_{n}}\right)\varepsilon_{n,1}\\\ \\\
\displaystyle\frac{d\gamma_{n,1}}{dt}=\frac{1}{2}\left({\frac{gk_{n,1}}{\omega_{n,1}}-\omega_{n,1}}\right)+\omega_{n,1}k_{n,1}^{2}a_{n}^{2}\left({1+\frac{\varepsilon_{n,2}}{2\varepsilon_{n,1}}\cos\theta_{n}}\right)\\\
\\\
\displaystyle\frac{d\gamma_{n,2}}{dt}=\frac{1}{2}\left({\frac{gk_{n,2}}{\omega_{n,2}}-\omega_{n,2}}\right)+\omega_{n,2}k_{n,2}^{2}a_{n}^{2}\left({1+\frac{\varepsilon_{n,1}}{2\varepsilon_{n,2}}\cos\theta_{n}}\right)\\\
\end{array}}\right\\}$ (35)

The last two equations of (35) may be added to give an equation for
$\theta_{n}=\gamma_{n,1}+\gamma_{n,2}$,

$\frac{d\theta_{n}}{dt}=\omega_{n}k_{n}^{2}a_{n}^{2}\left\\{{1+\frac{\varepsilon_{n,1}^{2}+\varepsilon_{n,2}^{2}}{2\varepsilon_{n,1}\varepsilon_{n,2}}\cos\theta_{n}}\right\\}-\omega_{n}\delta^{2}$
(36)

The resulting three equations in $\varepsilon_{n,1}$, $\varepsilon_{n,2}$, and
$\theta_{n}$ may be reduced to a single equation in $\varepsilon_{n,1}$ by
introducing the parameters

$T_{n}=k_{n}^{2}a_{n}^{2}\omega t$ (37)

and

$\alpha_{n}=1-\frac{\delta^{2}}{k_{n}^{2}a_{n}^{2}}.$ (38)

It may then be shown [1],

$\frac{d\varepsilon_{n,1}^{{}^{2}}}{dT_{n}}=\frac{d\varepsilon_{n,2}^{2}}{dT_{n}}=\varepsilon_{n,1}\varepsilon_{n,2}\sin\theta_{n}\quad,$
(39)
$\varepsilon_{n,1}\varepsilon_{n,2}\cos\theta_{n}+\alpha_{n}\varepsilon_{n,1}^{2}=\rho_{n}=const,$
(40)

and

$\varepsilon_{n,1}^{2}-\varepsilon_{n,2}^{2}=2\alpha_{n}\rho_{n}\left({1-v_{n}}\right)=const.$
(41)

We then obtain

$\left({\frac{d\varepsilon_{n,1}^{2}}{dT_{n}}}\right)^{2}=\left({1-\alpha_{n}^{2}}\right)\varepsilon_{{}_{n,1}}^{4}+2\alpha_{n}v_{n}\rho_{n}\varepsilon_{{}_{n,1}}^{2}-\rho_{n}^{2}.$
(42)

Equation (42) shows that the growth rate of the sidebands is expressible as a
quadratic in $\varepsilon_{n,1}^{2}$. We first consider the case
$\alpha_{n}^{2}=1$, when (42) reduces to a linear function in
$\varepsilon_{n,1}^{2}$. Since (38) implies $\alpha_{n}<1$, we consider
$\alpha_{n}=-1$. Since (42) must be satisfied for arbitrary initial values of
${d\varepsilon_{n,1}}/{dT_{n}}$ and $\varepsilon_{n,1}^{2}$, it can be shown
that $\rho_{n}v_{n}<0$ so that (42) may be solved directly, with
$\varepsilon_{n,1}^{2}\sim T_{n}$ so that this case must correspond to
instability.  We now suppose $\alpha_{n}^{2}\neq 1$ and consider four cases
for the form of the quadratic function (42), as shown in Figure 1.

Figure 1: Growth rate of sidebands
correspo$e_{1}e_{2}\left({\frac{d\varepsilon_{n,1}^{2}}{dt}}\right)^{2}$nding
to $n^{th}$ harmonic.

In the case $1-\alpha_{{}_{n}}^{2}<0$, unbounded sideband growth cannot occur,
as there exists only a finite interval $e_{1}\leq\varepsilon_{n,1}^{2}\leq
e_{2}$ where the growth rate
$\left({{d\varepsilon_{n,1}^{2}}/{dT_{n}}}\right)^{2}>0$. If the initial
conditions at time $T_{n}=T_{n,0}$ place
$\varepsilon_{n,1}^{2}\left({T_{n,0}}\right)$ within this interval, then the
amplitude $\varepsilon_{n,1}^{2}\left({T_{n}}\right)$ must be greater at a
slightly greater time $T$. Then $\varepsilon_{n,1}^{2}\left({T_{n}}\right)$
will increase until it reaches the value $e_{2}$, when
$\left({{d\varepsilon_{n,1}^{2}}/{dT_{n}}}\right)^{2}=0$ and further growth is
suppressed. If the initial conditions at $T_{n}=T_{n,0}$ place
$\varepsilon_{n,1}^{2}\left({T_{n,0}}\right)$ outside this interval, then the
amplitude $\varepsilon_{n,1}^{2}\left({T_{n}}\right)$ will decrease at
slightly greater time $T$, and unbounded growth will not occur.

We therefore consider the case $1-\alpha_{{}_{n}}^{2}\geq 0$ which, from (38)
corresponds to

$0\leq\delta<\sqrt{2}k_{n}a_{n},$ (43)

providing the frequency range for the classical Benjamin-Feir instability in
the case $n=1$, when $k_{1}=k$ and $a_{1}=a$. If a higher harmonic satisfies
(43), then (42) shows there exists an infinite interval
$e_{1}<\varepsilon_{n,1}^{2}$ where the growth rate
$\left({{d\varepsilon_{n,1}^{2}}/{dT_{n}}}\right)^{2}>0$. For each harmonic,
if $1-\alpha_{n}^{2}>0$, three cases arise depending on the number of
intercepts on the positive $\varepsilon_{n,1}^{2}$ axis (cases 2-4 in Figure
1). Arguing as in case 1, the initial condition
$\varepsilon_{n,1}^{2}\left({T_{n,0}}\right)>e_{2}$ insures unbounded growth
in cases 2 and 3, while any initial condition leads to unbounded growth in
case 4. With these conditions, the wave is unstable regardless of the sign of
$\varepsilon_{n,1}\left({T_{n,0}}\right)$, and the asymptotic growth rate of
both sidebands is equal from (41).

We can characterize the conditions leading to unbounded growth due to the
interaction between the primary wave and the $n^{th}$ harmonic of the
disturbance by noting that (42) may be written in the form

$\left({\frac{d\varepsilon_{n,1}^{2}}{dT_{n}}}\right)^{2}=\left({1-\alpha_{n}^{2}}\right)\left\\{{\left({\varepsilon_{{}_{n,1}}^{2}-A_{N}}\right)^{2}-B_{n}^{2}}\right\\},$
(44)

where

$A_{n}=-\frac{\alpha_{n}v_{n}\rho_{n}}{1-\alpha_{n}^{2}}\quad{\rm and}\quad
B_{n}=\frac{\rho_{n}\left({1-\alpha_{n}^{2}+\alpha_{n}^{2}v_{n}^{2}}\right)^{1\mathord{\left/{\vphantom{12}}\right.\kern-1.2pt}2}}{|1-\alpha_{n}^{2}|},$

so that in Figure 1

$e_{2}=\max\left\\{{A_{n}+B_{n},A_{n}-B_{n}}\right\\},$ (45)

and the condition for unbounded growth in cases 2,3, and 4 may be written as

$\varepsilon_{n,1}\left({T_{n,0}}\right)>e_{2}$ (46)

If (43) and (46) both hold, then unbounded growth of the $n^{th}$ harmonic of
the disturbance will occur. Furthermore, the sign of
$\varepsilon_{n,1}\left({T_{n}}\right)$ will not change from its sign at
initial time $T_{0}$, and inspection of (35) shows that
$\varepsilon_{n,1}\left({T_{n}}\right)$ and
$\varepsilon_{m,1}\left({T_{m,0}}\right)$ must have the same sign. Therefore,
it is not possible for unbounded sideband growth to cancel, when a pair of
sidebands for a single harmonic is considered.

##  4 Conditions for instability: Collective effects of all harmonics

In Section 3, conditions were given on the disturbance and the initial
conditions so that interaction between the primary wave and the $n^{th}$
harmonic of the disturbance produce unbounded growth of the sideband
amplitudes $\varepsilon_{n,1}\left({T_{n}}\right)$ and
$\varepsilon_{n,2}\left({T_{n}}\right)$. These conditions were dependent on
the coefficients of the Stokes wave,
$\eta\left({x,t}\right)=\sum_{n=1}^{\infty}{a_{n}\cos\zeta_{n}}$. Longuet-
Higgins showed that the Stokes coefficients are decreasing up to very high
orders of $n$ [2,3]. In that representation, each coefficient is represented
by a series where the leading term agrees with a formula of Wilton [5],

$a_{n}=\frac{n^{n}}{n!}s_{n},$ (47)

with $s_{n}$ a parameter related to the wave height. The first few terms of
the Stokes wave may then be written

$y=s_{1}\cos kx+2s_{2}\cos 2kx+\textstyle\frac{27}{6}s_{3}\cos
3kx+\textstyle\frac{64}{24}s_{4}\cos 4kx+\textstyle\frac{3125}{120}s_{5}\cos
5x+...$

Using Stokes’ representation [4],

$y=a\cos kx+\textstyle\frac{1}{2}ak^{2}\cos
2kx+\textstyle\frac{3}{8}a^{3}k^{2}\cos
3kx+\textstyle\frac{1}{3}a^{4}k^{3}\cos
4kx+\textstyle\frac{125}{384}a^{5}k^{4}\cos 5x+....$

and comparing coefficients, we have

$s_{1}=a,\quad s_{2}=\textstyle\frac{1}{4}a^{2}k,\quad
s_{3}=\textstyle\frac{1}{12}a^{3}k^{2},\quad
s_{4}=\textstyle\frac{1}{8}a^{3}k^{2},\quad
s_{5}=\textstyle\frac{24}{625}a^{3}k^{2}.$

Now consider the collective contribution of all harmonics in (26) and (27). As
noted in Section 3, values of $\delta\left(n\right)$ providing the marginal
stability case for a particular harmonic decrease with $n.$ Therefore, if a
particular harmonic of the disturbance has frequency sufficiently close to the
corresponding harmonic of the primary wave, unbounded sideband growth will
occur for all lower order harmonics, so long as the initial condition (46) is
met. Furthermore, if (46) holds for $n=M$, but not for $n=M+1$, then unbounded
growth will indeed occur, as the coefficients of the higher harmonics will
decay in time and will not suppress the instability.

Figure 2: The maximum value of $\delta$ as a function of wave steepness ak so
that all harmonics up to the $N^{th}$ harmonic contribute to instability.
$ak\delta N=2N=1N=3N=4$.

Now suppose that (43) holds for some harmonic. Then it must hold for all lower
order harmonics, and we may consider the sequence of harmonics where (43)
holds. For these harmonics, since only cases 2-4 in Figure 1 need be
considered,

$-1<\alpha_{n}<1$

or, equivalently,

$0<\frac{\delta^{2}}{n^{2}k^{2}a_{n}^{2}}<2$

From (47),

$0<\frac{\left({n!}\right)^{2}\delta^{2}}{n^{2}k^{2}n^{2n}s_{n}^{2}}<2$

which may be written

$0<\delta<\frac{\sqrt{2}\,k\,n^{n}s_{n}}{\left({n-1}\right)!}.$ (48)

For the first few harmonics, the marginal stability condition is,

$\displaystyle\begin{array}[]{lcl}n=1:&&0<\delta<\displaystyle\frac{\sqrt{2}\,k\,n^{n}s_{n}}{\left({n-1}\right)!}=\sqrt{2}ak\\\
\\\
n=2:&&0<\delta<\displaystyle\frac{\sqrt{2}\,k\,n^{n}s_{n}}{\left({n-1}\right)!}=4\sqrt{2}\,k\,s_{2}=\sqrt{2}\,a^{2}k^{2}\\\
\\\
n=3:&&0<\delta<\displaystyle\frac{\sqrt{2}\,k\,n^{n}s_{n}}{\left({n-1}\right)!}=\displaystyle\frac{27\sqrt{2}\,ks_{3}}{4}=\displaystyle\frac{9\sqrt{2}}{16}\,a^{3}k^{3}\\\
\\\
n=4:&&0<\delta<\displaystyle\frac{\sqrt{2}\,k\,n^{n}s_{n}}{\left({n-1}\right)!}=\displaystyle\frac{256\sqrt{2}\,ks_{4}}{6}=\displaystyle\frac{16\sqrt{2}}{3}\,a^{4}k^{4}\\\
\\\
n=5:&&0<\delta<\displaystyle\frac{\sqrt{2}\,k\,n^{n}s_{n}}{\left({n-1}\right)!}=\displaystyle\frac{3125\sqrt{2}\,ks_{4}}{24}=5\sqrt{2}\,a^{5}k^{5}\\\
\\\ \end{array}$ (59)

Since $ak\ll 1$, the value of $\delta$ corresponding to the marginal
instability decreases with $n$. This is expected, since disturbance
frequencies sufficiently close to that of the primary wave so that sideband
amplitude $\varepsilon_{n,1}$ becomes unbounded, will be sufficiently small so
that sideband amplitudes for all lower order harmonics are also unbounded.
Therefore, (48) may be considered a generalization of the Benjamin-Feir
criterion in the sense that, if it is satisfied, all harmonics up to order $n$
contribute to unbounded growth. This conclusion is subject to initial
conditions, represented by (46). Indeed, if (48) is satisfied for a (possibly
infinite) set of harmonics, instability will occur only if (46) is satisfied
for at least one value of $n$.

Values for the first five coefficients using wavelength
${2\pi}\mathord{\left/{\vphantom{{2\pi}k}}\right.\kern-1.2pt}k=64$ cm and
$a=0.1$ are compiled in Table 1 using Wilton’s formula for the coefficients.
These parameters give $ak=0.098\ll 1$. The last column provides the marginal
instability criterion given in (48).

$n$ | $s_{n}$ | $a_{n}=\left({\frac{n^{n}}{n!}}\right)s_{n}$ | $\delta=\frac{\sqrt{2}\,k\,n^{n}s_{n}}{\left({n-1}\right)\,!}$
---|---|---|---
1 | $a$ | 0.1 | 0.013884009
2 | $a^{2}k/4$ | 0.000490874 | 0.000136306
3 | $a^{3}k^{2}/12$ | 0.003681554 | 0.001533442
4 | $a^{4}k^{3}/8$ | $1.26165\times 10^{-7}$ | $7.0067\times 10^{-8}$
5 | $24a^{5}k^{4}/625$ | $9.28965\times 10^{-10}$ | $6.4488\times 10^{-10}$
Table 1: Sample coefficients of Stokes wave and disturbance frequency
corresponding to marginal instability for the nth harmonic.

Equation (47) shows that the instability criterion (48) depends on the wave
steepness $a^{n}k^{n-1}$. In Figure 2, we illustrate values of
$\delta\left(n\right)$ as a function of steepness for $1\leq n\leq 4$ Note
that the value of $\delta\left(n\right)$ decreases with $n$ for all values of
$ak<.44$. While $\delta\left(1\right)=\delta\left(2\right)$ for $ak=0.44$, it
must be remembered that the small amplitude assumption $ak<<1$ is in place, so
that values of ak where this equality occurs are arguably outside the realm
applicable to this analysis.

Figure 3: Growth rate ${d}\ln\left({\varepsilon_{n,1}}\right)/{dt}$ as a
function of ${\delta}/{ak}$ for cases $n=1,2,3$.

We now consider the growth of the amplitude of the disturbance. Following [1],
we solve (44) directly for ${d\varepsilon_{n,1}^{2}}/{dT_{n}}$ by separating
variables, and recalling the asymptotic equivalence of
$\varepsilon_{n,1}\left({T_{n}}\right)$ and
$\varepsilon_{n,2}\left({T_{n}}\right)$. We obtain

$\varepsilon_{n,1}\sim\exp\left[{\frac{\delta}{2}\sqrt{2k_{n}^{2}a_{n}^{2}-\delta^{2}}\,\,\omega
T_{n}}\right].$ (60)

Applying (37),

$\ln\varepsilon_{n,1}=\frac{\delta}{2}\omega\,a_{n}k_{n}\sqrt{2-\left({\frac{\delta}{a_{n}k_{n}}}\right)^{2}}\,\,\,t,$
(61)

and then (47), with $k_{n}=nk$,

$\ln\varepsilon_{n,1}=\frac{\delta}{2}\omega\,\left({\frac{n^{n}}{n!}s_{n}}\right)nk\sqrt{2-\left({\frac{n!\delta}{n^{n+1}ks_{n}}}\right)^{2}}\,\,\,t.$
(62)

If we select $s_{n}=a^{n}k^{n-1}$, noting that this parameter goes to zero
with the wave height as required, we obtain

$\ln\varepsilon_{n,1}=\frac{\omega
a^{2}k^{2}}{2}\,\left({\frac{n^{n+1}}{n!}}\right)\left({\frac{\delta}{ak}}\right)\sqrt{2-\left({\frac{(n-1!}{n^{n}(ak)^{n-1}}}\right)^{2}\left({\frac{\delta}{ak}}\right)^{2}}\,\,\,t.$
(63)

Figure 3 shows a graph of ${d}\ln\left({\varepsilon_{n,1}}\right)/{dt}$ versus
${\delta}/{ak}$ where the case $n=1$ corresponds to Figure 1 in [1] and,
following that development, we have employed the normalization
${wa^{2}k^{2}}/{2}=1$. For $n>1$, the factor ak appears and Figure 1 uses
$ak=0.1\ll 1$ to display the growth rate of the amplitude for the first two
harmonics. The pattern persists for higher harmonics, and (63) gives the
maximum growth rate occurring at

$\delta=\frac{n\left({ak^{n-1}}\right)}{\left({n-1}\right)!}.$ (64)

Note that in the case $n=1$, (64) gives the maximum growth rate occurring at
the known value of $\delta=ak$, while the maximum growth rate of the second
harmonic occurs at $\delta=2ak$, shown in Figure 3 for the case $ak=0.1.$

Assuming appropriate initial conditions and sufficiently small $\delta$ so
that a set of initial harmonics contribute to resonance, the collective
asymptotic growth rate represented by the $N^{th}$ partial sum will be
approximated by the sum of the growth rates of the individual harmonics. From
(62), we have

$\sum\limits_{n=1}^{N}{\frac{d\varepsilon_{n,1}}{dt}=\sum\limits_{i=1}^{N}{\frac{\delta}{2}\omega
a_{n}k_{n}\sqrt{2n^{2}-\left({\frac{\delta}{a_{n}k_{n}}}\right)^{2}}\,\,\,\exp\left\\{{\frac{\delta}{2}\omega
a_{n}k_{n}\sqrt{2n^{2}-\left({\frac{\delta}{a_{n}k_{n}}}\right)^{2}}}\right\\}t}}.$
(65)

which, for sufficiently large $N$, represents the asymptotic growth rate for
the fully nonlinear Stokes wave.

## References

  * [r0] [1] Benjamin, T.B. & Feir, J.E. 1967 The disintegration of wave trains on deep water. Part 1. Theory. J. Fluid Mech. 27, 417-430.
  * [r1] [2] Longuet-Higgins, M.S. 1975 Some new relations between Stokes’s coefficients in the theory of gravity waves. J. Inst. Maths. Applics. 22, 261-273.
  * [r3] [3] Longuet-Higgins, M.S. 1985 The asymptotic behaviour of the coefficients in Stokes’s series for surface gravity waves., J. Inst. App. Maths. 34, 269-277.
  * [r4] [4] Stokes, G.G. need reference
  * [r5] [5] Wilton, J.R. 1914 On deep water waves. Phil. Mag. 27 (12), 385-394.