# Preferential orientation of floaters drifting in water waves

Wietze Herreman1<EMAIL_ADDRESS>Basile Dhote1
Lucile Danion1
Frédéric Moisy1  1Université Paris-Saclay, CNRS, FAST, 91405, Orsay, France.

###### Abstract

Elongated floaters drifting in propagating water waves slowly rotate towards a
preferential state of orientation. Short and heavy floaters tend to align
longitudinally, along the direction of wave propagation, whereas long and
light floaters align transversely, parallel to the wave crests and troughs. We
investigate this phenomenon for homogeneous parallelepiped floaters by
combining laboratory experiments with numerical simulations and asymptotic
theory. For floaters small with respect to wavelength and for low amplitude
waves, we show that the floater orientation is controlled by the non-
dimensional number $F=kL_{x}^{2}/\beta L_{z}$, with $k$ is the wavenumber,
$\beta$ the floater-to-water density ratio, and $L_{x}$ and $L_{z}$ the
floater length and thickness. Theory places the longitudinal-transverse
transition at the critical value $F_{c}=60$, in fair agreement with the
experiments. Using a simplified physical model, we elucidate the physical
origin of the preferential orientation. Through its motion, the floater probes
the velocity gradients along its surface. Next to a small mean displacement
(Stokes drift), this results in a net torque which, for short floaters, always
favors the longitudinal orientation. This net torque arises from a phase
correlation between the instantaneous buoyancy torque and the instantaneous
yaw angle of the floater, a mechanism analogous to the Kapitza pendulum. The
transverse equilibrium of longer floaters has a different origin and arises
from the variation of the submersion depth along their long axis. This varying
submersion significantly increases the torque in the trough positions, when
the tips are more submersed, and always pushes towards the transverse
orientation.

###### keywords:

##  1 Introduction

The motion of a floating body in gravity waves is a classical problem in fluid
mechanics with evident applications in the domain of naval engineering
(Faltinsen, 1993; Newman, 2018; Falnes & Kurniawan, 2020). At first order in
wave magnitude, waves cause harmonic oscillations of the floating body in all
six degrees of freedom, both linear (heave, surge, sway) and angular (pitch,
roll, yaw). At second order, waves also cause a mean drift force and yaw
moment on the body that affect surge, sway and yaw angle on long time-scales.
For small isotropic floaters, this mean motion reduces to the classical Stokes
drift in the direction of the wave propagation (Stokes, 1847; van den Bremer &
Breivik, 2018; Calvert et al., 2021), a problem that received considerable
interest for the modeling of pollutant transport in the oceans (Suaria et al.,
2021; Yang et al., 2023; Sutherland et al., 2023). For larger floaters of
arbitrary shape, such as ships and floating structures, an angular drift can
change the floater’s orientation with respect to the wave incidence and this
slow reorientation can in turn modify its linear drift. The combined linear
and angular drifts are key features in sea keeping and maneuvering, and their
modeling has been the subject of numerous works (Faltinsen, 1993; Skejic &
Faltinsen, 2008; Newman, 2018). In this paper, we are interested in the slow,
second order yaw motion of elongated floaters drifting in gravity waves and
how a preferential state of orientation spontaneously emerges.

Research on this topic started more than a century ago, with Suyehiro (1921)
who first reported how small boat models displaced by propagating gravity
waves rotate towards a preferential state of orientation with respect to the
direction of incidence. Suheyiro believed that this was due to gyrostatic
torques and hence of purely solid-mechanical origin. Newman disagreed with
this and proposed in 1967 his famous article on the mean drift force and yaw
moment due to waves (Newman, 1967). Starting from a global momentum and torque
balance, he expresses the second order drift force and yaw moment in terms of
the far-field hydrodynamic wave potential (Kochin functions). Using slender
body theory, he was able to calculate this far-field potential and hence to
derive the first explicit formula for the second order yaw moment. This mean
torque compared reasonably well with the experimental data of Spens & Lalangas
(1962) (reproduced in Newman (1967)).

Figure 1: Experimental setup and chronophotographies. (a) Side view. Waves are
generated in a water channel, of length 3 m and filled at height $H=10$ cm.
(b) Top View. Floater of length $L_{x}$, making an angle $\psi$ with the
direction of wave propagation. (c-d) Chronophotographies, obtained by
superimposing images acquired every wave period, for a wave length
$\lambda=29$ cm and wave slope $\epsilon=ak=0.16$. The short floater (length
$L_{x}=60$ mm) gradually aligns in the direction of wave propagation (c),
while the long floater ($L_{x}=100$ mm) aligns parallel to the wave crests
(d).

A key prediction of Newman’s theory is that slender floating structures of
length shorter than the wave-length should always be stable in beam seas,
meaning that they will prefer to have their long axis parallel to the wave
crests. In figure 1 we show two chronophotographs from our laboratory
experiments that disagree with this prediction. Here, small homogeneous
parallelepipeds of centimer scale are left adrift in a propagating wave in a
3-m long water tank, and pictures synchronized with the wave period are taken
from above (details are given in section 2). In just a few wave periods, we
observe that long floaters align parallel to the wave crests (‘beam sea’,
$\psi\rightarrow 90^{o}$), in agreement with Newman’s prediction, but shorter
floaters align along the direction of wave propagation (‘head sea’,
$\psi\rightarrow 0^{o}$); here $\psi$ is the yaw angle between the floater
long axis and the direction of wave propagation. Hence, it seems that
preferential orientation, and by extension the mean second order yaw moment,
may depend more subtly on the floater length and density. The main objective
of this article is to clarify what physically governs this preference and to
locate where the longitudinal-transverse transition occurs for floaters small
with respect to the wavelength.

Since Newman’s pioneering work, other groups have studied the second order yaw
moment acting on floating bodies. The work of Salvesen (1974) is a
simplification of Newman’s model in which the floater is considered as a weak
scatterer of the incoming wave. In Kashiwagi (1992) and Kashiwagi & Ohkusu
(1993), Parseval’s theorem is used to evaluate the mean drift force and yaw
moment, instead of the method of stationary phase used by Newman. An entirely
different, near-field approach, of direct pressure integration was proposed by
Faltinsen (1980). These different methods are compared in Skejic & Faltinsen
(2008), and show consistent results. Chen (2007) proposed a third method, the
so-called middle field formulation, to compute mean wave loads on structures.
This method, in which the potential is determined using a boundary element
method, is implemented in the software pack Hydrostar (Bureau-Veritas, 2016)
that is commonly used in naval engineering applications. The experimental
study of Le Boulluec et al. (2008) on drifting cargo containers briefly
mentions the fact that containers can drift either in longitudinal positions
(head-seas) or transverse position (beam-seas). Recently, Yasukawa et al.
(2019) compared theory to new experimental measurements on a particular ship
model. According to the authors, it remains difficult to make the theory and
experiments for the mean yaw moment match well.

In this article, we propose a combined experimental, numerical and theoretical
study on the preferential orientation of a model floater (homogeneous
parallelepiped) drifting in a propagating gravity wave. Our model is based on
the Froude-Krylov assumption and hence ignores all wave diffraction and wave
radiation effects (no added mass, no radiation impedance). In spite of this
simplicity, this model predicts a mean yaw moment in the limit of small
floater length/wavelength ratio that is fully compatible with Newman’s
classical result (Newman, 1967) and nearly in perfect agreement with mean yaw
moment calculations of Chen (2007). Moreover, our simplified model also allows
us to provide a simple physical picture for the origin of this preferential
orientation.

The article is structured as follows. In section 2, we present a systematic
series of experiments investigating the preferential orientation of floaters
of varying length and density. In section 3, we define a simplified model in
which we calculate the non-linear motion of the floater using a Froude-Krylov
approximation. We show that numerical solutions of this model reproduce well
the observed state of preferential orientation in our experiments. In section
4, we propose an asymptotic solution to our Froude-Krylov model in the double
limit of small wave and small floater size. This yields the following,
idealized evolution equation for the average yaw angle $\overline{\psi}$,

$\ddot{\overline{\psi}}\approx-\epsilon^{2}\sin\overline{\psi}\cos^{3}\overline{\psi}\left(1-\frac{F}{F_{c}}\right).$
(1)

This equation depends on two non-dimensional numbers: the wave-slope
$\epsilon=ka$ (with $k$ the wavenumber and $a$ the wave amplitude) and the
number $F$, defined as

$F=\frac{kL_{x}^{2}}{\beta L_{z}},$ (2)

with $L_{x},L_{z}$ the floater length and thickness and $\beta$, the floater-
to-water density ratio. We recognise $\bar{h}=\beta L_{z}$ as the equilibrium
submersion depth or draft of our parallelepiped floaters. This idealised,
dissipation-less evolution equation certainly does not capture all the
complexity of realistic slow yawing, but it provides insight into the
preferential state of orientation. It shows that, for $F<F_{c}$, the
longitudinal position $\overline{\psi}=0^{o}$ is stable and hence preferred,
whereas for $F>F_{c}$, the transverse position $\overline{\psi}=90^{o}$ is
stable. Our theory predicts a transition at $F_{c}=60$, in fair agreement with
the experimental value $F_{c}\simeq 45\pm 10$.

Our theory follows a ’near-field approach’, but different from that of
Faltinsen (1980), in the fact that it uses the Froude-Krylov approximation,
which allows more analytical evaluation. In section 5, we finally derive the
same model using a more simple physical procedure. This allows us to better
understand what physically controls the preferential orientation. The
longitudinal equilibrium position is due to an effective torque that appears
as the floater probes low and high pressure regions while going through its
first order motion. We show that this part of the torque creates a motion that
is analogous to that of classical Kapitza pendulum, a pendulum whose anchor
point is rapidly vibrated (Kapitza, 1951; Landau & Lifschitz, 1960; Butikov,
2001). The transverse state of orientation is on the other hand due to the
fact that long floaters have a variable submersion along their length. This
variable submersion significantly enhances the instantaneous yaw moment in
trough positions that always favors the transverse position. In Newman’s
model, only this second part of the torque is present, which explains why he
found elongated structures to be stable in beam seas. Finally, in Sec. 6, we
compare the analytical formula for the mean yaw moment in our theory to
previous results obtained by Newman (1967) and Chen (2007). The agreement is
surprisingly good, in particular when considering that we use a Froude-Krylov
assumption in our model.

##  2 Experiments

A series of laboratory experiments with centimeter-scale floaters of varying
size and density have been performed in a water flume. The experimental setup,
sketched in figure 1(a-b), consists in a tank of length 3 m, width 0.38 m,
filled with water at height $H=0.1$ m. Waves are generated by a wavemaker
consisting in a paddle oscillating at frequency $\omega/2\pi$ between 1 and 4
Hz, and are absorbed at the other end of the channel by a sloping plate.

We determine the wave profile $\zeta(x,t)$ by imaging the instantaneous
contact line through the lateral channel wall and using a line detection
algorithm. The wave profile is well described by a simple sinus law,
$\zeta=a\cos(kx+\varphi(t))$, with $\varphi(t)$ the wave phase, from which we
measure the wave amplitude $a$ and wave number $k=2\pi/\lambda$ with precision
$\simeq 5\%$. The measured wave number matches the dispersion relation in the
gravity regime, $\omega^{2}=gk\tanh(kH)$, with wave lengths $\lambda$ in the
range $10-82$ cm. Experiments for wavelengths of the order of the channel
width are discarded because resonant transverse sloshing modes are excited. We
work in the weakly nonlinear wave regime, for wave slopes $\epsilon=ak$
between 0.02 and 0.23.

The floaters are homogeneous rectangular parallelepipeds with length $L_{x}$
between 20 and 140 mm, width $L_{y}=10$ mm and height $L_{z}$ between 5 and 10
mm, cut from expanded PVC or polystyrene boards of various densities. We
denote $M$ the floater mass, $\rho_{s}=M/(L_{x}L_{y}L_{z})$ its density, and
$\beta=\rho_{s}/\rho$ the floater-to-water density ratio. The floater size is
large enough for the capillary effects to be negligible: the vertical
component of the capillary force is less than 10% of the floater weight.

Figure 2: (a) Yaw angle $\psi$ as a function of the time normalized by the
wave period, for floaters of various lengths $L_{x}$. The time origin is
chosen such that $\psi(0)=45^{\mathrm{o}}$. Wave frequency $f=2.3$ Hz,
wavelength $\lambda=0.29$ m, wave slope $ak=0.16$, floater waterline depth
$\bar{h}=2.2$ mm. (b) Normalized angular velocity $\dot{\psi}/\omega$ of the
floater as a function of $L_{x}$.

For each run, a floater is gently deposited at the surface of water at a
distance $x_{0}\simeq 0.8$ m from the wave maker, with an initial yaw angle
$\psi_{0}$ of approximately $45^{\mathrm{o}}\pm 15^{\mathrm{o}}$ from the $x$
axis (controlling precisely $\psi_{0}$ is difficult because the phase of the
wave is not known at the time the floater is released). The floaters are
imaged by a camera located above the wave tank. Using a tracking algorithm
(library Tracker in Python), we measure the center of mass $x_{c}(t)$ and the
yaw angle $\psi(t)$ of the floater on each frame. For each run, the floater is
left adrift for about 1 m, and only trajectories staying approximately along
the center line of the channel are retained to discard possible interaction
with the side walls.

At first order, the floater motion is a combination of a back-and-forth
oscillation of its center of mass, of amplitude given by the typical
horizontal excursion of the fluid particle trajectories ($\Delta
x=a/\tanh(kH)$ for waves in finite depth) and angular oscillations.
Superimposed to these fast oscillations are a slow drift of the center of mass
in the direction of the wave propagation (Stokes drift) and a slow drift of
the yaw angle, either towards the longitudinal ($\psi=0^{\mathrm{o}}$, denoted
‘L’) or transverse ($\psi=90^{\mathrm{o}}$, denoted ‘T’) orientation. To
filter out the fast oscillations and focus on the slow dynamics of $\psi$, we
synchronize the image acquisition with the wave maker oscillation, as
illustrated in the chronophotographies of figure 1(c-d).

We first consider a set of floaters with a density ratio
$\beta=\rho_{s}/\rho=0.44$ and thickness $L_{z}=5$ mm (expanded PVC foam
boards “Forex”), and investigate the influence of the floater length $L_{x}$,
for a fixed wavelength $\lambda=290$ mm and wave amplitude $a=7.4$ mm (wave
slope $\epsilon=ka=0.16$). The time evolution of the yaw angle $\psi$ is shown
in figure 2(a). Because of the uncertainty on the initial angle $\psi_{0}$, we
shift the time origin so that $\psi$ is $45^{\mathrm{o}}$ at $t=0$ for each
run. The curves clearly separate in two groups, with small floaters
($L_{x}\leq 60$ mm) tending to $\psi=0^{\mathrm{o}}$ (longitudinal) and long
floaters ($L_{x}\geq 80$ mm) tending to $\psi=90^{\mathrm{o}}$ (transverse).
While short floaters precisely align in the longitudinal direction, with
$\psi\simeq 0\pm 5^{\mathrm{o}}$ at large time, long floaters show larger
variations around $\psi\simeq 90\pm 30^{\mathrm{o}}$. These larger excursions
can be due to the fact that long floaters are more subject to interactions
with the side walls and to residual inhomogeneities in the mean flow.

Figure 3: (a) Asymptotic floater orientation as a function of $L_{x}$ and
wavelength $\lambda$, for floaters with density $\beta=0.44$, width $L_{y}=10$
mm and thickness $L_{z}=5$ mm. Red circles: longitudinal; Blue triangles:
transverse. Black squares indicate erratic oscillations of the floater angle
or non-reproducible experiments. The separation line is
$L_{x}=\sqrt{\ell\lambda}$, with $\ell\simeq 16\pm 3$ mm. (b) Same data in the
plan $(F,\epsilon)$, demonstrating the independence of the orientation with
the wave steepness $\epsilon$. The long dashed line shows the experimental
transition at $F_{c}\simeq 50$, and the short dashed line the theoretical
prediction at $F_{c}=60$.

The reorientation dynamics is faster for very short or very long floaters:
they reach their asymptotic orientation after approximately 5 wave periods
only, while the convergence is much slower (at least 20 wave periods) for
intermediate lengths. This convergence rate is illustrated in figure 2(b) for
various floater lengths, showing the angular velocity $\dot{\psi}$ measured at
$t=0$ normalized by the wave frequency $\omega$ (this ratio measures the
fraction of complete turn performed by the floater during one wave period).
For this particular floater density and thickness, $\dot{\psi}/\omega$ crosses
zero for $L_{x}\simeq 75$ mm, which defines the critical length $L_{xc}$
separating the longitudinal and transverse orientations. Because of the slow
dynamics at the crossover, the orientation is very sensitive to any
experimental uncertainty for $L_{x}$ close to $L_{xc}$, such as the precise
choice of the the initial angle $\psi_{0}$, inhomogeneities in the streaming
flow, or small defects in the wetting line.

We have systematically determined the asymptotic yaw angle for various floater
lengths $L_{x}$, wave lengths $\lambda$ and amplitudes $a$. The preferential
orientation of the floaters is first summarized in the plan $(L_{x},\lambda)$
in figure 3(a). When the asymptotic angle is $\psi\simeq 0\pm
10^{\mathrm{o}}$, floaters are labeled as “longitudinal” (red circles), and
when $\psi\simeq 90\pm 30^{\mathrm{o}}$ they are labeled as “transverse” (blue
triangles). Floaters with indistinct orientation are marked with a black
square. This diagram shows a clear separation between the longitudinal and
transverse orientations, with a transition line well described by a square-
root law, $L_{xc}\simeq\sqrt{\ell\lambda}$, with $\ell\simeq 16\pm 3$ mm a
fitting parameter.

Figure 4: Critical values of $F=kL_{x}^{2}/{\beta L_{z}}$ for the L-T
transition as a function of the wavelength, for floaters of various densities
$\beta$ and thicknesses $L_{z}$. Solid line: average value $F\simeq 45\pm 10$.
Dashed line: theoretical prediction $F_{c}=60$.

This square-root law nicely conforms to the prediction of the asymptotic
theory presented in the next section, which demonstrates that the preferential
orientation is independent of the wave slope $\epsilon=ka$ and governed by the
non-dimensional number $F=kL_{x}^{2}/\beta L_{z}$, with a longitudinal-
transverse transition at $F_{c}=60$. To check these predictions, we plot in
figure 3(b) the same data in the $(F,\epsilon)$ plane. We observe a clear
separation between the longitudinal and transverse orientations, delimited by
the line $F\simeq 50$, in fair agreement with the theory.

To further test the theory, we performed additional experiments with different
floater densities $\beta$ and thicknesses $L_{z}$, covering a range of
immersion depths $\overline{h}=\beta L_{z}$ between 1.8 and 5.3 mm. For each
wavelength $\lambda$, we determine the critical floater length $L_{xc}$
separating the longitudinal and transverse orientation, defined as the average
between the largest longitudinal and smallest transverse floaters (the
uncertainty on $L_{xc}$ is 10 mm). Figure 4 summarizes the values of $F$ at
the transition as a function of the wavelength, yielding again a constant
value, $F\simeq 45\pm 10$. We can conclude that the two key predictions of the
theory (floater orientation governed by $F$ and independent of the wave slope
$\epsilon$) are well verified by our experiments.

##  3 Inviscid Froude Krylov model

To model the preferential state of orientation, we could in principle compute
the mean yaw moment using existing models, either using the far-field approach
of Newman (1967), the near-field approach of Faltinsen (1980) or the middle-
filed approach of Chen (2007). Here, we derive a more basic numerical model
for idealised floater motion, that is based on the Froude-Krylov assumption.
We demonstrate that this model reproduces well our experimental observations
and hence, that it contains much of the essential physics.

###  3.1 Simplifying assumptions

The idealised model that we will use to explain preferential orientation
relies on the following simplifying assumptions:

  1. 1.

The incoming wave is a linear, inviscid potential gravity wave in deep water –
The linearity assumption is reasonably satisfied in the experiments (maximum
wave slope $\epsilon=ka=0.23$); the deep-water assumption is not fully
satisfied ($\tanh(kH)\simeq 0.65$ for the largest wave-length), but this is
not essential to the theory.

  2. 2.

Viscosity is negligible – The viscous stress on the floater is of the order
$\eta a\omega/\delta_{s}$, where $\delta_{s}=(\nu/\omega)^{1/2}$ is the
thickness of the Stokes boundary layer. This viscous stress is much smaller
than the wave pressure variations, $p\sim\rho a\omega^{2}/k$, for $\nu
k^{2}/\omega\ll 1$, which is well satisfied in the experiments: $\nu
k^{2}/\omega<2\times 10^{-4}$.

  3. 3.

Capillarity is negligible – The capillary length, $\ell_{c}=\sqrt{\rho
g/\gamma}\simeq 2$ mm (with $\gamma$ the surface tension), is smaller than the
characteristic floater size in our experiments. Note however that the meniscus
at the rim of the floater likely creates an effective shape and may also
weakly change the equilibrium immersion depth $\overline{h}$.

  4. 4.

Steady streaming flows are negligible – The steady streaming flows is a
nonlinear, mean flow correction of order $O(\epsilon^{2})$ that comes along
with the linear wave. This mean flow can affect reorientation, but only if it
is inhomogenous at the scale of the floater. Steady streaming flows typically
vary spatially on the scale $k^{-1}$. Hence, with $\delta=kL\ll 1$, we expect
a weak effect of streaming on the floater orientation.

  5. 5.

Wave scattering and wave emission are neligible (Froude-Krylov approximation)
– In principle, floaters placed in an incoming wave act as obstacles and
scatter this incoming wave. They can also move differentially with respect to
the flow and hence emit waves. However, we can neglect these effects when the
floater is small with respect to the wavelength and when differential motion
is weak:

$\delta=kL\ll
1\quad,\quad\frac{||\boldsymbol{u}-\boldsymbol{v}||}{||\boldsymbol{u}||}\ll
1,$ (3)

with $\boldsymbol{u}$ the fluid velocity and $\boldsymbol{v}$ floater
velocity. We therefore assume that the floater only moves due to pressure
forces and torques associated with the incoming wave.

The Froude-Krylov approximation is the most important assumption in our model.
By neglecting the wave emission, we filter out radiative losses and added mass
effects. As a result, free “bobbing” oscillations of the floater around the
equilibrium position are not damped in our model. In practice, we avoid these
free oscillations in our calculations using a particular initialisation
strategy that will be detailed below. Another consequence of the absence of
dissipation is that, as for an undamped pendulum, the equilibrium states must
be either unstable or marginally stable. In our simulations, the yaw angle
$\psi$ will oscillate around the stable fixed point, rather then converging
towards it as in the experiments.

Figure 5: (a) Sketch of the floater and notations: laboratory frame
$(O,x,y,z)$, moving floater frame
$(C,\widetilde{x},\widetilde{y},\widetilde{z})$, and Euler angles $\theta$
(pitch), $\phi$ (rall) and $\psi$ (yaw). In the simulation, we numerically
calculate the pressure force and torque on the submerged surface $S_{sub}$
using rectangular meshes on each face and a mask function $f$ that indicates
whether the point on the face is submerged or not. (b) Free vertical
oscillations of $z_{c}$ at the bobbing frequency $\sqrt{1/\beta\delta_{z}}$ in
a numerical test-case without incoming wave

###  3.2 Equations of motion

We derive equations of motion for a small parallepiped floater drifting on a
propagating gravity wave on infinitely deep water. We introduce a laboratory
frame of reference $(O,x,y,z)$, with
$(\boldsymbol{e}_{x},\boldsymbol{e}_{y},\boldsymbol{e}_{z})$ basis vectors
(figure 5). By convention, the origin $O$ is on the equilibrium fluid surface
and the wave propagates along $x$ and is $y$-invariant.

We use as incoming gravity wave the classical, linear potential wave solution
on infinitely deep water

$\zeta=a\sin(kx-\omega t),\quad\left\\{\begin{array}[]{rcr}u_{x}&=&a\omega
e^{kz}\sin(kx-\omega t)\\\ u_{z}&=&-a\omega e^{kz}\cos(kx-\omega
t)\end{array}\right.,\quad p=p_{0}-\rho gz+\rho gae^{kz}\sin(kx-\omega t).$
(4)

Here $\zeta$ is the surface elevation, $u_{x},u_{z},p$ the velocity components
and pressure, $p_{0}$ the atmospheric pressure, and $\omega=\sqrt{gk}$. In the
following, we non-dimensionalise space, time, velocity and pressure using the
scales

$[\boldsymbol{r}]=k^{-1}\quad,\quad[t]=(gk)^{-1/2}\quad,\quad[\boldsymbol{u}]=g^{1/2}k^{-1/2}\quad,\quad[p]=\rho
gk^{-1}.$ (5)

In this unit system, Eqs. (4) become

$\zeta=\epsilon\sin(x-t),\quad\left\\{\begin{array}[]{rcr}u_{x}&=&\epsilon
e^{z}\sin(x-t)\\\ u_{z}&=&-\epsilon e^{z}\cos(x-t)\end{array}\right.,\quad
p=p_{0}-z+\epsilon e^{z}\sin(x-t),$ (6)

and the wave slope $\epsilon=ka\ll 1$ is the only remaining parameter. In line
with the Froude-Krylov assumption, we ignore the flow and pressure corrections
caused by differential motion of the floater: only the incoming wave affects
the floater motion in this model.

The floater is a rectangular parallelepiped with non-dimensonal density
$\beta$, non-dimensional length, width and height
$(\delta_{x},\delta_{y},\delta_{z})=(kL_{x},kL_{y},kL_{z})$, and we assume
that $\delta_{x,y,z}\ll 1$. We introduce a material frame
$(C,\widetilde{x},\widetilde{y},\widetilde{z})$ co-moving with the floater,
with $C$ at the center of mass and the unit vectors
$(\widetilde{\boldsymbol{e}}_{x}(t),\widetilde{\boldsymbol{e}}_{y}(t),\widetilde{\boldsymbol{e}}_{z}(t))$
aligned with the long, medium and short axes (figure 5). The instantaneous
position of the floater is determined by the 3 coordinates
$x_{c}(t),y_{c}(t),z_{c}(t)$, with

$\boldsymbol{r}_{c}(t)=\boldsymbol{OC}(t)=x_{c}(t)\boldsymbol{e}_{x}+y_{c}(t)\boldsymbol{e}_{y}+z_{c}(t)\boldsymbol{e}_{z},$
(7)

and its orientation by the three Euler angles of roll $\varphi(t)$, pitch
$\theta(t)$ and yaw $\psi(t)$. These Euler angles define the transform that
links the laboratory frame to the floater frame, and in our angle convention
(see Appendix A) we have

$\left[\begin{array}[]{c}\boldsymbol{e}_{x}\\\ \boldsymbol{e}_{y}\\\
\boldsymbol{e}_{z}\end{array}\right]=\underbrace{\left[\begin{array}[]{ccc}c_{\psi}c_{\theta}&\left(c_{\psi}s_{\theta}s_{\varphi}-s_{\psi}c_{\varphi}\right)&\left(c_{\psi}s_{\theta}c_{\varphi}+s_{\psi}s_{\varphi}\right)\\\
s_{\psi}c_{\theta}&\left(s_{\psi}s_{\theta}s_{\varphi}+c_{\psi}c_{\varphi}\right)&\left(s_{\psi}s_{\theta}c_{\varphi}-c_{\psi}s_{\varphi}\right)\\\
-s_{\theta}&c_{\theta}s_{\varphi}&c_{\theta}c_{\varphi}\end{array}\right]}_{R^{T}}\left[\begin{array}[]{c}\widetilde{\boldsymbol{e}}_{x}\\\
\widetilde{\boldsymbol{e}}_{y}\\\
\widetilde{\boldsymbol{e}}_{z}\end{array}\right].$ (8a) Here and further we
denote in short $c_{\psi}=\cos\psi$, $s_{\psi}=\sin\psi$ and similar for the
other angles. This transform is easily inverted because $R$ is an orthogonal
matrix $R^{-1}=R^{T}$. Components of a vector $\boldsymbol{A}$ and the
coordinates of the laboratory and floater frame are also connected by this
matrix, $\left[\begin{array}[]{c}A_{x}\\\ A_{y}\\\
A_{z}\end{array}\right]=R^{T}\left[\begin{array}[]{c}\widetilde{A}_{x}\\\
\widetilde{A}_{y}\\\
\widetilde{A}_{z}\end{array}\right]\quad,\quad\left[\begin{array}[]{c}x-x_{c}\\\
y-y_{c}\\\
z-z_{c}\end{array}\right]=R^{T}\left[\begin{array}[]{c}\widetilde{x}\\\
\widetilde{y}\\\ \widetilde{z}\end{array}\right].$ (8b)

By definition, the floater moves as a solid. This means that an arbitrary
point $\boldsymbol{r}$ of the floater has velocity

$\boldsymbol{v}=\boldsymbol{v}_{c}(t)+\boldsymbol{\Omega}(t)\times(\boldsymbol{r}-\boldsymbol{r}_{c}(t)),$
(9)

with $\boldsymbol{v}_{c}(t)=\dot{\boldsymbol{r}}_{c}(t)$ the translation
velocity of the center of mass and $\boldsymbol{\Omega}(t)$ the instantaneous
rotation velocity. We decompose $\boldsymbol{v}_{c}(t)$ in the lab-frame and
$\boldsymbol{\Omega}(t)$ in the floater frame and kinematically link these
components to the derivatives of the coordinate functions by the relations

$\left[\begin{array}[]{c}\dot{x}_{c}\\\ \dot{y}_{c}\\\
\dot{z}_{c}\end{array}\right]=\left[\begin{array}[]{c}{v}_{c,x}\\\
{v}_{c,y}\\\
{v}_{c,z}\end{array}\right]\quad,\quad\left[\begin{array}[]{c}\dot{\varphi}\\\
\dot{\theta}\\\
\dot{\psi}\end{array}\right]=\left[\begin{array}[]{ccc}1&\sin\varphi\tan\theta&\cos\varphi\tan\theta\\\
0&\cos\varphi&-\sin\varphi\\\
0&\dfrac{\sin\varphi}{\cos\theta}&\dfrac{\cos\varphi}{\cos\theta}\end{array}\right]\left[\begin{array}[]{c}\widetilde{\Omega}_{x}\\\
\widetilde{\Omega}_{y}\\\ \widetilde{\Omega}_{z}\end{array}\right].$ (10)

The formula for the angles is non-trivial and depends on the convention used
to define the angles (details given in Appendix A).

We now define the floater dynamics. From Newton’s law and the angular momentum
theorem for the floater, we have

$m\dot{\boldsymbol{v}}_{c}=\boldsymbol{F}\
,\quad\frac{d}{dt}\left(\boldsymbol{I}\cdot\boldsymbol{\Omega}\right)=\boldsymbol{K},$
(11)

with $m=\beta\delta_{x}\delta_{y}\delta_{z}$ the non-dimensional mass of the
floater and $\boldsymbol{I}$ the non-dimensional inertia tensor. The principal
moments of inertia of the rectangular parallelepiped are

$\widetilde{I}_{xx}=\frac{m(\delta_{y}^{2}+\delta_{z}^{2})}{12},\quad\widetilde{I}_{yy}=\frac{m(\delta_{z}^{2}+\delta_{x}^{2})}{12},\quad\widetilde{I}_{zz}=\frac{m(\delta_{x}^{2}+\delta_{y}^{2})}{12}.$
(12)

In our model, the floater is subject to gravity and to pressure forces and
torques, given in non-dimensional form by

$\boldsymbol{F}=\int_{S_{sub}}(p-p_{0})\,d\boldsymbol{S}-m\boldsymbol{e}_{z}\
,\quad\boldsymbol{K}=\int_{S_{sub}}(\boldsymbol{r}-\boldsymbol{r}_{c})\times(p-p_{0})\,d\boldsymbol{S},$
(13)

with $d\boldsymbol{S}$ orientated towards the floater, and $S_{sub}$ the
(time-dependent) submerged surface of the floater. In the Froude-Krylov
approximation, the pressure $p$ is that of equation (6) which contains a
hydrostratic part and a dynamical part associated with the purely
hydrodynamic, incoming wave (pressure corrections due to scattered and emitted
waves are ignored in this model).

The previous surface integral formulation is used in the numerical
simulations, but in the theory, we have found it easier to use an alternative,
equivalent formulation of $\boldsymbol{F}$ and $\boldsymbol{K}$ in terms of
volume integrals over the submerged volume $V_{sub}$. In equation (13), we can
replace $d\boldsymbol{S}=-d\boldsymbol{S}_{ext}$ so that the surface element
is pointing from the floater towards the fluid. Using then the divergence
theorem and the fact that
$\partial_{t}\boldsymbol{u}+\boldsymbol{e}_{z}=-\boldsymbol{\nabla}p$ for the
incoming linear wave we obtain the alternative formula

$\displaystyle\boldsymbol{F}$ $\displaystyle=$
$\displaystyle\int_{V_{sub}}\boldsymbol{a}\,dV+\left(\int_{V_{sub}}\,dV-m\right)\boldsymbol{e}_{z}$
(14a) $\displaystyle\boldsymbol{K}$ $\displaystyle=$
$\displaystyle\int_{V_{sub}}(\boldsymbol{r}-\boldsymbol{r}_{c})\times\boldsymbol{a}\,dV+\int_{V_{sub}}(\boldsymbol{r}-\boldsymbol{r}_{c})\times\boldsymbol{e}_{z}\,dV,$
(14b)

with $\boldsymbol{a}=\partial_{t}\boldsymbol{u}$ the local fluid acceleration.
These formulas show better the leading Archimedes balance that defines the
floater equilibrium in absence of waves ($\boldsymbol{a}=0)$. Notice also that
with $a_{y}=0$ for the incoming $y$-invariant wave, we also have $F_{y}=0$: in
this model, floaters will never be displaced along $y$ and hence $y_{c}$
remains constant.

We decompose Newton’s law in the laboratory frame and the angular momentum
equation in the floater frame. Using the fact that the material frame rotates
along with the floater, i.e. that
$\dot{\widetilde{\boldsymbol{e}}}_{i}(t)=\boldsymbol{\Omega}(t)\times\widetilde{\boldsymbol{e}}_{i}(t)$,
we obtain

$\left[\begin{array}[]{c}m\dot{v}_{x,c}\\\ m\dot{v}_{y,c}\\\
m\dot{v}_{z,c}\end{array}\right]=\left[\begin{array}[]{c}F_{x}\\\ F_{y}\\\
F_{z}\end{array}\right]\quad,\quad\left[\begin{array}[]{c}\widetilde{I}_{xx}\dot{\widetilde{\Omega}}_{x}\\\
\widetilde{I}_{yy}\dot{\widetilde{\Omega}}_{y}\\\
\widetilde{I}_{zz}\dot{\widetilde{\Omega}}_{z}\end{array}\right]=\left[\begin{array}[]{c}{\widetilde{K}}_{x}+(\widetilde{I}_{yy}-\widetilde{I}_{zz})\widetilde{\Omega}_{y}{\widetilde{\Omega}}_{z}\\\
{\widetilde{K}}_{y}+(\widetilde{I}_{zz}-\widetilde{I}_{xx})\widetilde{\Omega}_{z}{\widetilde{\Omega}}_{x}\\\
{\widetilde{K}}_{z}+(\widetilde{I}_{xx}-\widetilde{I}_{yy})\widetilde{\Omega}_{x}{\widetilde{\Omega}}_{y}\end{array}\right].$
(15)

We recognise the Euler equations for the rotation speed. Equations (15)
combined with the kinematic relations (10) define a first order system of 12
differential equations. If we provide an initial state for the floater
position and speed, we can numerically integrate this system forward in time.

In the absence of waves, $\epsilon=0$, we expect the floaters to be in
equilibrium, with $\boldsymbol{F}=0$ and $\boldsymbol{K}=0$. Due to symmetry
of the floater, we expect the Archimedes torque to vanish for a perfectly
leveled, flat floater. From the Archimedes force balance, we can calculate the
equilibrium submersion depth as $\overline{h}=\beta\delta_{z}$. The floater
center $C$ is exactly $\delta_{z}/2$ above the bottom face of the floater.
Hence, at equilibrium, we have floater coordinates

$\overline{x}_{c},\ \overline{y}_{c}\ ,\
\overline{\psi}\quad\text{arbitrary}\quad,\quad\overline{z}_{c}=\left(\frac{1}{2}-\beta\right)\delta_{z}\quad,\quad\overline{\theta}=\overline{\varphi}=0.$
(16)

Here and further, we use bars to denote equilibrium positions.

###  3.3 Numerical simulations

To obtain a numerical solution to the specified equations of motion, we use
the standard Runge-Kutta 4th order explicit numerical scheme. The numerical
calculation of the surface integrals (13) that define $\boldsymbol{F}$ and
$\boldsymbol{K}$ is non-trivial because the submerged surface $S_{sub}$ varies
in time. We use the following procedure to compute these integrals (see sketch
of figure 5). On each of the six faces of the floater surface, we define two-
dimensional rectangular meshes that contain
$\widetilde{x},\widetilde{y},\widetilde{z}$ coordinates on those faces. To
evaluate the force components ${F}_{x},{F}_{y},{F}_{z}$ at a given time $t$,
we use a loop that visits all six faces. On each face, we then first calculate
the lab-frame $({x},{y},{z})$ coordinates of the points on that face, using
the coordinate transform (8b) and the present position
$x_{c}(t),y_{c}(t),z_{c}(t),\varphi(t),\theta(t),\psi(t)$. With these lab-
frame coordinates, we can then evaluate the pressure $(p-p_{0})$ on that face
using (6). To handle the fact that the faces can be totally, partially or not
submerged, we introduce an indicator function
$f=(1+\tanh((\zeta(x,t)-z)/l)/2$, with $l$ a length over which the interface
is smoothed. This function is equal to $1$ in the liquid and $0$ in the air
and hence, the product $(p-p_{0})f$ is only non-zero on the submerged points
of that face. Using a two-dimensional quadrature rule and the numerical values
of $(p-p_{0})f$ on the face, we can then compute the surface integral on that
face. Each face gives a local contribution to the force in the direction of
the local inward normal $d\boldsymbol{S}$, so after having visited all 6
faces, we obtain the components
$\widetilde{F}_{x},\widetilde{F}_{y},\widetilde{F}_{z}$ in the floater frame.
Using the transform (8a) we then finally obtain the force components
${F}_{x},{F}_{y},{F}_{z}$ in the lab frame. The torque components
$\widetilde{K}_{x},\widetilde{K}_{y},\widetilde{K}_{z}$ are calculated in the
same way.

We have done several static and dynamical tests in the absence of waves
($\epsilon=0$). In the static tests, we validated the calculation of
$\boldsymbol{F}$ and $\boldsymbol{K}$ on floaters that were submerged and
rotated to positions for which we could easily compute the force and torque
analytically. In the dynamical tests, we used the code to reproduce sustained
free oscillations of the floaters. When we release the floater slightly off an
equilibrium position (16), we expect free “bobbing” oscillations in the
vertical $z_{c}$ or angular $\theta,\varphi$ coordinates, of non-dimensional
frequencies (Falnes & Kurniawan, 2020)

$\omega_{z}=\sqrt{\frac{1}{\beta\delta_{z}}},\quad\omega_{\theta}=\sqrt{\frac{1}{\beta\delta_{z}}\left(\frac{\delta_{x}^{2}+6\beta(\beta-1)\delta_{z}^{2}}{\delta_{x}^{2}+\delta_{z}^{2}}\right)},\quad\omega_{\varphi}=\sqrt{\frac{1}{\beta\delta_{z}}\left(\frac{\delta_{y}^{2}+6\beta(\beta-1)\delta_{z}^{2}}{\delta_{y}^{2}+\delta_{z}^{2}}\right)}.\quad$
(17)

Figure 5(b) shows an example of timeseries for free vertical oscillations in
$z_{c}$, for a floater with $\beta=0.5$ released slightly above its
equilibrium position $z_{c}=0$. As illustrated in the figure, the expected
oscillatory motion $z_{c}=A\cos(\omega_{z}t)$ is accurately reproduced by our
code.

The free oscillations in $z_{c},\theta,\varphi$ are useful to test the code
but do not reflect the behavior of the floaters in our experiment. Restricting
to floaters much smaller than the wavelength, $\delta\ll 1$, implies that the
bobbing frequencies (17) are much larger than the incoming wave frequency
($\omega=1$ in non-dimensional units). Accordingly, such fast bobbing
oscillations are never resonant, and are expected to be rapidly damped in the
experiments, either by viscous friction or radiation loss. Since no damping is
included in the simulations, we need to minimize these parasitic bobbing
excitations. We use for this the following strategy: At time $t=0$, we place
the floater at its equilibrium position (16), and gradually ramp up the wave
amplitude in time, by replacing $\epsilon$ by $\epsilon(1-\exp(-t/T))$ in the
definition of pressure and surface height (6). Practice shows that with
$T=15\times 2\pi$, we can keep the fast bobbing oscillations of the floater
motion at low amplitude while capturing the slow dynamics induced by the wave
motion.

Figure 6: Time series for yaw angle for numerically simulated floaters with
$\beta=0.44$, $L_{y}=10$ mm, $L_{z}=5$ mm and $L_{x}$ varying from $30$ to
$120$ mm. The incoming wave has wavelength $\lambda=0.29$ m and
$\epsilon=ka=0.16$, as in the experiments of figure 2. The L-T transition
occurs near $L_{xc}\approx 75$ mm, in good agreement with the experiments.

We now consider numerical solutions for the floater motion in presence of an
incoming wave. In figure 6, we show time series of the yaw angle $\psi(t)$,
for the same experimental conditions used in figure 2: floater size $L_{y}=10$
mm, $L_{z}=5$ mm and $L_{x}$ varying from $30$ to $120$ mm, density ratio
$\beta=0.44$, wavelength $\lambda=0.29$ m and wave steepness
$\epsilon=ka=0.16$, initial yaw angle $\psi_{0}=45^{o}$. After a transient of
typically 15 wave periods given by the ramp in the wave amplitude, the yaw
angle $\psi$ shows fast oscillations (of period $2\pi$), that corresponds to
the back and forth motion of the floaters, superimposed to slow oscillations
around either the longitudinal position $\psi=0^{o}$ for short floaters or the
transverse position $\psi=90^{o}$ for long floaters. The fast oscillations
were not visible in figure 2(a) where the yaw angle measurement was
synchronised at the wave frequency. The slow oscillations reflect the marginal
stability of the fixed points in this dissipationless model. Apart from these
differences, the numerical curves show a L-T transition somewhere in between
$L_{x}=70$ mm and $80$ mm, in excellent agreement with the experimental
transition estimated at $L_{xc}=75$ mm for this set of floaters.

Figure 7: Preferential state of orientation in the $(F,\epsilon)$ plane
according to the numerical simulations. More than 500 numerical simulations
have been performed for varying
$\beta,\delta_{x},\delta_{y},\delta_{z},\epsilon$, showing an excellent
agreement with the theoretical prediction $F_{c}=60$.

We have repeated the simulations for more than 500 floaters, with parameters
varied in broad ranges and even slightly beyond the $\epsilon,\delta\ll 1$
limit for which our model is designed:
$\beta\in[0.03,0.97],\delta_{x}\in[0.004,5.43],\delta_{y}\in[0.01,0.36],\delta_{z}\in[0.007,0.29],\epsilon\in[0.01,0.33]$.
We summarize the preferential floater orientation in figure 7 using the same
representation $(F,\epsilon)$ as in the experiments (Fig. 3). The L-T
transition is clearly located at $F_{c}=60$ with no dependence in $\epsilon$,
in excellent agreement with the asymptotic theory developed in the next
section.

##  4 Asymptotic description of floater motion

The main objective of this section is to use asymptotic theory to derive the
nonlinear evolution equation (1) for the slow motion of the yaw angle $\psi$
in the limit of $\epsilon,\delta\ll 1$, and to prove that the criterion (2)
describes the L-T transition.

Let us start by specifying how the floater motion is decomposed in this
theory. Without wave, $\epsilon=0$, the floater is in its equilibrium position
(16). With a low amplitude wave, $\epsilon\ll 1$, deviations from equilibrium
can be parametrized as

$\left\\{\begin{array}[]{rcl}x_{c}&=&\overline{x}_{c}(\tau)+x_{c}^{\prime}(t)+O(\epsilon^{2})\\\
y_{c}&=&\overline{y}_{c}\\\
z_{c}&=&\overline{z}_{c}+z_{c}^{\prime}(t)+O(\epsilon^{2})\end{array}\right.\quad,\quad\left\\{\begin{array}[]{rcl}\theta&=&0+\theta^{\prime}(t)+O(\epsilon^{2})\\\
\varphi&=&0+\varphi^{\prime}(t)+O(\epsilon^{2})\\\
\psi&=&\overline{\psi}(\tau)+\psi^{\prime}(t)+O(\epsilon^{2})\end{array}\right.$
(18)

Next to ”mean” parts (bars) that are of order $O(1)$, we find smaller
$O(\epsilon)$ ”oscillations” that vary on the rapid time-scale $t$ of the wave
(primes). We also admit that $\overline{x}_{c},\overline{\psi}$ can vary on a
long time-scale $\tau=t/\epsilon$. The coordinate $\overline{y}_{c}$ needs no
further consideration as it remains constant in this Froude-Krylov model. In
the calculation, we will first find the $O(\epsilon)$ deviations
$x_{c}^{\prime}(t),z_{c}^{\prime}(t),\varphi^{\prime}(t),\theta^{\prime}(t),\psi^{\prime}(t)$,
that carry the harmonic, linear response of the floater to the incoming wave.
With this we can compute the second order, $O(\epsilon^{2})$ torque that
defines the evolution equations of the yaw angle $\overline{\psi}(\tau)$.

The fact that roll and pitch angles $\varphi,\theta$ always remain close to
$0$ allows to simplify (linearize) all the dependencies on these angles:
$s_{\theta}\approx\theta^{\prime}+O(\epsilon^{3}),c_{\theta}\approx
1+O(\epsilon^{2}),s_{\varphi}\approx\varphi^{\prime}+O(\epsilon^{3}),c_{\varphi}\approx
1+O(\epsilon^{2})$. The transform formula (8) then reduces to

$\left[\begin{array}[]{c}\boldsymbol{e}_{x}\\\ \boldsymbol{e}_{y}\\\
\boldsymbol{e}_{z}\end{array}\right]=\underbrace{\left[\begin{array}[]{ccc}c_{\psi}&-s_{\psi}&\left(c_{\psi}\theta^{\prime}+s_{\psi}\varphi^{\prime}\right)\\\
s_{\psi}&c_{\psi}&\left(s_{\psi}\theta^{\prime}-c_{\psi}\varphi^{\prime}\right)\\\
-\theta^{\prime}&\varphi^{\prime}&1\end{array}\right]}_{\mathcal{R}^{T}}\left[\begin{array}[]{c}\widetilde{\boldsymbol{e}}_{x}\\\
\widetilde{\boldsymbol{e}}_{y}\\\
\widetilde{\boldsymbol{e}}_{z}\end{array}\right]+O(\epsilon^{2})$ (19)

and similar for the transform formula for vector components and coordinates in
both frames [see equation (8b)]. Up to $O(\epsilon^{2})$, we can define the
inverse transforms with the transposed matrix $\mathcal{R}$.

Let us now write a preliminary version of the evolution equations that
determine these variables. For the first order deviations
$x_{c}^{\prime},z_{c}^{\prime},\varphi^{\prime},\theta^{\prime}$ we have

$m\ddot{x}_{c}^{\prime}=F_{x}^{\prime}\quad,\quad
m\ddot{z}_{c}^{\prime}=F_{z}^{\prime}\quad,\quad\widetilde{I}_{xx}\ddot{\varphi}^{\prime}=\widetilde{K}_{x}^{\prime}\quad,\quad\widetilde{I}_{yy}\ddot{\theta}^{\prime}=\widetilde{K}_{y}^{\prime}$
(20)

The forces $F_{x}^{\prime},F_{y}^{\prime}$ and torques
$\widetilde{K}_{x}^{\prime},\widetilde{K}_{z}^{\prime}$ contain the
linearized, $O(\epsilon)$ part of the total force and torque, and we explain
below how they are calculated in practice. The evolution equation for the yaw
angle $\psi$ needs to be defined at order $O(\epsilon^{2})$ and this requires
some attention. With small $\theta,\varphi$, we have
$s_{\varphi}\approx\varphi^{\prime}$,
$\widetilde{\Omega}_{x}\approx\dot{\varphi}^{\prime}$ and
$\widetilde{\Omega}_{y}\approx\dot{\theta}^{\prime}$ at $O(\epsilon)$. From
Eqs. (10) and (15) we then get that the equations

$\dot{\psi}=\widetilde{\Omega}_{z}+\varphi^{\prime}\,\dot{\theta}^{\prime}+O(\epsilon^{3})\quad,\quad\widetilde{I}_{zz}\dot{\widetilde{\Omega}}_{z}={\widetilde{K}}_{z}+(\widetilde{I}_{xx}-\widetilde{I}_{yy})\dot{\varphi}^{\prime}\dot{\theta}^{\prime}.$
(21)

define the order $O(\epsilon^{2})$ description of yaw angle. These equations
contain many different terms and so the description of the second order motion
of the yaw angle seems rather complex. We remark the following simplification.
Using the transform (19), the fact that
$\widetilde{K}_{x}=\widetilde{K}_{x}^{\prime}+O(\epsilon^{2})$,
$\widetilde{K}_{y}=\widetilde{K}_{y}^{\prime}+O(\epsilon^{2})$ and the
evolution equations (20), we can rewrite the torque component
${\widetilde{K}}_{z}$ as

$\displaystyle
K_{z}=-\theta^{\prime}\widetilde{K}_{x}+\varphi^{\prime}\widetilde{K}_{y}+{\widetilde{K}}_{z}\quad\Rightarrow\quad{\widetilde{K}}_{z}=K_{z}+\widetilde{I}_{xx}\ddot{\varphi}^{\prime}\theta^{\prime}-\widetilde{I}_{yy}\varphi^{\prime}\ddot{\theta}^{\prime}.$
(22)

If we inject this expression of ${\widetilde{K}}_{z}$ into the second equation
of (21) and use the first one to eliminate
$\widetilde{\Omega}_{z}=\dot{\psi}-\varphi^{\prime}\,\dot{\theta}^{\prime}$,
we obtain the following equation for $\psi$:

$\displaystyle\widetilde{I}_{zz}\ddot{\psi}$ $\displaystyle=$ $\displaystyle
K_{z}+\underbrace{\frac{d}{dt}\left(\widetilde{I}_{xx}\dot{\varphi}^{\prime}\theta^{\prime}-\widetilde{I}_{yy}\varphi^{\prime}\dot{\theta}^{\prime}+\widetilde{I}_{zz}\varphi^{\prime}\,\dot{\theta}^{\prime}\right)}_{\text{second
order harmonics}\ \sim\ e^{\pm i2t}}+O(\epsilon^{3}).$ (23)

In the right hand side, we find $K_{z}$, the vertical torque component in the
lab-frame, next to a term $d/dt(\ldots)$ that is a time-derivative of
$O(\epsilon^{2})$ products. In the evolution equation, this term can only
produce harmonics with frequency $\pm 2$, but no terms that are stationary. As
a consequence, this group of terms cannot create mean motion in
$\overline{\psi}$. Of order $O(\epsilon^{2})$, this term $d/dt(\ldots)$ will
also have no effect on the $O(\epsilon)$ rapid motion of $\psi^{\prime}$. In
summary, this just means that we can derive the equations for the yaw angle
variables $\overline{\psi}$ and $\psi^{\prime}$ from the balance
$\widetilde{I}_{zz}\ddot{\psi}\approx K_{z}$ and this is a very significant
simplification with respect to what we initially had in (21).

To separate the rapid from the slow motion, we still need to split the torque
$K_{z}$ into a rapidly varying $O(\epsilon)$ fluctuating part and slowly
varying $O(\epsilon^{2})$ mean part. We denote this as

$K_{z}=\overline{K}_{z}+K_{z}^{\prime}+O(\epsilon^{3}).$ (24)

The part $K_{z}^{\prime}$ is of order $O(\epsilon)$ and oscillates on the
time-scale of the wave. The part $\overline{K}_{z}$ is of order
$O(\epsilon^{2})$ and is stationary on the rapid time-scale of the wave.
Calculating this torque up to second order is the main difficulty in the
asymptotic theory, but once we have it, the equations of motion for the yaw
angle variables are as simple as

$\displaystyle\widetilde{I}_{zz}\ddot{\psi}^{\prime}=K_{z}^{\prime}\quad,\quad\widetilde{I}_{zz}\ddot{\overline{\psi}}=\overline{K}_{z}.$
(25)

A preliminary version of the necessary evolution equations has now been
specified, but we still need to calculate the force and torque components
$F_{x}^{\prime},F_{z}^{\prime},\widetilde{K}_{x}^{\prime},\widetilde{K}_{y}^{\prime},K_{z}^{\prime},\overline{K}_{z}$
analytically. We start from the volume formulation (14) of $\boldsymbol{F}$
and $\boldsymbol{K}$. Analytical evaluation of these volume integrals is not
simple for several reasons: (i) the submerged volume $V_{sub}$, is
continuously changing in time and this motion needs to be parametrised in
floater frame coordinates $(\widetilde{x},\widetilde{y},\widetilde{z})$, (ii)
integrands are easily expressed in lab-frame coordinates $(x,y,z)$, but need
to be expressed in floater frame coordinates. Both problems can be handled by
making use of the assumption that the floater is small with respect to the
wavelength. This allows to Taylor expand fields near the floater and in
practice, to calculate the integrals.

Let us first explain how we handle problem (i), how we parametrise the volume
integrals. In all calculations, we use

$\int_{V_{sub}}\ldots
dV=\int_{-\delta_{x}/2}^{\delta_{x}/2}\int_{-\delta_{y}/2}^{\delta_{y}/2}\int_{-\delta_{z}/2}^{\widetilde{\zeta}(\widetilde{x},\widetilde{y},t)}\ldots
d\widetilde{x}\,d\widetilde{y}\,d\widetilde{z}.$ (26)

We introduce here a new function,
$\widetilde{\zeta}(\widetilde{x},\widetilde{y},t)$, that defines the height of
the water surface as seen in the floater frame. This way of calculating the
integral implicitly supposes that the top face of the floater is never
partially submerged ($\widetilde{\zeta}<\delta_{z}/2)$ and that the bottom
face is always totally submerged ($\widetilde{\zeta}>-\delta_{z}/2)$ through
the motion. This is a sensible approximation for low $\epsilon$ and
intermediate values of $\beta$. When $\beta$ gets too close to $0$ (very light
floaters) or $1$ (equal density floaters) we expect some discrepancies.
Considering that the floater is supposed small with respect to the wavelength,
we now construct $\widetilde{\zeta}(\widetilde{x},\widetilde{y},t)$ as a
polynomial approximation, locally adapted near the floater. We start by Taylor
expanding the surface elevation in the lab-frame $\zeta(x,t)$ around $x_{c}$,

$\zeta(x,t)=\zeta_{c}+(x-x_{c})\partial_{x}\zeta_{c}+\frac{(x-x_{c})^{2}}{2}\partial^{2}_{xx}\zeta_{c}+\frac{(x-x_{c})^{3}}{6}\partial^{3}_{xxx}\zeta_{c}+\ldots$
(27)

The index c is used to express that the field is evaluated at the center of
the floater $C$, e.g. $\zeta_{c}=\zeta(x_{c},t)=\epsilon\sin(x_{c}-t)$,
$\partial_{x}\zeta_{c}=\partial_{x}\zeta|_{x=x_{c}}=\epsilon\cos(x_{c}-t)$,
etc. We inject this Taylor expansion in the equation $z=\zeta(x,t)$ that
defines the free surface and we replace both $z\approx
z_{c}+(-\theta^{\prime}\widetilde{x}+\varphi^{\prime}\widetilde{y}+\widetilde{z})$
and $x-x_{c}\approx c_{\psi}\widetilde{x}-s_{\psi}\widetilde{y}+O(\epsilon)$,
the leading order parts of the coordinate transform formula (19). Reorganising
this equation into
$\widetilde{z}=\widetilde{\zeta}(\widetilde{x},\widetilde{y},t)$ allows to
identify

$\displaystyle\widetilde{\zeta}(\widetilde{x},\widetilde{y},t)$
$\displaystyle=$ $\displaystyle\underbrace{-\overline{z}_{c}}_{O(\delta)}\ \
\underbrace{-z_{c}^{\prime}+\zeta_{c}}_{O(\epsilon)}+\underbrace{(\theta^{\prime}+c_{\psi}\partial_{x}\zeta_{c})\widetilde{x}+(-\varphi^{\prime}-s_{\psi}\partial_{x}\zeta_{c})\widetilde{y}}_{O(\epsilon\delta)}$
$\displaystyle\underbrace{+\frac{1}{2}\left(c_{\psi}\widetilde{x}-s_{\psi}\widetilde{y}\right)^{2}\partial^{2}_{xx}\zeta_{c}}_{O(\epsilon\delta^{2})}+\underbrace{\frac{1}{6}\left(c_{\psi}\widetilde{x}-s_{\psi}\widetilde{y}\right)^{3}\partial^{3}_{xxx}\zeta_{c}}_{O(\epsilon\delta^{3})}+\,O(\epsilon^{2},\epsilon\delta^{4})$

For clarity, we group the terms that have same order of magnitude in terms of
powers of $\delta$ and $\epsilon$. We do not always need all the terms in this
expansion but at some moments we do. With the function
$\widetilde{\zeta}(\widetilde{x},\widetilde{y},t)$ known, we can split the
integration over the submerged volume $V_{sub}$ in two parts:

$\int_{V_{sub}}\ldots
dV=\underbrace{\int_{-\delta_{x}/2}^{\delta_{x}/2}\int_{-\delta_{y}/2}^{\delta_{y}/2}\int_{-\delta_{z}/2}^{-\overline{z}_{c}}\ldots
d\widetilde{x}\,d\widetilde{y}\,d\widetilde{z}}_{\int_{V_{sub}^{(0)}}\ldots\
dV=\
O(\delta^{3})}+\underbrace{\int_{-\delta_{x}/2}^{\delta_{x}/2}\int_{-\delta_{y}/2}^{\delta_{y}/2}\int_{-\overline{z}_{c}}^{\widetilde{\zeta}}\ldots
d\widetilde{x}\,d\widetilde{y}\,d\widetilde{z}}_{\int_{V_{sub}^{(1)}}\ldots\
dV=\ O(\epsilon\delta^{3})}.$ (29)

The integral over $V_{sub}^{(0)}$ covers the equilibrium submerged volume. The
integral over $V_{sub}^{(1)}$ covers the $O(\epsilon)$ perturbation of
submerged volume caused by the wave. In the theory, we only need an
$O(\epsilon)$ approximation of this $V_{sub}^{(1)}$ integral and then we can
use

$\displaystyle\int_{V_{sub}^{(1)}}f\ dV$ $\displaystyle\approx$
$\displaystyle\int_{-\delta_{x}/2}^{\delta_{x}/2}\int_{-\delta_{y}/2}^{\delta_{y}/2}f(\widetilde{x},\widetilde{y},-\overline{z}_{c})\left(\widetilde{\zeta}(\widetilde{x},\widetilde{y},t)+\overline{z}_{c}\right)d\widetilde{x}\,d\widetilde{y}.$
(30)

as an approximation, where $f$ is an arbitrary field.

Let us now explain how we handle problem (ii), i.e., how we express the
integrands in the floater frame coordinates. In the force and torque formula
(14), we find the fields

$a_{x}=\partial_{t}u_{x}=-\epsilon e^{z}\cos(x-t)\quad,\quad
a_{z}=\partial_{t}u_{z}=-\epsilon e^{z}\sin(x-t)$ (31)

These fields depend on the lab-frame coordinates $x,z$ but need to be
expressed in terms of $\widetilde{x},\widetilde{y},\widetilde{z}$ variables if
we want to compute the integrals analytically. Also here, the idea is to
replace $a_{x},a_{z}$ with polynomial approximations, valid near the floater.
Using Taylor series, we have

$\displaystyle\boldsymbol{a}$ $\displaystyle=$
$\displaystyle\boldsymbol{a}_{c}+(x-x_{c})\partial_{x}\boldsymbol{a}_{c}+(z-z_{c})\partial_{z}\boldsymbol{a}_{c}$
$\displaystyle+\frac{1}{2}(x-x_{c})^{2}\partial_{xx}^{2}\boldsymbol{a}_{c}+\frac{1}{2}(z-z_{c})^{2}\partial^{2}_{zz}\boldsymbol{a}_{c}+(x-x_{c})(z-z_{c})\partial^{2}_{xz}\boldsymbol{a}_{c}+O(\epsilon\delta^{3})$

Replacing
$x-x_{c}=c_{\psi}\widetilde{x}-s_{\psi}\widetilde{y}+\left(c_{\psi}\theta^{\prime}+s_{\psi}\varphi^{\prime}\right)\widetilde{z}$
and
$z-z_{c}=-\theta^{\prime}\widetilde{x}+\varphi^{\prime}\widetilde{y}+\widetilde{z}$
from (19) we get

$\boldsymbol{a}=\boldsymbol{a}^{(1)}+\boldsymbol{a}^{(2)}+O(\epsilon^{3}).$
(33a) There is an $O(\epsilon)$ part $\displaystyle\boldsymbol{a}^{(1)}$
$\displaystyle=$
$\displaystyle\underbrace{\boldsymbol{a}_{c}}_{O(\epsilon)}+\underbrace{(c_{\psi}\widetilde{x}-s_{\psi}\widetilde{y})\partial_{x}\boldsymbol{a}_{c}+\widetilde{z}\partial_{z}\boldsymbol{a}_{c}}_{O(\epsilon\delta)}$
$\displaystyle+\underbrace{\frac{1}{2}(c_{\psi}\widetilde{x}-s_{\psi}\widetilde{y})^{2}\partial_{xx}^{2}\boldsymbol{a}_{c}+\frac{\widetilde{z}^{2}}{2}\partial_{zz}^{2}\boldsymbol{a}_{c}+(c_{\psi}\widetilde{x}-s_{\psi}\widetilde{y})\widetilde{z}\partial_{xz}^{2}\boldsymbol{a}_{c}}_{O(\epsilon\delta^{2})}$
and an $O(\epsilon^{2})$ part $\displaystyle\boldsymbol{a}^{(2)}$
$\displaystyle=$
$\displaystyle\underbrace{(c_{\psi}\theta^{\prime}+s_{\psi}\varphi^{\prime})\widetilde{z}\partial_{x}\boldsymbol{a}_{c}+(-\theta^{\prime}\widetilde{x}+\varphi^{\prime}\widetilde{y})\partial_{z}\boldsymbol{a}_{c}}_{O(\epsilon^{2}\delta)}$
$\displaystyle+\bigg{[}(c_{\psi}\widetilde{x}-s_{\psi}\widetilde{y})(c_{\psi}\theta^{\prime}+s_{\psi}\varphi^{\prime})\widetilde{z}\partial^{2}_{xx}\boldsymbol{a}_{c}+(-\theta^{\prime}\widetilde{x}+\varphi^{\prime}\widetilde{y})\widetilde{z}\partial^{2}_{zz}\boldsymbol{a}_{c}$
$\displaystyle\
\underbrace{\quad\quad+\left[(c_{\psi}\widetilde{x}-s_{\psi}\widetilde{y})(-\theta^{\prime}\widetilde{x}+\varphi^{\prime}\widetilde{y})+(c_{\psi}\theta^{\prime}+s_{\psi}\varphi^{\prime})\widetilde{z}^{2}\right]\partial_{xz}^{2}\boldsymbol{a}_{c}\bigg{]}.}_{O(\epsilon^{2}\delta^{2})}$

This second order part $\boldsymbol{a}^{(2)}$ is of no importance for the
harmonic, linear response of the floater to the wave. However, it does
contribute to the second order mean torque $\overline{K}_{z}$.

We now have explained all the mathematical tools that are needed to calculate
the force and torque components. Using the notations of the previous
paragraphs, we use the following formula to compute the $O(\epsilon)$
fluctuating forces and torques that appear in the equations of motion (20):

$\displaystyle F_{x}^{\prime}$ $\displaystyle=$
$\displaystyle\left(\int_{V_{sub}^{(0)}}a_{x}^{(1)}dV\,\right)_{(\overline{x}_{c},\overline{z}_{c},\overline{\psi})}$
(34a) $\displaystyle F_{z}^{\prime}$ $\displaystyle=$
$\displaystyle\left(\int_{V_{sub}^{(0)}}a_{z}^{(1)}\,dV+\int_{V_{sub}^{(1)}}dV\right)_{(\overline{x}_{c},\overline{z}_{c},\overline{\psi})}$
(34b) $\displaystyle\widetilde{K}^{\prime}_{x}$ $\displaystyle=$
$\displaystyle\left(\int_{V_{sub}^{(0)}}\left(\widetilde{y}a_{z}^{(1)}+s_{\psi}\widetilde{z}a_{x}^{(1)}-\widetilde{z}\varphi^{\prime}\right)dV+\int_{V_{sub}^{(1)}}\widetilde{y}dV\right)_{(\overline{x}_{c},\overline{z}_{c},\overline{\psi})}$
(34c) $\displaystyle\widetilde{K}^{\prime}_{y}$ $\displaystyle=$
$\displaystyle\left(\int_{V_{sub}^{(0)}}\left(-\widetilde{x}a_{z}^{(1)}+c_{\psi}\widetilde{z}a_{x}^{(1)}-\widetilde{z}\theta^{\prime}\right)dV+\int_{V_{sub}^{(1)}}(-\widetilde{x})dV\right)_{(\overline{x}_{c},\overline{z}_{c},\overline{\psi})}$
(34d)

To obtain the formula for these torque components, we have used
$\boldsymbol{r}-\boldsymbol{r}_{c}=\widetilde{x}\widetilde{\boldsymbol{e}}_{x}+\widetilde{y}\widetilde{\boldsymbol{e}}_{y}+\widetilde{z}\widetilde{\boldsymbol{e}}_{z}$
and the fact that
$\boldsymbol{a}=c_{\psi}a_{x}\widetilde{\boldsymbol{e}}_{x}-s_{\psi}a_{x}\widetilde{\boldsymbol{e}}_{y}+a_{z}\widetilde{\boldsymbol{e}}_{z}+O(\epsilon^{2})$
in the floater frame, according to the transform (19). In practice, the
calculation of these integrals is not particularly difficult as we only have
to integrate polynomials. The subscript
${(\overline{x}_{c},\overline{z}_{c},\overline{\psi})}$ expresses that the
formula, obtained after integration, needs to be evaluated at the equilibrium
position $x_{c}\approx\overline{x}_{c}$, $z_{c}\approx\overline{z}_{c}$ and
$\psi\approx\overline{\psi}$.

Remain the torque components $\overline{K}_{z}$ and $K_{z}^{\prime}$ that need
to be calculated up to respective orders $O(\epsilon^{2})$ and $O(\epsilon)$.
Using the definition of $K_{z}$ and the coordinate transform formula for
$y-y_{c}$, we have

$\displaystyle K_{z}$ $\displaystyle=$
$\displaystyle-\int_{V_{sub}}(y-y_{c})a_{x}\,dV=-\int_{V_{sub}}\left(s_{\psi}\widetilde{x}+c_{\psi}\widetilde{y}+\left(s_{\psi}\theta^{\prime}-c_{\psi}\varphi^{\prime}\right)\widetilde{z}\right)a_{x}\,dV+O(\epsilon^{3}).$

Using the previous notations, we separate this torque in two parts
$K_{z}=K_{z}^{(1)}+K_{z}^{(2)}+O(\epsilon^{3})$ of respective orders
$O(\epsilon)$ and $O(\epsilon^{2})$. We have

$\displaystyle K_{z}^{(1)}$ $\displaystyle=$
$\displaystyle-\int_{V_{sub}^{(0)}}\left(s_{\psi}\widetilde{x}+c_{\psi}\widetilde{y}\right)a_{x}^{(1)}dV$
(35a) $\displaystyle K_{z}^{(2)}$ $\displaystyle=$
$\displaystyle-\int_{V_{sub}^{(0)}}\left(s_{\psi}\theta^{\prime}-c_{\psi}\varphi^{\prime}\right)\widetilde{z}a_{x}^{(1)}dV\
-\int_{V_{sub}^{(0)}}\left(s_{\psi}\widetilde{x}+c_{\psi}\widetilde{y}\right)a_{x}^{(2)}dV\
-\int_{V_{sub}^{(1)}}\left(s_{\psi}\widetilde{x}+c_{\psi}\widetilde{y}\right)a_{x}^{(1)}dV.$

These torques $K_{z}^{(1)}$ and $K_{z}^{(2)}$ are not yet the torques
$K_{z}^{\prime}$ and $\overline{K}_{z}$ that enter the evolution equations of
$\psi^{\prime}$ and $\overline{\psi}$. The fluctuating torque is just

$K_{z}^{\prime}=\Big{(}K_{z}^{(1)}\Big{)}_{(\overline{x}_{c},\overline{z}_{c},\overline{\psi})}$
(36)

which is equation (35a) evaluated at the equilibrium position. The
$O(\epsilon^{2})$ mean torque $\overline{K}_{z}$ is more subtle and has two
parts

$\displaystyle\overline{K}_{z}$ $\displaystyle=$
$\displaystyle\overline{\Big{(}K_{z}^{(1)}\Big{)}}_{(\overline{x}_{c}+x_{c}^{\prime},\overline{z}_{c}+z_{c}^{\prime},\overline{\psi}+\psi^{\prime})}+\overline{\left({K}_{z}^{(2)}\right)}_{(\overline{x}_{c},\overline{z}_{c},\overline{\psi})}$
(37)

The bars in the right hand side denote a time-average. The second term is just
the time-average of ${K}_{z}^{(2)}$ evaluated at the equilibrium position. The
first term is a ”drift”- like contribution that can be understood in the same
way as we understand the Stokes drift phenomenon (Stokes, 1847). As the
floater moves along with the wave, goes through its first order motion, it
explores regions of slightly higher and lower velocity. On average, this can
result in a net displacement (Stokes drift) and here, for our spatially
extended floaters, it also results in a weak $O(\epsilon^{2})$ mean torque. In
practice, we calculate this ”drift”-like torque by using a Taylor series
around the equilibrium position
$(\overline{x}_{c},\overline{z}_{c},\overline{\psi})$. Up to order
$O(\epsilon^{2})$ we may use

$\displaystyle\overline{K}_{z}$ $\displaystyle\approx$
$\displaystyle\underbrace{\overline{\left(K_{z}^{(1)}\right)}_{(\overline{x}_{c},\overline{z}_{c},\overline{\psi})}}_{0}+\overline{\left(x_{c}^{\prime}\frac{\partial
K_{z}^{(1)}}{\partial x_{c}}+z_{c}^{\prime}\frac{\partial
K_{z}^{(1)}}{\partial z_{c}}+\psi^{\prime}\frac{\partial
K_{z}^{(1)}}{\partial\psi}\right)}_{(\overline{x}_{c},\overline{z}_{c},\overline{\psi})}+\overline{\left({K}_{z}^{(2)}\right)}_{(\overline{x}_{c},\overline{z}_{c},\overline{\psi})}.$
(38)

The first term vanishes because we are taking the time-average of
$K_{z}^{\prime}$ that is by constructing a quantity that fluctuates on the
time-scale of the wave. The second group caries the mean effective torque and
requires the first order motion $x_{c}^{\prime},z_{c}^{\prime},\psi^{\prime}$.

The integrals defining
$F_{x}^{\prime},F_{z}^{\prime},\widetilde{K}_{x}^{\prime},\widetilde{K}_{y}^{\prime},K_{z}^{\prime},\overline{K}_{z}$
or now clearly defined but one question remains. We still need to estimate to
which order in $\delta$ we must push all the Taylor series that we use to
approximate $\widetilde{\zeta}$ and the fields $\boldsymbol{a}^{(1)}$,
$\boldsymbol{a}^{(2)}$ in the integrands. Considering the equations of motion
(20) and (25), and the fact that $m=O(\delta^{3})$ and
$\boldsymbol{I}=O(\delta^{5})$, we deduce that we needed to calculate

$\displaystyle F^{\prime}_{x},F^{\prime}_{z}$ $\displaystyle\ \ \text{up to
order}\ O(\epsilon\delta^{4})$
$\displaystyle\widetilde{K}^{\prime}_{x},\widetilde{K}^{\prime}_{y},K_{z}^{\prime}$
$\displaystyle\ \ \text{up to order}\ O(\epsilon\delta^{6})$ (39)
$\displaystyle\overline{K}_{z}$ $\displaystyle\ \ \text{up to order}\
O(\epsilon^{2}\delta^{6}).$

The idea is that this should give access to the leading, $O(\epsilon)$
description of
$x_{c}^{\prime},z_{c}^{\prime},\varphi^{\prime},\theta^{\prime},\psi^{\prime}$,
$O(\epsilon^{2})$ description of $\overline{\psi}$, but also to floater shape-
related corrections of respective orders $O(\epsilon\delta)$ and
$O(\epsilon^{2}\delta)$. In the limit $\delta\ll 1$, these shape-corrections
seem much smaller but this is without considering that these
$O(\epsilon\delta)$ and $O(\epsilon^{2}\delta)$ terms are actually taking the
form of $O(\epsilon\delta_{x}^{2}/\beta\delta_{z})$ and
$O(\epsilon^{2}\delta_{x}^{2}/\beta\delta_{z})$ terms. For strongly elongated
floater, we can easily have

$\frac{\delta_{x}^{2}}{\beta\delta_{z}}\sim 1$ (40)

and in that case, the shape related corrections may become as large or even
larger than the so-called leading order terms. This is precisely what happens
at the L-T transition and a theoretical pain in the neck: we really need to
push the calculation of the forces and torques to the orders of (39),
otherwise we will miss the terms that stabilise the transverse equilibrium.
This, on the contrary is too much and results in very long, unpractical
formula. In our view, the best way, in between calculating no shape-related
corrections and all shape related correction, is to add an extra assumption.
In the experiment, all the floaters are flat and this means that

$\delta_{x},\delta_{y}\gg\delta_{z}$ (41)

is a fair assumption. By exploiting this information in the calculation of the
forces and torques we can ignore (i) all $O(\delta_{z})$ terms with respect to
$O(1)$ terms, (ii) all $O(\delta_{z}^{2})$ terms with respect to
$O(\delta_{x}^{2},\delta_{y}^{2})$ terms. This greatly simplifies the theory
and produces a physically relevant result. We can in fact, even further
simplify the model by taking into account that our floaters are also very
elongated $\delta_{x}\gg\delta_{y}$ in our experiments. This will be done at
the end, but in our calculation, we have found it useful to keep
$\delta_{x},\delta_{y}$ at similar magnitude as it produces more symmetrical
formula that allow supplementary checks. For now, we continue our presentation
with the flat floater assumption (41).

Once
$F_{x}^{\prime},F_{z}^{\prime},\widetilde{K}_{x}^{\prime},\widetilde{K}_{y}^{\prime},{K}_{z}^{\prime}$
are found, we immediately have the differential equations for the first order
motion
$x_{c}^{\prime},z_{c}^{\prime},\varphi^{\prime},\theta^{\prime},\psi^{\prime}$.
Solving these differential equations is rather trivial and yields the harmonic
response

$\displaystyle{x}^{\prime}_{c}$ $\displaystyle\approx$
$\displaystyle\epsilon\cos(\overline{x}_{c}-t)$ (42a) $\displaystyle
z_{c}^{\prime}$ $\displaystyle\approx$
$\displaystyle\epsilon\left(1-\frac{1}{24}\left(\overline{c}^{2}_{\psi}\delta_{x}^{2}+\overline{s}_{\psi}^{2}\delta_{y}^{2}\right)\right)\sin(\overline{x}_{c}-t).$
(42b) $\displaystyle\varphi^{\prime}$ $\displaystyle\approx$
$\displaystyle-\epsilon\overline{s}_{{\psi}}\left(1-\left(\overline{s}_{{\psi}}^{2}\dfrac{\delta_{y}^{2}}{40}+\overline{c}_{{\psi}}^{2}\dfrac{\delta_{x}^{2}}{24}\right)\right)\cos(\overline{x}_{c}-t)$
(42c) $\displaystyle\theta^{\prime}$ $\displaystyle\approx$
$\displaystyle-\epsilon\overline{c}_{{\psi}}\left(1-\left(\overline{c}_{{\psi}}^{2}\dfrac{\delta_{x}^{2}}{40}+\overline{s}_{{\psi}}^{2}\dfrac{\delta_{y}^{2}}{24}\right)\right)\cos(\overline{x}_{c}-t)$
(42d) $\displaystyle\psi^{\prime}$ $\displaystyle\approx$
$\displaystyle\epsilon\left(\frac{\delta_{x}^{2}-\delta_{y}^{2}}{\delta_{x}^{2}+\delta_{y}^{2}}\right)\overline{s}_{{\psi}}\overline{c}_{{\psi}}\sin(\overline{x}_{c}-t),$
(42e)

Let us interpret these equations. At leading $O(\epsilon)$ order, we find in
$x_{c}^{\prime}$ and $z_{c}^{\prime}$ that the floater oscillates around its
mean position, just as a material particle on the wave-surface would do.
Although this is less trivial to see, the leading $O(\epsilon)$ expressions
for the angles $\varphi^{\prime},\theta^{\prime}$ are such that the flat
floater rotates so to align with the local wave slope (imagine the rotation of
a surfboard on a long wavelength wave that passes). The yaw angle
$\psi^{\prime}$ rocks back and forth around the mean value and thus the more
for sufficiently elongated floaters with $\delta_{x}\gg\delta_{y}$ and for
$\overline{\psi}$ close to $45^{o}$. Next the leading $O(\epsilon)$ terms, we
also have some smaller, shape-related corrections of order
$O(\epsilon\delta_{x}^{2},\epsilon\delta_{y}^{2})$ in the
$z_{c}^{\prime},\varphi^{\prime},\theta^{\prime}$ variables. These corrections
may seem utterly small but we really can not ignore them. Physically, they are
due the fact that the water surface is not perfectly flat at the scale of the
floater. This waterline curvature induces small modification in the buoyancy
force that affect the force $F_{z}^{\prime}$ and the torques
$\widetilde{K}_{x},\widetilde{K}_{y}$. In particular the
$O(\epsilon\delta_{x}^{2})$ correction in $z_{c}^{\prime}$ is essential in
this problem: without it, we just cannot explain why long floaters prefer the
transverse equilibrium. In section 5 we provide a more physical understanding
of what this correction actually captures.

With the first order motion specified, we can continue to next order. The
evaluation of the second order mean torque $\overline{K}_{z}$ is a difficult
and technical part in this calculation (some elements are provided in Appendix
B). More important is that for flat floaters with
$\delta_{x},\delta_{y}\gg\delta_{z}$, we obtain the following nonlinear
equation of motion for the slowly varying yaw angle

$\displaystyle\ddot{\overline{\psi}}$ $\displaystyle=$
$\displaystyle\epsilon^{2}\left[\overline{s}_{\psi}\overline{c}_{\psi}^{3}\left(-\frac{(\delta_{x}^{2}-\delta_{y}^{2})\delta_{x}^{2}}{(\delta_{x}^{2}+\delta_{y}^{2})^{2}}+\frac{\dfrac{\delta_{x}^{4}}{60}-\dfrac{\delta_{x}^{2}\delta_{y}^{2}}{48}}{\beta\delta_{z}(\delta_{x}^{2}+\delta_{y}^{2})}\right)-\overline{s}_{\psi}^{3}\overline{c}_{\psi}\left(-\frac{(\delta_{y}^{2}-\delta_{x}^{2})\delta_{x}^{2}}{(\delta_{x}^{2}+\delta_{y}^{2})^{2}}+\frac{\dfrac{\delta_{y}^{4}}{60}-\dfrac{\delta_{x}^{2}\delta_{y}^{2}}{48}}{\beta\delta_{z}(\delta_{x}^{2}+\delta_{y}^{2})}\right)\right]$
(43)

We can separate a common factor $\overline{s}_{\psi}\overline{c}_{\psi}$ in
the right hand side and because of this term, we know that longitudinal
($\overline{\psi}=0^{o}$) and transverse ($\overline{\psi}=90^{o}$) positions
are equilibria. We can also verify how this equation incorporates a
fundamental symmetry: if we exchange $\delta_{x}\leftrightarrow\delta_{y}$ and
rotate the angle $\overline{\psi}\rightarrow\overline{\psi}+\pi/2$, then the
rectangular floater is in the same physical state. When
$\delta_{x}=\delta_{y}$, a new fixed point emerges at
$\overline{\psi}=45^{o}$, but the idea of preferential orientation then looses
its interest.

The equation for the slow evolution of yaw angle can be used to discuss the
stability of the L and T states. However, this discussion becomes even simpler
with the extra assumption of very elongated floaters:
$\delta_{x}\gg\delta_{y}\gg\delta_{z}$. Ignoring all $O(\delta_{y}^{2})$ terms
with respect to $O(\delta_{x}^{2})$ terms, we can simplify (43) further to
find the equation of motion (2) already mentioned in the introduction

$\displaystyle\delta_{x}\gg\delta_{y}\gg\delta_{z}:\quad\quad\ddot{\overline{\psi}}$
$\displaystyle=$
$\displaystyle-\epsilon^{2}\overline{s}_{\psi}\overline{c}_{\psi}^{3}\bigg{(}1-\underbrace{\frac{\delta_{x}^{2}}{60\beta\delta_{z}}}_{F/F_{c}}\bigg{)}.$
(44)

This confirms that the non-dimensional number

$F=\frac{\delta_{x}^{2}}{\beta\delta_{z}}=\frac{kL_{x}^{2}}{\beta L_{z}}$ (45)

is the control parameter that sets the preferential state of orientation. When
$F=F_{c}=60$, the right hand side changes change sign and this is indicative
of a global change in stability. To see this better, we rewrite the equation
in conservative form

$\displaystyle\quad\quad\ddot{\overline{\psi}}$ $\displaystyle=$
$\displaystyle=-\frac{dV(\overline{\psi})}{d\overline{\psi}},\qquad
V(\overline{\psi})=\frac{\epsilon^{2}}{4}\left(-1+\frac{F}{F_{c}}\right)\overline{c}_{\psi}^{4}.$
(46)

This equation tells us that the slow motion of the yaw angle is analogous to
that of a particle in a potential well
$V(\overline{\psi})\sim\cos^{4}\overline{\psi}$. For short floaters with
$F<F_{c}=60$, the longitudinal position of $\overline{\psi}=0^{o}$ is the
minimum of the potential and hence the stable state. For long floaters with
$F>F_{c}$, the situation flips and the transverse equilibrium,
$\overline{\psi}=90^{o}$ becomes the potential minimum and the stable state,
in excellent agreement with the experiments (figure 3) and the numerical
simulations (figure 7).

##  5 Simplified model and physical interpretation

The asymptotic approach already provided some physical insights, but the
calculation of the second order torque $\overline{K}_{z}$ is so technical that
much of the simpler physics remains hidden. In this section, we show how to
find the evolution equation for $\overline{\psi}$ much more rapidly, by
bringing in some physical ideas earlier on in the calculation.

We recall that both the rapid and slow motion of yaw angle $\psi$ are
controlled by the torque component

$K_{z}=\boldsymbol{e}_{z}\cdot\int_{V_{sub}}(\boldsymbol{r}-\boldsymbol{r}_{c})\times\boldsymbol{a}\
dV=-\int_{V_{sub}}(y-y_{c})\,a_{x}\ dV,$ (47)

with $a_{x}=-\epsilon e^{z}\cos(x-t)$. Evaluating this integral is the main
difficulty in the theory and there are numerous contributions that all need
careful consideration (see Appendix B). We have seen at the end of the
previous section that the limit of strongly elongated floaters
$\delta_{x}\gg\delta_{y}\gg\delta_{z}$ is clearly adapted to capture the
preferential orientation phenomenon, so let us exploit this information
directly. If the floater is indeed thin (along $\widetilde{z}$) and not wide
(along $\widetilde{y}$), we can simply ignore the $\widetilde{y}$ and
$\widetilde{z}$ variations in this torque integral and simplify it to

$K_{z}\approx-\int_{-\delta_{x}/2}^{+\delta_{x}/2}(y-y_{c})\,a_{x}\,\delta_{y}\underbrace{\left(\widetilde{\zeta}+\frac{\delta_{z}}{2}\right)}_{\widetilde{h}(\widetilde{x},t)}d\widetilde{x}.$
(48)

For this formula to make sense, we must express $(y-y_{c})a_{x}$ in terms of
floater frame coordinates and ignore the $\widetilde{y}$ and $\widetilde{z}$
variations. We also need to find the local submersion depth
$\widetilde{h}(\widetilde{x},t)=\widetilde{\zeta}+({\delta_{z}}/{2})$ along
the long $\widetilde{x}$-axis of the floater. To get this done, we use a
simplified coordinate transform. From (19) and ignoring all dependencies along
$\widetilde{y}$ and $\widetilde{z}$, we find that

$x\approx x_{c}+c_{\psi}\,\widetilde{x},\quad y\approx
y_{c}+s_{\psi}\,\widetilde{x},\quad z\approx
z_{c}-\theta^{\prime}\widetilde{x}$ (49)

along the long axis of the floater. The lever arm is then $y-y_{c}\approx
s_{\psi}\,\widetilde{x}$. The dependence of the field $a_{x}$ on
$\widetilde{x}$ can be found by injecting this coordinate transform in the
theoretical expression. Then, making use of a Taylor expansion we get

$\displaystyle a_{x}$ $\displaystyle\approx$ $\displaystyle-\epsilon
e^{z_{c}-\theta^{\prime}\widetilde{x}}\cos(x_{c}+c_{\psi}\widetilde{x}-t)$
(50) $\displaystyle\approx$
$\displaystyle-\epsilon\left(1+z_{c}-\theta^{\prime}\widetilde{x}\right)\left(\cos(x_{c}-t)-c_{\psi}\widetilde{x}\,\sin(x_{c}-t)\right)+\ldots$

up to $O(\epsilon^{2})$. To find the local submersion depth
$\widetilde{h}(\widetilde{x},t)$, we ignore the $\widetilde{y}$-dependencies
in the general definition of $\widetilde{\zeta}$ in (4). We also use the fact
that $-\overline{z}_{c}+(\delta_{z}/2)=\beta L_{z}=\overline{h}$ is the
equilibrium submersion depth. In this way, we find that the local submersion
depth $\widetilde{h}(\widetilde{x},t)$ along the floater long axis is

$\widetilde{h}(\widetilde{x},t)\approx\overline{h}+(-z_{c}^{\prime}+\epsilon\sin(x_{c}-t))+(\theta^{\prime}+\epsilon
c_{\psi}\cos(x_{c}-t)\widetilde{x}-\frac{\epsilon}{2}c_{\psi}^{2}\widetilde{x}^{2}\sin(x_{c}-t).$
(51)

The wave surface at the scale of the floater is here approximated by a second
order polynomial and this is sufficient in this simplified model (we do not
need the third derivate terms). In this expression, we need to insert the
first order motion $z_{c}^{\prime}$ and $\theta^{\prime}$. We already know the
first order motion from the asymptotic theory, but let us show how we can
alternatively find this motion using some simpler physical arguments. In the
pressure force and torque, the buoyancy terms are the largest ones, much
larger than the dynamical pressure terms. Hence, we estimate that the floater
is moving in such a way that it keeps its submersed volume nearly constant in
time. This requires

$\frac{1}{\delta_{x}}\int_{-\delta_{x}/2}^{+\delta_{x}/2}\widetilde{h}(\widetilde{x},t)d\widetilde{x}\approx\overline{h}$
(52)

Evaluating the integral using the expression (51), we get the result

$z_{c}^{\prime}\approx\epsilon\sin(x_{c}-t)\left(1-\frac{\delta_{x}^{2}}{24}c_{\psi}^{2}\right).$
(53)

This indeed corresponds to the expression of $z_{c}^{\prime}$ that we have
obtained from a more formal treatment of the equations of motion, in the limit
$\delta_{x}\gg\delta_{y}$, see equation (42b). Interestingly, we also recover
the shape-related $O(\epsilon\delta_{x}^{2})$ correction that is quite crucial
in the model. To explain the angular motion $\theta^{\prime}$, we can use a
similar physical argument. Due to buoyancy, the floater will rotate so as to
align with the local surface slope. The local normal on the wave surface is
$\boldsymbol{n}\approx\boldsymbol{\nabla}(z-\zeta(x,t))|_{x=x_{c}}=\boldsymbol{e}_{z}-\epsilon\cos({x}_{c}-t)\boldsymbol{e}_{x}$
and then needs to be orthogonal to the long axis
$\widetilde{\boldsymbol{e}}_{x}\approx
c_{\psi}\boldsymbol{e}_{x}+s_{\psi}\boldsymbol{e}_{y}-\theta^{\prime}\boldsymbol{e}_{z}$.
Hence, we get

$\boldsymbol{n}\cdot\widetilde{\boldsymbol{e}}_{x}\approx
0\quad\Rightarrow\quad\theta^{\prime}=-\epsilon c_{\psi}\cos(x_{c}-t).$ (54)

This approximation is sufficient, the extra $O(\epsilon\delta_{x}^{2})$
correction in $\theta^{\prime}$ of (42d) is in fact not so important.
Replacing these expressions of $z_{c}^{\prime}$ and $\theta^{\prime}$ into
(51), we find the following approximation for the local shape of the waterline
at the floater position

$\widetilde{h}(\widetilde{x},t)\approx\overline{h}+\epsilon
c_{\psi}^{2}\left(\frac{\delta_{x}^{2}}{24}-\frac{\widetilde{x}^{2}}{2}\right)\sin(x_{c}-t).$
(55)

This result can be interpreted as follows: the waterline as seen from the
floater center always takes the shape of a parabola, symmetrical around
$\widetilde{x}=0$.

Combining all the elements together, we end up with the following simplified
formula for the mean yaw moment $K_{z}$:

$\displaystyle K_{z}$ $\displaystyle\approx$
$\displaystyle-\int_{-\delta_{x}/2}^{+\delta_{x}/2}\underbrace{\widetilde{x}s_{\psi}}_{\text{local
lever
arm}}\,\underbrace{\left(-\epsilon\left(1+z_{c}-\theta^{\prime}\widetilde{x}\right)\left(\cos(x_{c}-t)-c_{\psi}\widetilde{x}\,\sin(x_{c}-t)\right)\right)}_{\text{local
force density $f_{x}$}}$ (56) $\displaystyle\hskip
56.9055pt\times\,\underbrace{\left(\overline{h}+\epsilon
c_{\psi}^{2}\left(\frac{\delta_{x}^{2}}{24}-\frac{\widetilde{x}^{2}}{2}\right)\sin(x_{c}-t)\right)}_{\text{local
submersion $\widetilde{h}$}}\,\delta_{y}\,d\widetilde{x}.$

This integral contains all the physics that explains the preferential
orientation. The torque $K_{z}$ is the result of the local force density
$f_{x}$ that varies along the floater’s long axis applied on the local level
arm. This force density is weighted by the local submersion depth
$\widetilde{h}$ and this effect is thus more important as the floater is
longer. We evaluate this integral up to $O(\epsilon^{2})$ and write the
evolution equation $\widetilde{I}_{zz}\ddot{\psi}=K_{z}$. Approximating
$\widetilde{I}_{zz}\approx m\delta_{x}^{2}/12$ for our elongated floater, we
find

$\ddot{\psi}\approx-\epsilon(1+z_{c})s_{\psi}c_{\psi}\sin(x_{c}-t)-\epsilon\theta^{\prime}s_{\psi}\cos(x_{c}-t)+\frac{\epsilon^{2}\delta_{x}^{2}}{30\beta\delta_{z}}s_{\psi}c_{\psi}^{3}\sin^{2}(x_{c}-t).$
(57)

We now inject in this equation the decomposition
$x_{c}=\overline{x}_{c}+x_{c}^{\prime}$ and
$z_{c}=\overline{z}_{c}+z_{c}^{\prime}$. In the first, $O(\epsilon)$ term of
(57), we use a Taylor expansion:

$\displaystyle-\epsilon(1+\overline{z}_{c}+z_{c}^{\prime})s_{\psi}c_{\psi}\sin(\overline{x}_{c}+x_{c}^{\prime}-t)$
$\displaystyle\quad=-\epsilon(1+\overline{z}_{c})s_{\psi}c_{\psi}\sin(\overline{x}_{c}-t)-\epsilon
z_{c}^{\prime}s_{\psi}c_{\psi}\sin(\overline{x}_{c}-t)-\epsilon
x_{c}^{\prime}s_{\psi}c_{\psi}\cos(\overline{x}_{c}-t)+O(\epsilon^{3}).$ (58)

As the floater is thin, we can approximate $\overline{z}_{c}\approx 0$. The
second and third terms of (57) are already of order $O(\epsilon^{2})$, so
there we can use
$\sin^{2}(x_{c}-t)\approx\sin^{2}(\overline{x}_{c}-t)+O(\epsilon)$ and
$\cos(x_{c}-t)\approx\cos(\overline{x}_{c}-t)+O(\epsilon)$. We then replace
the first order deviations by

$x_{c}^{\prime}\approx\epsilon\cos(\overline{x}_{c}-t)\quad,\quad
z_{c}^{\prime}\approx\epsilon\sin(\overline{x}_{c}-t)\quad,\quad\theta^{\prime}=-\epsilon
c_{\psi}\cos(\overline{x}_{c}-t).$ (59)

The expression of $x_{c}^{\prime}$ can be found by integrating
$\dot{x}^{\prime}_{c}=u_{x}|_{z=0}$, which just means that, at lowest order,
the floater translates as a fluid material particle on the surface. After
these reductions, we obtain

$\ddot{\psi}\approx-\epsilon
s_{\psi}c_{\psi}\sin(\overline{x}_{c}-t)+\epsilon^{2}s_{\psi}c_{\psi}\left(-1+\frac{\delta_{x}^{2}}{30\beta\delta_{z}}c_{\psi}^{2}\right)\sin^{2}(\overline{x}_{c}-t)$
(60)

as evolution equation for $\psi$. We now also inject the decomposition
$\psi=\overline{\psi}+\psi^{\prime}$ and use a Taylor expansion to replace

$s_{\psi}c_{\psi}=\overline{s}_{\psi}\overline{c}_{\psi}+\psi^{\prime}(\overline{c}_{\psi}^{2}-\overline{s}_{\psi}^{2})+O(\epsilon^{2}).$
(61)

This yields

$\displaystyle\ddot{\overline{\psi}}+\ddot{\psi}^{\prime}$
$\displaystyle\approx$
$\displaystyle-\epsilon\overline{s}_{\psi}\overline{c}_{\psi}\sin(\overline{x}_{c}-t)$
(62)
$\displaystyle-\epsilon(\overline{c}_{\psi}^{2}-\overline{s}_{\psi}^{2})\psi^{\prime}\sin(\overline{x}_{c}-t)+\epsilon^{2}\overline{s}_{\psi}\overline{c}_{\psi}\left(-1+\frac{\delta_{x}^{2}}{30\beta\delta_{z}}\overline{c}_{\psi}^{2}\right)\sin^{2}(\overline{x}_{c}-t).$

At order $O(\epsilon)$, we identify the equation for the fast yaw angle
excursion
$\ddot{\psi}^{\prime}=-\epsilon\overline{s}_{\psi}\overline{c}_{\psi}\sin(\overline{x}_{c}-t)$,
that has the solution

$\psi^{\prime}\approx\epsilon\overline{s}_{\psi}\overline{c}_{\psi}\sin(\overline{x}_{c}-t).$
(63)

This expression is identical to that of the asymptotic model (42e) in the
limit $\delta_{x}\gg\delta_{y}$. Injecting this expression of $\psi^{\prime}$
back into the equation, we isolate the second order evolution equation for
$\overline{\psi}$ by averaging over time and find

$\ddot{\overline{\psi}}\approx\epsilon^{2}\overline{s}_{\psi}\overline{c}_{\psi}^{3}\bigg{(}1-\underbrace{\frac{\delta_{x}^{2}}{60\beta\delta_{z}}}_{F/F_{c}}\bigg{)}.$
(64)

Hence, we have shown that the simplified model reproduces exactly the more
formal asymptotic theory.

Figure 8: Position, orientation and force distribution on the floater in wave
crests and troughs. Here $\delta_{x}=0.5,\delta_{z}=0.01$, $\beta=0.2$,
$\epsilon=0.2$ and $\overline{x}_{c}=0$, $\overline{\psi}=45^{o}$. (a,d) Side
views on vertical $x-z$ plane. The circle shows the trajectory of the center
of mass. The inset figures suggest the instantaneous, parabolic shape of the
waterline near the floater. (b,c,e,f) top views on $x-y$ plane. The arrows
show the instantaneous force distribution along the floater that creates the
instantaneous torque $K_{z}$. In (b,e), without considering the variable
submersion depth, using $\overline{h}$. In (c,f) taking into account the
variable submersion $\widetilde{h}$.

The main advantage of the simplified model is that it allows a better
understanding of the physics that causes reorientation. Let us reconsider the
torque $K_{z}$ in (56), expressed as the product of the local force density
$f_{x}$ by the local level arm weighted by the local submersion depth
$\widetilde{h}$. The force density $f_{x}$ is illustrated as vector arrows in
figure 8, showing the floater at two times, $t=\pi/2$ and $t=3\pi/2$ (wave
crest and trough); animations are available as Supplementary Materials. The
parameters are $\beta=0.2$, $\delta_{x}=0.5,\delta_{z}=0.01$ and
$\epsilon=0.2$, corresponding to a non-dimensional number
$F=\delta_{x}^{2}/\beta\delta_{z}=125$, larger than the critical value
$F_{c}=60$: this floater will prefer the transverse position at late times. In
the second column [Figs. 8(b,e)], the force distribution is weighted by the
equilibrium, spatially uniform submersion depth $\overline{h}$, whereas in the
third column [Figs. 8(c,f)] it is weighted by the true, varying depth
$\widetilde{h}$. Comparing both cases allows us to illustrate the main
physical mechanisms at the origin of the longitudinal-transverse transition.

We first consider the case where the depth variation is ignored [Figs.
8(b,e)], an approximation acceptable for small floaters only. At the wave
crest, the yaw angle is slightly larger than the mean value $\overline{\psi}$
whereas at the trough, it is slightly smaller than $\overline{\psi}$. The
lever arm, $\widetilde{x}\sin\psi$, is therefore larger at the crest. We also
see that the local force density $f_{x}$ is slightly larger at the crest than
at the trough. Both effects impact the instantaneous torque in the same way.
At the crest, the floater will experience a negative (clockwise) torque
$K_{z}<0$ that is slightly larger than the positive (counter-clockwise) torque
$K_{z}>0$ in the troughs, explaining why the part $-\widetilde{x}\sin\psi
f_{x}\overline{h}$ in the integrand of $K_{z}$ is slowly pushing the floater
towards the longitudinal position.

Figures 8(c,f) illustrates why including the varying submersion in the
weighting of the force distribution changes this conclusion in the case of
long floaters. We immediately see that there is clear influence of the
variable submersion at the tips of the floater: the weighted force density
$f_{x}\widetilde{h}$ is significantly changing in magnitude. As shown in Figs.
8(a,d), at the wave crest, the extremities of the floater are less submerged,
whereas in the troughs, they are more submerged. This locally changing
submersion implies that the instantaneous torque is significantly decreased at
the tips of the floater when they are at crest and increased at the tips when
the floater is in a trough. The result for this floater with $F=125>F_{c}$ is
that the positive (counter-clockwise) torque $K_{z}>0$ acting on the floater
at the troughs is significantly larger than the negative torque at the crests,
resulting in a slow rotation towards the transverse position.

It is finally interesting to note that the yaw angle motion of small floaters
is analogous to that of the Kapitza pendulum, a pendulum with an oscillating
anchor point (Kapitza, 1951; Landau & Lifschitz, 1960; Butikov, 2001). For
small floaters we have seen that the variation of the immersion depth can be
ignored and in that case, we have the evolution equation

$F\ll F_{c}\ :\ \ddot{\psi}\approx-\epsilon
s_{\psi}c_{\psi}\sin(\overline{x}_{c}-t)-\epsilon^{2}s_{\psi}c_{\psi}\sin^{2}(\overline{x}_{c}-t).$
(65)

Using the fact that $s_{\psi}c_{\psi}=(1/2)\sin 2\psi$ and changing notation
$2\psi=\alpha$, we can rewrite this as

$\ddot{\alpha}+\bigg{(}\underbrace{-\frac{\epsilon^{2}}{2}}_{\overline{g}}+\underbrace{\epsilon\sin(\overline{x}_{c}-t)+O(\epsilon^{2})}_{g^{\prime}(t)}\bigg{)}\sin\alpha\approx
0.$ (66)

We recognize the equation of motion for the angle $\alpha$ of a Kapitza
pendulum, written in the frame of reference attached to the anchor point. The
Kapitza pendulum, analogue to our floater, would be exposed to a weak
$O(\epsilon^{2})$ downward external gravity $\overline{g}$ and a larger
$O(\epsilon)$ oscillatory acceleration $g(t)$ that is due to the motion of the
anchor point. Using the same multiple time-scale techniques as we have
previously used, we can find that this pendulum has two equilibria. The lower
position $\overline{\alpha}=0^{o}$ is stable and the top position,
$\overline{\alpha}=180^{o}$ is unstable. Owing to the relation
$\overline{\psi}=\overline{\alpha}/2$, this result is entirely equivalent to
saying that short floaters prefer longitudinal positions,
$\overline{\psi}=0^{o}$, and avoid transverse positions,
$\overline{\psi}=90^{o}$.

##  6 Mean yaw moment: comparison with the literature

We have formulated a theoretical model for the idealized, second order motion
of the yaw angle, but in the existing literature, the focus is not so much on
the slow yaw motion, but rather on the mean yaw moment $\overline{K}_{z}$.
Several articles provide quantitative results for this mean yaw moment, either
in the form of an explicit equation (equation (55) in Newman (1967)) or in the
form of figures that result from numerical computations (Newman, 1967; Skejic
& Faltinsen, 2008; Chen, 2007; Yasukawa et al., 2019). In this section, we
compare the second order yaw moment that we have calculated from our Froude-
Krylov model to some existing results, focusing on the limit
$\delta_{x}\gg\delta_{y}\gg\delta_{z}$ of a strongly elongated floater.

To allow comparison, let us start by writing the dimensional yaw moment
according to our theory. By multiplying the right hand side of (1) with the
dimensional moment of inertia $\beta\rho L_{x}^{3}L_{y}L_{y}/12$ and
$\omega^{2}=gk$, we obtain

$\overline{K}_{z}=\frac{1}{12}\rho ga^{2}k^{3}L_{x}^{3}L_{y}\left(-\beta
L_{z}+\frac{kL_{x}^{2}}{60}\right)\sin\overline{\psi}\cos^{3}\overline{\psi}.$
(67)

This formula only applies to parallelepiped floaters that are short with
respect to the wavelength. As explained before, part of the torque
(contribution $-\beta L_{z}$ in the parentheses) favors a longitudinal floater
position (head-seas). This torque clearly depends on how deep the floater is
submerged as the draft of our floater is $\overline{h}=\beta L_{z}$. The other
part of the torque (contribution $+kL_{x}^{2}/60$ in the parentheses) favors a
transverse floater position (beam-seas) and interestingly, it does not depend
on the draft.

Newman (1967) derived an analytical formula [see his equation (55)] for the
mean yaw moment on a slender floating structure with rectangular planform,
such as our parallelepipeds. Written in our notations (wavenumber
$K\rightarrow k$, width $B\rightarrow L_{y}$, length $L\rightarrow L_{x}$,
angle of incidence $\beta=-\overline{\psi}$), this mean yaw moment is

$\overline{K}_{z}^{\text{\, Newman}}=\frac{1}{2}\rho
gka^{2}L_{x}^{2}L_{y}\sin\overline{\psi}\,j_{1}\left(\frac{1}{2}kL_{x}\cos\overline{\psi}\right)j_{2}\left(\frac{1}{2}kL_{x}\cos\overline{\psi}\right).$
(68)

Here $j_{1}$ and $j_{2}$ are spherical Bessel functions. In this formula,
diffraction and radiation are taken into account so it holds for any $kL_{x}$.
In the limit of small floater length, we can replace the spherical Bessel
functions with their small argument asymptotic expansions. From Abramowitz &
Stegun (1948), equation (10.1.2), we have $j_{1}(x)j_{2}(x)\approx x^{3}/45$
for small $x$ (rather than $x^{3}/3$ as written in Newman (1967)), so (68)
reduces to

$kL_{x}\cos\overline{\psi}\ll 1~{}:\quad\overline{K}_{z}^{\text{\,
Newman}}\approx\frac{1}{12}\rho
ga^{2}k^{3}L_{x}^{3}L_{y}\left(\frac{kL_{x}^{2}}{60}\right)\sin\overline{\psi}\cos^{3}\overline{\psi}.$
(69)

Interestingly, this equation contains exactly one term of our mean yaw moment
(67), the one that favors the transverse position. The other term, related to
the draft $\overline{h}=\beta L_{z}$, is absent, probably because Newman used
a slender body approximation which may hold as long as
$kL_{x}^{2}/\overline{h}\gg 1$. This missing term explains why Newman’s
formula does not predict a stable longitudinal position for short floaters.
The fact that our asymptotic theory based on the Froude-Krylov assumption
recovers the short floater limit of Newman’s theory is somehow surprising: in
our approach, we neglect wave diffraction and radiation, whereas the existence
of a radiated wave is essential in Newman’s far field theory. This suggests
that the diffracted/radiated wave correction in the near field of small
floaters does not contribute at leading order to the mean yaw moment acting on
such small floaters.

Figure 9: (a) Dimensionless mean yaw moment as a function of $kL_{x}$, and (b)
zoom at small $kL_{x}$. Red squares: Data from the boundary element
simulations of Chen (2007), computed for a 300-m long floating structure. Blue
line: present Froude-Krylov theory (67), valid in the limit $kL_{x}\leq 1$,
when diffraction and radiation is negligible. The longitudinal-transverse
transition, where $\overline{K}_{z}=0$ (or $F=F_{c}$), is approximately the
same for both methods.

In his middle-field formulation, Chen (2007) introduced a numerical approach
to calculate first order motion and second order load on quite general
floating structures. The potential field around the floating body is
calculated using the boundary element method and includes diffracted and
radiated waves. In one numerical application, Chen considers a very large
floating platform, called FPSO (Floating Production Storage Offloading)-unit,
that is close to a parallelepiped with dimensions $L_{x}=300$ m by $L_{y}=50$
m and submerged over $\overline{h}=\beta L_{z}=25$ m – huge compared to our
centimeter scale parallelepiped floaters –, placed in waves with angle of
incidence $\overline{\psi}=-165^{o}$. In common sea conditions, wavelengths
are usually shorter than $300$ m, so we expect diffraction and wave radiation
corrections, ignored in our theory, to be important. In figure 9(a), we
compare the mean yaw moment calculated by Chen’s BEM approach (red squares)
and our Froude-Krylov theory (67) (blue line). As $kL_{x}$ increases, the mean
yaw moment calculated by Chen is first negative, before changing sign and
oscillating at larger $kL_{x}$, a signature of diffraction and radiation
effects. As emphasized by the zoom in figure 9(b), the $-(kL_{x})^{4}$ trend
at small $kL_{x}$ is very well reproduced by our theory. In spite of the
strong departure of our formula from the simulation at larger $kL_{x}$, it
reproduces surprisingly well the change of sign of $\overline{K}_{z}$, marking
the longitudinal-transverse transition. Although this might be coincidental,
this suggests that the criterium $kL_{x}^{2}/\overline{h}\simeq 60$ for the
transition still applies at moderate $kL_{x}$, even when diffraction and
radiation effects are not negligible.

##  7 Conclusion

In this paper we have shown that the orientation of a small elongated floater
drifting in a propagating gravity wave is governed by the non-dimensional
number $F=kL_{x}^{2}/\bar{h}$, where $\bar{h}=\beta L_{z}$ is the equilibrium
submersion depth. Short and deeply immersed floaters align longitudinally,
along with the direction of propagation, whereas long and weakly immersed
floaters prefer to align transversely, along with the wave crests or troughs.

To model the preferential orientation of these drifting elongated floaters, we
introduced a Froude-Krylov theory that assumes a linear potential incoming
wave in infinitely deep water, ignoring viscous and capillary effects and all
feedbacks of the floater on the wave. We numerically solved the equations of
motion and demonstrated that this model contains the dominant physical
ingredients that are needed to reproduce the preferential state of
orientation. In the limit of small wave amplitude and small floaters,
$\epsilon,\delta\ll 1$, we derived an asymptotic solution of the Froude-Krylov
model. This classical and rather technical asymptotic method allows us to find
the second order evolution equation (2) for the slow motion of the yaw angle,
and places the L-T transition at $F_{c}=60$, which is reasonably close to the
experimental value $F_{c}\simeq 45\pm 10$.

To gain deeper insight into the physics of the floater orientation, we have
finally shown that the same evolution equation (2) for the slow motion of the
yaw angle can be found using a simpler approach in the limit of strongly
elongated floaters. This allowed us to identify that the preferential
orientation is the result of local gradients in flow magnitude that the
floater experiences through its motion. Short floaters see little variation in
submersion depth along their long axis and experience a mean torque that
favors the longitudinal position. This mean torque arises from a phase
correlation between the oscillating buoyancy force and the oscillating lever
arm, a feature shared with the classical Kapitza pendulum. For longer
floaters, the variation of the submersion along the floater has a strong
effect on the instantaneous torque, that is significantly decreased in crest
positions (the tips are less submerged) and increased in trough positions (the
tips are more submerged). Since in the trough position, the instantaneous
torque always pushes towards the transverse position, long floaters prefer to
take transverse positions.

We have compared our mean yaw moment formula to previously published results.
Compared to Newman’s theory, we have identified an additional contribution to
the mean yaw moment that varies linearly with the draft $\overline{h}$. The
longitudinal-transverse transition for short floaters is due to this
contribution, explaining why Newman’s prediction, that slender structures are
stable in beam-seas, does not apply for very short floaters. Comparing our
Froude-Krylov theory to Chen’s boundary element calculations applied to a
large floating structure, we obtain an excellent agreement for the mean yaw
moment in the small floater length/ wavelength limit, and a fair prediction
for the longitudinal-transverse transition. Away from this limit, diffraction
and radiation are no longer negligible and our simplified theory breaks down.
It would be interesting to extend these comparisons to smaller floating
structures such as sailing boats or small yachts.

We can finally draw some parallels between our work and recent studies in
relation to the problem of plastic waste transport by waves. The effect of
shape on the mean motion of non-spherical objects in wave flows has been
investigated in several studies, but limited to fully submerged, neutrally
buoyant ellipsoids (DiBenedetto et al., 2018; DiBenedetto & Ouellette, 2018;
DiBenedetto et al., 2019; Clark et al., 2020; DiBenedetto et al., 2022). A
similar preferential orientation phenomenon is observed there too, but its
physical origin is very different: it results from the pressure and viscous
stress distribution on their surface (Jeffery, 1922), while for floating
elongated bodies it results from the non-uniform pressure forces and spatially
varying submersion. This raises the question of the drift and preferential
orientation of very small, millimeter scale, floating anisotropic particles,
for which both viscous and buoyancy effects are expected.

## Acknowledgments

We are grateful to J. Andriamampianina, A. Eddi, M. Le Boulluec, L. Martin-
Witkowski and M. Rabaud for fruitful discussions, and X. Chen for sharing his
numerical data on the mean yaw moment. We thank A. Aubertin, L. Auffray, J.
Amarni and R. Pidoux for experimental help.

##  Appendix A Kinematics of rotation

We detail in this Appendix the Euler angle convention used in this article,
and find the kinematic relation that links
$\dot{\varphi}(t),\dot{\theta}(t),\dot{\psi(}t)$ to the components
$\widetilde{\Omega}_{x}(t),\widetilde{\Omega}_{y}(t),\widetilde{\Omega}_{z}(t)$
of the instantaneous rotation vector in the floater frame.

The instantaneous orientation of the floater is defined with three Euler
angles $\theta(t)$ pitch, $\varphi(t)$ roll and $\psi(t)$ yaw. These angles
relate to three successive rotations that bring the laboratory frame
$({\boldsymbol{e}}_{x},{\boldsymbol{e}}_{y},{\boldsymbol{e}}_{z})$ to the
floater frame
$(\widetilde{\boldsymbol{e}}_{x}(t),\widetilde{\boldsymbol{e}}_{y}(t),\widetilde{\boldsymbol{e}}_{z}(t))$.
Imagine looking at the wave from above, from the ${\boldsymbol{e}}_{z}$ axis.
The yaw angle $\psi(t)$ is the angle over which we need to rotate the
laboratory frame to align the ${\boldsymbol{e}}_{x}^{\prime}$-axis of a new
frame, with the front of the floater. This defines a first intermediate frame
$({\boldsymbol{e}}^{\prime}_{x},{\boldsymbol{e}}^{\prime}_{y},{\boldsymbol{e}}^{\prime}_{z})$
as

$\left[\begin{array}[]{c}\boldsymbol{e}_{x}^{\prime}\\\
\boldsymbol{e}_{y}^{\prime}\\\
\boldsymbol{e}_{z}^{\prime}\end{array}\right]=\underbrace{\left[\begin{array}[]{ccc}\cos\psi&\sin\psi&0\\\
-\sin\psi&\cos\psi&0\\\
0&0&1\end{array}\right]}_{R_{\psi}}\left[\begin{array}[]{c}\boldsymbol{e}_{x}\\\
\boldsymbol{e}_{y}\\\ \boldsymbol{e}_{z}\end{array}\right].$ (70)

$R_{\psi}$ is a first rotation matrix. We now change our point of view and
look at the floater from the side, from the $\boldsymbol{e}_{y}^{\prime}$
direction. The pitch angle $\theta(t)$ is the angle over which we need to
rotate the
$({\boldsymbol{e}}^{\prime}_{x},{\boldsymbol{e}}^{\prime}_{y},{\boldsymbol{e}}^{\prime}_{z})$
frame to find a new ${\boldsymbol{e}}_{x}^{\prime\prime}$ axis that aligns
with the nose of the boat. Hence, we have a second intermediate frame
$({\boldsymbol{e}}^{\prime\prime}_{x},{\boldsymbol{e}}^{\prime\prime}_{y},{\boldsymbol{e}}^{\prime\prime}_{z})$
defined as

$\left[\begin{array}[]{c}\boldsymbol{e}_{x}^{\prime\prime}\\\
\boldsymbol{e}_{y}^{\prime\prime}\\\
\boldsymbol{e}_{z}^{\prime\prime}\end{array}\right]=\underbrace{\left[\begin{array}[]{ccc}\cos\theta&0&-\sin\theta\\\
0&1&0\\\
\sin\theta&0&\cos\theta\end{array}\right]}_{R_{\theta}}\left[\begin{array}[]{c}\boldsymbol{e}_{x}^{\prime}\\\
\boldsymbol{e}_{y}^{\prime}\\\ \boldsymbol{e}_{z}^{\prime}\end{array}\right].$
(71)

$R_{\theta}$ is a second rotation matrix. In a third and final rotation, we
imagine looking at the boat from the front, from the
$\boldsymbol{e}_{x}^{\prime\prime}$ direction. The roll angle $\varphi(t)$,
rotates the
$({\boldsymbol{e}}^{\prime\prime}_{x},{\boldsymbol{e}}^{\prime\prime}_{y},{\boldsymbol{e}}^{\prime\prime}_{z})$
frame to the floater frame. Hence, we have

$\left[\begin{array}[]{c}\widetilde{\boldsymbol{e}}_{x}\\\
\widetilde{\boldsymbol{e}}_{y}\\\
\widetilde{\boldsymbol{e}}_{z}\end{array}\right]=\underbrace{\left[\begin{array}[]{ccc}1&0&0\\\
0&\cos\varphi&\sin\varphi\\\
0&-\sin\varphi&\cos\varphi\end{array}\right]}_{R_{\varphi}}\left[\begin{array}[]{c}\boldsymbol{e}_{x}^{\prime\prime}\\\
\boldsymbol{e}_{y}^{\prime\prime}\\\
\boldsymbol{e}_{z}^{\prime\prime}\end{array}\right],$ (72)

and $R_{\phi}$ is a third rotation matrix. All combined and using the fact
that the rotation matrices are orthogonal, we can explicit the passage from
the laboratory frame to the floater frame as

$\left[\begin{array}[]{c}\widetilde{\boldsymbol{e}}_{x}\\\
\widetilde{\boldsymbol{e}}_{y}\\\
\widetilde{\boldsymbol{e}}_{z}\end{array}\right]=\underbrace{R_{\varphi}R_{\theta}R_{\psi}}_{R}\left[\begin{array}[]{c}\boldsymbol{e}_{x}\\\
\boldsymbol{e}_{y}\\\
\boldsymbol{e}_{z}\end{array}\right],\qquad\left[\begin{array}[]{c}\boldsymbol{e}_{x}\\\
\boldsymbol{e}_{y}\\\
\boldsymbol{e}_{z}\end{array}\right]=\underbrace{R_{\psi}^{T}R_{\theta}^{T}R_{\varphi}^{T}}_{R^{T}}\left[\begin{array}[]{c}\widetilde{\boldsymbol{e}}_{x}\\\
\widetilde{\boldsymbol{e}}_{y}\\\
\widetilde{\boldsymbol{e}}_{z}\end{array}\right].$ (73)

Evaluating these matrix products explicitly we obtain formula (8a).

We now explain how to find (10) that links
$\dot{\psi}(t),\dot{\theta}(t),\dot{\varphi}(t)$ to
$\widetilde{\Omega}_{x}(t),\widetilde{\Omega}_{y}(t),\widetilde{\Omega}_{z}(t)$.
The floater frame rotates at instantaneous speed $\boldsymbol{\Omega}(t)$
which can be expressed using the differential equation
$\dot{\widetilde{\boldsymbol{e}}}_{i}(t)=\boldsymbol{\Omega}(t)\times\widetilde{\boldsymbol{e}}_{i}(t)$
for $i=x,y,z$. In other terms, we have

$\frac{d}{dt}\left[\begin{array}[]{c}{\widetilde{\boldsymbol{e}}}_{x}\\\
{\widetilde{\boldsymbol{e}}}_{y}\\\ {\widetilde{\boldsymbol{e}}}_{z}\\\
\end{array}\right]=\underbrace{\left[\begin{array}[]{ccc}0&\widetilde{\Omega}_{z}&-\widetilde{\Omega}_{y}\\\
-\widetilde{\Omega}_{z}&0&\widetilde{\Omega}_{x}\\\
\widetilde{\Omega}_{y}&-\widetilde{\Omega}_{x}&0\end{array}\right]}_{M}\left[\begin{array}[]{c}{\widetilde{\boldsymbol{e}}}_{x}\\\
{\widetilde{\boldsymbol{e}}}_{y}\\\ {\widetilde{\boldsymbol{e}}}_{z}\\\
\end{array}\right].$ (74)

Considering the transform (8a), we can also compute the time-derivative of the
vectors of the floater frame by deriving the rotation matrix. We then have

$\frac{d}{dt}\left[\begin{array}[]{c}{\widetilde{\boldsymbol{e}}}_{x}\\\
{\widetilde{\boldsymbol{e}}}_{y}\\\ {\widetilde{\boldsymbol{e}}}_{z}\\\
\end{array}\right]=\frac{d}{dt}\left(R_{\varphi}R_{\theta}R_{\psi}\right)\left[\begin{array}[]{c}\boldsymbol{e}_{x}\\\
\boldsymbol{e}_{y}\\\
\boldsymbol{e}_{z}\end{array}\right]=\frac{d}{dt}\left(R_{\varphi}R_{\theta}R_{\psi}\right)\left(R_{\psi}^{T}R_{\theta}^{T}R_{\varphi}^{T}\right)\left[\begin{array}[]{c}{\widetilde{\boldsymbol{e}}}_{x}\\\
{\widetilde{\boldsymbol{e}}}_{y}\\\ {\widetilde{\boldsymbol{e}}}_{z}\\\
\end{array}\right].$ (75)

Comparing with (74) implies

$\left(R_{\varphi}^{\prime}R_{\theta}R_{\psi}\dot{\varphi}+R_{\varphi}R_{\theta}^{\prime}R_{\psi}\dot{\theta}+R_{\varphi}R_{\theta}R^{\prime}_{\psi}\dot{\psi}\right)R_{\psi}^{T}R_{\theta}^{T}R_{\varphi}^{T}=M.$
(76)

We denote $R_{\varphi}^{\prime}=\partial_{\varphi}R_{\phi}$ and similar for
the other matrices. Considering the orthogonality of the rotation matrices,
this simplifies to
$R_{\varphi}^{\prime}R_{\varphi}^{T}\dot{\phi}+R_{\varphi}R_{\theta}^{\prime}R_{\theta}^{T}R_{\phi}^{T}\dot{\theta}+R_{\varphi}R_{\theta}R^{\prime}_{\psi}R_{\psi}^{T}R_{\theta}^{T}R_{\phi}^{T}\dot{\psi}=M$.
After evaluating all the matrix products, we then deduce that

$\left[\begin{array}[]{c}\widetilde{\Omega}_{x}\\\ \widetilde{\Omega}_{y}\\\
\widetilde{\Omega}_{z}\end{array}\right]=\left[\begin{array}[]{ccc}1&0&-\sin\theta\\\
0&\cos\varphi&\cos\theta\sin\varphi\\\
0&-\sin\varphi&\cos\theta\cos\varphi\end{array}\right]\left[\begin{array}[]{c}\dot{\varphi}\\\
\dot{\theta}\\\ \dot{\psi}\end{array}\right]$ (77)

Notice that this matrix is singular, when $\theta=\pm\pi/2$. In our
applications, we do not encounter such large pitch angles. Hence, we can
always invert this relation to find the system of equations (10).

##  Appendix B Extra details of the asymptotic calculation

The computation of
$F^{\prime}_{x},F^{\prime}_{z},\widetilde{K}^{\prime}_{x},\widetilde{K}^{\prime}_{y},K^{\prime}_{z}$
and the simplifications due to use of the flat floater assumption yield the
following differential equations for the first order motion

$\displaystyle\ddot{x}^{\prime}_{c}$ $\displaystyle\approx$
$\displaystyle-\epsilon\cos(\overline{x}_{c}-t)$ (78a)
$\displaystyle\ddot{z}_{c}^{\prime}+\frac{1}{\beta\delta_{z}}z_{c}^{\prime}$
$\displaystyle\approx$
$\displaystyle\epsilon\left[\frac{1}{\beta\delta_{z}}-1-\frac{1}{24\beta\delta_{z}}\left(\overline{c}_{\psi}^{2}\delta_{x}^{2}+\overline{s}_{\psi}^{2}\delta_{y}^{2}\right)\right]\sin(\overline{x}_{c}-t)$
(78b)
$\displaystyle\ddot{\varphi}^{\prime}+\frac{1}{{\beta\delta_{z}}}\varphi^{\prime}$
$\displaystyle\approx$
$\displaystyle-\epsilon\overline{s}_{{\psi}}\left[\frac{1}{{\beta\delta_{z}}}-1-\frac{1}{\beta\delta_{z}}\left(\overline{s}_{{\psi}}^{2}\dfrac{\delta_{y}^{2}}{40}+\overline{c}_{{\psi}}^{2}\dfrac{\delta_{x}^{2}}{24}\right)\right]\cos(\overline{x}_{c}-t)$
(78c)
$\displaystyle\ddot{\theta}^{\prime}+\frac{1}{{\beta\delta_{z}}}\theta^{\prime}$
$\displaystyle\approx$
$\displaystyle-\epsilon\overline{c}_{{\psi}}\left[\frac{1}{{\beta\delta_{z}}}-1-\frac{1}{\beta\delta_{z}}\left(\overline{c}_{{\psi}}^{2}\dfrac{\delta_{x}^{2}}{40}+\overline{s}_{{\psi}}^{2}\dfrac{\delta_{y}^{2}}{24}\right)\right]\cos(\overline{x}_{c}-t)$
(78d) $\displaystyle\ddot{\psi}^{\prime}$ $\displaystyle\approx$
$\displaystyle-\epsilon\left(\frac{\delta_{x}^{2}-\delta_{y}^{2}}{\delta_{x}^{2}+\delta_{y}^{2}}\right)\overline{s}_{{\psi}}\overline{c}_{{\psi}}\sin(\overline{x}_{c}-t).$
(78e)

The $z^{\prime}_{c}$, $\varphi^{\prime}$ and $\theta^{\prime}$ equations are
that of forced harmonic oscillators with natural frequencies
$\sqrt{1/\beta\delta_{z}}$. Notice that for flat floaters, with
$\delta_{z}\ll\delta_{x},\delta_{y}$, we indeed have nearly the same natural
oscillation frequencies
$\omega_{z}\approx\omega_{\varphi}\approx\omega_{\theta}\approx\sqrt{1/\beta\delta_{z}}$
[see the general Eqs. (17)]. Considering that $\delta_{z}\ll 1$, we have
$\sqrt{1/\beta\delta_{z}}\gg 1$, implying that the incoming wave is never
resonantly forcing the $z^{\prime}_{c}$, $\varphi^{\prime}$ and
$\theta^{\prime}$ oscillations. Hence, the solution for the motion is the sum
of free oscillations at the bobbing frequency and a harmonic response. In the
theoretical calculation, we discard all free high-frequency oscillations as
such motions are strongly damped by radiative dissipation in reality.

The second order mean torque $\overline{K}_{z}$ is defined in (38) and it
split in two parts
$\overline{K}_{z}=\overline{K}_{z}^{(1)}+\overline{K}_{z}^{(2)}$. To find the
first part, we must reconsider the calculation of the torque

$K_{z}^{(1)}=-\int_{V_{sub}^{(0)}}\left(s_{\psi}\widetilde{x}+c_{\psi}\widetilde{y}\right)a_{x}^{(1)}dV\approx\widetilde{I}_{zz}\left(-\epsilon\Lambda
s_{\psi}c_{\psi}\,e^{z_{c}}\sin(x_{c}-t)\right)$ (79)

where
$\Lambda=({\delta_{x}^{2}-\delta_{y}^{2}})/({\delta_{x}^{2}+\delta_{y}^{2}})$.
To get the first and second order parts in this torque, we must evaluate this
formula not a the equilibrium position, but on the trajectory of the floater
$(x_{c},z_{c},\psi)\approx(\overline{x}_{c}+x_{c}^{\prime},\overline{z}_{c}+z_{c}^{\prime},\overline{\psi}+\psi^{\prime})$.
This generates the $O(\epsilon^{2})$ mean ”drift”-like torque that we
expressed in (38) using a Taylor expansion. Let us detail this step here. We
have

$\displaystyle\widetilde{I}_{zz}^{-1}K_{z}^{(1)}\Big{|}_{(\overline{x}_{c}+x_{c}^{\prime},\overline{z}_{c}+z_{c}^{\prime},\overline{\psi}+\psi^{\prime})}$
(80)
$\displaystyle\approx-\underbrace{\epsilon\Lambda\overline{s}_{\psi}\overline{c}_{\psi}\,e^{\overline{z}_{c}}\sin(\overline{x}_{c}-t)}_{K_{z}^{\prime}}$
$\displaystyle\quad-\epsilon\Lambda
e^{\overline{z}_{c}}\left[(\overline{c}_{\psi}^{2}-\overline{s}_{\psi}^{2})\,{\psi^{\prime}\sin(\overline{x}_{c}-t)}+\overline{s}_{\psi}\overline{c}_{\psi}\,{\left(z_{c}^{\prime}\sin(\overline{x}_{c}-t)+x_{c}^{\prime}\cos(\overline{x}_{c}-t)\right)}\right]+O(\epsilon^{3}).$

We simplify $e^{\overline{z}_{c}}=1+O(\delta_{z})$ in the flat floater limit
and inject the first order expressions of
$x_{c}^{\prime},z_{c}^{\prime},\psi^{\prime}$. Averaging over time and using

$\overline{\psi^{\prime}\sin(\overline{x}_{c}-t)}=\frac{\epsilon\Lambda}{2}\overline{s}_{\psi}\overline{c}_{\psi}\quad,\quad\overline{z_{c}^{\prime}\sin(\overline{x}_{c}-t)+x_{c}^{\prime}\cos(\overline{x}_{c}-t)}\approx\epsilon,$
(81)

we find

$\displaystyle\widetilde{I}_{zz}^{-1}\overline{K}^{(1)}_{z}\approx-\epsilon^{2}\Lambda\overline{s}_{\psi}\overline{c}_{\psi}\left[1+\frac{\Lambda}{2}\left(\overline{c}_{\psi}^{2}-\overline{s}_{\psi}^{2}\right)\right]$
(82)

as first part in the mean torque. This part of the torque always favors the
longitudinal state.

The second part of the mean torque is the sum of three terms
$\overline{K}_{z}^{(2)}=\overline{T}_{1}+\overline{T}_{2}+\overline{T}_{3}$.
This notation refers here to the three separate integrals that are visible in
(LABEL:Kz2_new). The first contribution turns out to be negligible in the flat
floater limit $\delta_{z}\ll\delta_{x},\delta_{y}$. We find

$\widetilde{I}_{zz}^{-1}\overline{T}_{1}\approx\overline{-\widetilde{I}_{zz}^{-1}\left(\int_{V_{sub}^{(0)}}\left(s_{\psi}\theta^{\prime}-c_{\psi}\varphi^{\prime}\right)\widetilde{z}a_{x}^{(1)}dV\right)_{(\overline{x}_{c},\overline{z}_{c},\overline{\psi})}}\approx
O(\epsilon^{2}\delta_{z})$ (83)

and this will always be smaller than the other terms of order
$O(\epsilon^{2})$ and $O(\epsilon^{2}\delta_{x}^{2}/\beta\delta_{z})$. The
second term is not negligible:

$\displaystyle\widetilde{I}_{zz}^{-1}\overline{T}_{2}$ $\displaystyle=$
$\displaystyle\overline{-\widetilde{I}_{zz}^{-1}\left(\int_{V_{sub}^{(0)}}\left(s_{\psi}\widetilde{x}+c_{\psi}\widetilde{y}\right)a_{x}^{(2)}dV\right)_{(\overline{x}_{c},\overline{z}_{c},\overline{\psi})}}$
(84) $\displaystyle\approx$
$\displaystyle\overline{-\widetilde{I}_{zz}^{-1}\left(\int_{V_{sub}^{(0)}}\left(s_{\psi}\widetilde{x}+c_{\psi}\widetilde{y}\right)(-\theta^{\prime}\widetilde{x}+\varphi^{\prime}\widetilde{y})\partial_{z}{a}_{x,c}\,dV\right)_{(\overline{x}_{c},\overline{z}_{c},\overline{\psi})}}\approx\,\frac{\epsilon^{2}\Lambda}{2}\overline{s}_{\psi}\overline{c}_{\psi}.$

We detail here the part of $a_{x}^{(2)}$ that creates the leading
contribution. This term captures the small torque that is created by small
vertical gradients in wave-magnitude ($\partial_{z}a_{x,c}$) along its
elongated surface and due to rotations ($\theta^{\prime}$ and
$\varphi^{\prime}$). The term alone creates a slow $O(\epsilon^{2})$ motion
that would rather be in favor of the transverse position. However, it can
never dominate the other $O(\epsilon^{2})$ terms in the
$\overline{K}^{(1)}_{z}$ mean torque.

The third contribution to $\overline{K}_{z}^{(2)}$ is the longest to evaluate.
In the flat floater limit, we find it as

$\displaystyle\widetilde{I}_{zz}^{-1}\overline{T}_{3}$ $\displaystyle=$
$\displaystyle\overline{-\widetilde{I}_{zz}^{-1}\left(\int_{V_{sub}^{(1)}}\left(s_{\psi}\widetilde{x}+c_{\psi}\widetilde{y}\right)a_{x}^{(1)}dV\right)_{(\overline{x}_{c},\overline{z}_{c},\overline{\psi})}}$
(85) $\displaystyle\approx$ $\displaystyle\overline{\frac{\epsilon
e^{z_{c}}\cos(x_{c}-t)}{\beta\delta_{z}(\delta_{x}^{2}+\delta_{y}^{2})}\Bigg{[}s_{\psi}\underbrace{\left(\theta^{\prime}+\epsilon
c_{\psi}\cos(x_{c}-t)\right)}\delta_{x}^{2}-c_{\psi}\underbrace{\left(\varphi^{\prime}+\epsilon
s_{\psi}\cos(x_{c}-t)\right)}\delta_{y}^{2}}$
$\displaystyle\quad\quad+\left(c_{\psi}^{3}s_{\psi}\left(-\frac{\delta_{x}^{4}}{40}+\frac{\delta_{x}^{2}\delta_{y}^{2}}{24}\right)-s_{\psi}^{3}c_{\psi}\left(-\frac{\delta_{y}^{4}}{40}+\frac{\delta_{x}^{2}\delta_{y}^{2}}{24}\right)\right)\epsilon\cos(x_{c}-t)\Bigg{]}\qquad\quad$
$\displaystyle+$ $\displaystyle\overline{\frac{\epsilon
e^{z_{c}}\sin(x_{c}-t)}{\beta\delta_{z}(\delta_{x}^{2}+\delta_{y}^{2})}\Bigg{[}s_{\psi}c_{\psi}(\delta_{x}^{2}-\delta_{y}^{2})\underbrace{(z_{c}^{\prime}-\epsilon\sin(x_{c}-t))}\hskip
71.13188pt}$
$\displaystyle\quad\quad+\left(c_{\psi}^{3}s_{\psi}\left(3\frac{\delta_{x}^{4}}{40}-2\frac{\delta_{x}^{2}\delta_{y}^{2}}{24}\right)-s_{\psi}^{3}c_{\psi}\left(3\frac{\delta_{y}^{4}}{40}-2\frac{\delta_{x}^{2}\delta_{y}^{2}}{24}\right)\right)\epsilon\sin(x_{c}-t)\bigg{]}.$

Physically, this term captures on its own the torque that is due to the
changing submersion of the floater (integral over $V_{sub}^{(1)}$). The
underbraced terms show more clearly how the first order motion
$z_{c}^{\prime},\theta^{\prime},\varphi^{\prime}$ influences this varying
submersion and hence this torque. Quite astonishingly, we observe that the
first term exactly vanishes after injecting the solutions of $\theta^{\prime}$
and $\varphi^{\prime}$, even with the tiny
$O(\epsilon\delta_{x}^{2},\epsilon\delta_{y}^{2})$ corrections. Hence, we can
deduce that these weak corrections in $\theta^{\prime}$ and $\varphi^{\prime}$
are in fact not so relevant in this problem. The second term does not vanish.
After injecting the $z_{c}^{\prime}$-motion, we find that the very small
$O(\epsilon\delta_{x}^{2},\epsilon\delta_{y}^{2})$ are crucially important.
From (42b), we get that

$z_{c}^{\prime}-\epsilon\sin(\overline{x}_{c}-t)=-\frac{\epsilon}{24}\left(\overline{c}^{2}_{\psi}\delta_{x}^{2}+\overline{s}_{\psi}^{2}\delta_{y}^{2}\right)\sin(\overline{x}_{c}-t).$
(86)

The right hand side may seem tiny, but without these terms, we would simply
find $\overline{T}_{3}=0$. Hence, these weak buoyancy-induced corrections of
the vertical motion are crucial and the only ones that can cause
$\overline{T}_{3}$ to be non-zero in this model, at this order. After taking
the time-average, we find

$\displaystyle\widetilde{I}_{zz}^{-1}\overline{T}_{3}$ $\displaystyle\approx$
$\displaystyle\frac{\epsilon^{2}}{\beta\delta_{z}(\delta_{x}^{2}+\delta_{y}^{2})}\left[\overline{s}_{\psi}\overline{c}_{\psi}^{3}\left(\frac{\delta_{x}^{4}}{60}-\frac{\delta_{x}^{2}\delta_{y}^{2}}{48}\right)-\overline{s}_{\psi}^{3}\overline{c}_{\psi}\left(\frac{\delta_{y}^{4}}{60}-\frac{\delta_{x}^{2}\delta_{y}^{2}}{48}\right)\right].$
(87)

Without considering the anisotropy of the floater, this term causes a
contribution of $O(\epsilon^{2}\delta)$ to the motion of $\overline{\psi}$
that in fact, seems smaller than the other mean torques of order
$O(\epsilon^{2})$. However, for strongly elongated floaters with
$\delta_{x}\gg\delta_{y}\gg\delta_{z}$, this term is rather of magnitude
$O(\epsilon^{2}\delta_{x}^{2}/\beta\delta_{z})$ and dominates the
$O(\epsilon^{2})$ terms. In our calculations, we were able to retrace that
this term $\overline{T}_{3}$ was entirely due to the existence of a non-zero
curvature of the water-line at the floater position (the term
$(1/2)(x-x_{c})^{2}\partial_{xx}^{2}\zeta_{c}$ in the Taylor expansion for
$\widetilde{\zeta}$) and it is this physical element that brought us to the
simplified model discussed in section 5.

In the end, we have summed up all the contributions to the second order mean
torque. Expressing
$\widetilde{I}_{zz}^{-1}\ddot{\overline{\psi}}=\overline{K}_{z}$, we then
found the equation of motion (43).

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