1. Introduction
Uniform long-crested water surface waves of permanent form on deep water, commonly known as Stokes waves (Stokes Reference Stokes1847), are unstable to sideband perturbations. The instability was first discovered for narrow-band collinear perturbations (Lighthill Reference Lighthill1965; Benjamin Reference Benjamin1967; Benjamin & Feir Reference Benjamin and Feir1967) and is commonly known as the Benjamin–Feir instability. The collinear narrow-band instability vanishes for finite depth
$kh \lt 1.363$
(Benjamin Reference Benjamin1967; Hasimoto & Ono Reference Hasimoto and Ono1972). Here,
$k = 2\pi /\lambda$
is the wavenumber,
$\lambda$
is the wavelength and
$h$
is the depth. The instability also extends to transverse perturbations which are unstable for both greater and smaller depths (Benney & Roskes Reference Benney and Roskes1969).
In the course of long-term evolution, following the initial instability, and after a stage of heavy modulations, there may be recurrence back to a uniform wave train. There is laboratory evidence that the recurrence can either be back to the original Stokes wave frequency or it can be accompanied by a downshift to a lower frequency (Lake et al. Reference Lake, Yuen, Rungaldier and Ferguson1977). The downshift has been observed together with wave breaking and with the growth of a continuous background spectrum (Melville Reference Melville1982; Tulin & Waseda Reference Tulin and Waseda1999).
Simple model equations predict that the Fermi–Pasta–Ulam–Tsingou recurrence, in which the initial uniform wave train recurs (Fermi, Pasta & Ulam Reference Fermi, Pasta and Ulam1955; Dauxois Reference Dauxois2008), also can happen for water waves (Yuen & Ferguson Reference Yuen and Ferguson1978b ; Janssen Reference Janssen1981). Although a temporal downshift is predicted by the non-dissipative Dysthe equation for long-crested waves (Dysthe Reference Dysthe1979; Janssen Reference Janssen1983; Lo & Mei Reference Lo and Mei1985), a permanent recurrence with downshift was previously thought to depend on dissipative effects. A permanent downshift has been captured by long-crested model equations equipped with various types of dissipation, including ad hoc models for wave breaking (Trulsen & Dysthe Reference Trulsen and Dysthe1990), models for viscous damping without breaking (Uchiyama & Kawahara Reference Uchiyama and Kawahara1994; Kato & Oikawa Reference Kato and Oikawa1995; Skandrani, Kharif & Poitevin Reference Skandrani, Kharif and Poitevin1996; Segur et al. Reference Segur, Henderson, Carter, Hammack, Li, Pheiff and Socha2005; Islas & Schober Reference Islas and Schober2011a , Reference Islas and Schoberb ), combinations of wind forcing and damping (Hara & Mei Reference Hara and Mei1991; Trulsen & Dysthe Reference Trulsen and Dysthe1992; Schober & Strawn Reference Schober and Strawn2015) and models for energy and momentum loss due to breaking (Tulin Reference Tulin1996; Tulin & Waseda Reference Tulin and Waseda1999). A review of long-crested downshift with various dissipation mechanisms was given by Carter, Henderson & Butterfield (Reference Carter, Henderson and Butterfield2019). Recently, Ratliff, Trichtchenko & Bridges (Reference Ratliff, Trichtchenko and Bridges2025) suggested that the permanent downshift in the recurrence cycle of long-crested Stokes waves can be achieved without dissipation.
On the other hand, downshifting has also been observed in the undamped evolution of a finite-width spectrum (Onorato et al. Reference Onorato, Osborne, Serio, Resio, Pushkarev, Zakharov and Brandini2002; Dysthe et al. Reference Dysthe, Trulsen, Krogstad and Socquet-Juglard2003; Onorato et al. Reference Onorato2009; Chalikov Reference Chalikov2012), however, we consider that to be an essentially different phenomenon partly due to the lack of recurrence to a uniform wave and partly due to the fact that a finite-width spectrum does not support the initial modulational instability (Alber Reference Alber1978).
Stokes waves are unstable to infinitesimal perturbations due to both four-wave and five-wave nonlinear resonance. In McLean et al. (Reference McLean, Ma, Martin, Saffman and Yuen1981) these are denoted instabilities of types I and II, respectively. The Benjamin–Feir instability is of narrow-band type I. These instabilities are nicely summarised in figure 1 of McLean et al. (Reference McLean, Ma, Martin, Saffman and Yuen1981) from which it is observed that type I is nearly collinear and mostly narrow band while type II is mainly transverse and only wide band. Type I transverse instability of Stokes waves has been reported in experiments (Toffoli et al. Reference Toffoli2013; Pinho & Babanin Reference Pinho and Babanin2015). Type II instability of Stokes waves gives rise to characteristic horseshoe patterns of transverse length equal to the longitudinal wavelength (Su Reference Su1982; Su et al. Reference Su, Bergin, Marler and Myrick1982). Transverse modulations of narrow-band type I and broad-band type II are easily distinguished by visual inspection of the wave field, as can be appreciated in photos from our experiments in figure 1. Pictures from all the experiments are shown in Appendix B.
Upper left: Transverse modulation (TM) of type I in run 8008. Upper right: TM of type I in run 8015. Lower left and right: Horseshoe patterns typical of TM of type II in run 8012.
The emphasis of this paper is on the weakly nonlinear and narrow-banded transverse instability of type I. For sufficiently narrow wave tanks there is numerical evidence that non-dissipative transverse modulations can lead to recurrence without downshift (Yuen & Ferguson Reference Yuen and Ferguson1978a ; Lo & Mei Reference Lo and Mei1987). For slightly wider wave tanks there is numerical evidence of a permanent non-dissipative downshift to transverse modes (Trulsen & Dysthe Reference Trulsen and Dysthe1997). Trulsen, Stansberg & Velarde (Reference Trulsen, Stansberg and Velarde1999) provided experimental evidence that the downshift can be accompanied by substantial transfer of energy from long-crested modes to transverse modes, which can easily be misinterpreted as dissipation in the case of insufficient measurements in the transverse direction. Accurate numerical simulations of short-crested waves by Fuhrman, Madsen & Bingham (Reference Fuhrman, Madsen and Bingham2006) further support the conclusions of Trulsen & Dysthe (Reference Trulsen and Dysthe1997).
Trulsen & Dysthe (Reference Trulsen and Dysthe1997) and Trulsen et al. (Reference Trulsen, Stansberg and Velarde1999) anticipated that the evolution with transverse modes depends on a width parameter
$\mu = k_\perp /k_c$
, where
$k_\perp$
is a permissible transverse wavenumber vector component, while
$k_c = 2\pi /\lambda$
is the characteristic wavenumber and
$\lambda$
is the characteristic wavelength of the original Stokes wave. In a wave tank of width
$b$
the permissible transverse wavenumber components will be discretised in multiples of
$\Delta k_\perp = \pi /b$
and the width parameter will be discretised in multiples of
$\Delta \mu = \Delta k_\perp /k_c = \lambda /(2b)$
. On deep water this evaluates to
\begin{equation} \Delta \mu = \frac {\Delta k_\perp }{k_c} = \frac {\lambda }{2b} = \frac {g T^2}{4 \pi b} ,\end{equation}
where
$T$
is the characteristic wave period and
$g$
is the acceleration due to gravity. Transverse type I instability can be excited in experiments when
$\Delta \mu$
is small enough to be inside the unstable domain. In the figures of McLean et al. (Reference McLean, Ma, Martin, Saffman and Yuen1981) their
$q$
along the second axis is identical to our
$\mu$
, and our
$\Delta \mu$
is the discretisation along that axis. Visual inspection of figure 1 in McLean et al. (Reference McLean, Ma, Martin, Saffman and Yuen1981) suggests that the transverse instability of type I is contained within
$\mu \lt 0.36$
. From the same figure we also see that unstable perturbations of type II are essentially limited to
$\mu \gt 1$
for the small steepnesses
$\epsilon = a_c k_c \lt 0.33$
employed in our experiments. Here, we define the characteristic amplitude to be
$a_c = \sqrt {2} \sigma$
, where
$\sigma$
is the standard deviation of the surface elevation, thus for uniform waves
$a_c$
is the amplitude and
$H = 2a_c$
is the wave height.
The single experiment reported by Trulsen et al. (Reference Trulsen, Stansberg and Velarde1999) had
$\Delta \mu = 0.074$
, with unseeded Stokes waves generated by the wave maker, and with transverse measurements limited to only one propagation distance along the tank, where only two probes were located across the tank. The original experiments of Lake et al. (Reference Lake, Yuen, Rungaldier and Ferguson1977) had
$\mu = 0.066$
(see Trulsen & Dysthe Reference Trulsen and Dysthe1997) with only one probe across the tank. Toffoli et al. (Reference Toffoli2013) reported experiments with
$\Delta \mu$
between 0.011 and 0.031, with seeded Stokes waves with transverse modulations generated by the wave maker, unfortunately without transverse measurements. Pinho & Babanin (Reference Pinho and Babanin2015) reported experiments with
$\Delta \mu$
between 0.0094 and 0.026, with unseeded Stokes waves generated by the wave maker, and with transverse measurements limited to only one propagation distance along the tank where nine probes were located across the tank. Numerical simulations with
$\mu = 0.098$
and with seeded perturbations of Stokes waves reported by Fernandez et al. (Reference Fernandez, Onorato, Monbaliu and Toffoli2014) showed substantial growth of oblique sidebands.
There is a lack of experiments resolving the evolution of several transverse modes of Stokes waves propagating along a tank of finite width. We here report such experiments, with
$\Delta \mu$
between 0.05 and 0.15, with unseeded Stokes waves generated by the wave maker, and with transverse measurements using five probes across the tank at four distances along the tank. This gives adequate resolution for the unstable mode dynamics within the domain of the type I instability, while it does not resolve the dynamics of the type II instability. The presence of both type I and II modulations were, however, observed visually. These tests were run in a 260 m long and 10.5 m wide tank, which is long enough to study the problem at a reasonable scale.
This paper deals with the evolution of uniform waves in a long tank for the limiting regime that the tank is just wide enough for unstable transverse modulations to modify the long range evolution compared with a narrow tank. We discover that the limiting width corresponds to
$\Delta \mu \approx 0.1$
, thus the tank should accommodate crest lengths of at least 10 wavelengths. In this paper we do not consider the opposite limit of wide basins with
$\Delta \mu \ll 0.1$
, or irregular waves for which the modulational instability is suppressed (Alber Reference Alber1978) such as Onorato et al. (Reference Onorato2009), Toffoli et al. (Reference Toffoli, Gramstad, Trulsen, Monbaliu, Bitner-Gregersen and Onorato2010) and Cherneva & Guedes Soares (Reference Cherneva and Guedes Soares2011) who employed irregular waves with small
$\Delta \mu = 0.016$
.
It is a curious coincidence, not necessarily related, that Gramstad & Trulsen (Reference Gramstad and Trulsen2007) concluded the Benjamin–Feir index is a good indicator of increased freak wave activity only when the wave field is long crested with characteristic crest length at least 10 characteristic wavelengths.
2. Experimental set-up
The experiments were carried out during the summer of 1998 in the towing tank at MARINTEK (later renamed SINTEF Ocean). The tank is
$260\,$
m long and
$b = 10.5\,$
m wide. The depth of the tank is
$h=10\,$
m for the first
$80\,$
m closest to the wave maker, and
$h=5\,$
m elsewhere, with a step-type bathymetric connection. The experimental set-up, using 20 probes arranged with five across the tank at four distances from the wave maker, is shown in figure 2. The
$x$
-axis runs along the tank measuring the distance from the wave maker. The
$y$
-axis is across with
$y=0$
at one side. The probes are at transverse positions
$y_l = l b/6$
for
$l = 1,2,3,4,5$
, between the lateral tank walls at
$y = 0$
and
$y = b$
. The wave tank is sufficiently long that waves reflected from the damping beach do not reach back within the time window used for analysis.
Experimental set-up with array of 20 probes. For runs 8000–8017 probes were located at
$x_1 = 10\,$
,
$x_2 = 40\,$
,
$x_3 = 80\,$
and
$x_4 = 120\,$
m. For runs 8018–8026 the probes and probe numbers at
$x = 40\,$
,
$80\,$
and
$120\,$
m were moved to
$80\,$
,
$120\,$
and
$160\,$
m, respectively.
The experiments consisted of generating long series of un-seeded uniform long-crested waves, with crests aligned in the
$y$
-direction, with given period
$T$
and wave height
$H$
. Unstable perturbations grew naturally and the resulting modulations were recorded by the 20 probes along and across the tank. A total of 27 experimental runs were carried out, of which four were repeated runs. Linear harmonic signals were used as input to the wave maker, i.e. the paddle oscillated like a single harmonic. As a result, free ‘parasitic’ overharmonic components will also propagate in the wave field at their own (much lower) speed, and be present after an additional time depending on the location. However, for the present study of modulations and downshift we do not consider that these effects disturb our findings, especially because time windows are chosen mostly before they arrive.
Table 1 gives a summary of the run numbers and the corresponding nominal period
$T$
and nominal wave height
$H$
. The nominal wave height is defined to be the peak-to-trough height of the generated uniform wave. We define the characteristic amplitude to be half the nominal wave height
$a_c = H/2$
, and we define the characteristic frequency to be given by the nominal period
$\omega _c = 2\pi /T$
. We let the characteristic wavenumber be given by the linear dispersion relation
$\omega _c^2 = g k_c \tanh k_c h$
, where
$g = 9.81\,$
m s−2 is the acceleration due to gravity and
$h$
is the depth. For all but runs 8004–8006 we can safely use the deep-water dispersion relation
$\omega _c^2 = g k_c$
. For this analysis we do not employ nonlinear corrections to the linear dispersion relation, leaving the nonlinear corrections to be taken care of by nonlinear evolution equations (e.g. Trulsen & Dysthe Reference Trulsen and Dysthe1997). In the table we show the resulting values for the dimensionless depth
$k_c h$
and steepness
$\epsilon = a_c k_c$
for the shallower depth
$h = 5\,$
m. The transverse parameter
$\Delta \mu$
from (1.1) is also shown. The largest fetch
$\epsilon ^2k_cx_4$
is an important parameter telling us how long the tank is in terms of nonlinear modulational evolution. For ease of comparison we employ the same definition of fetch as in Trulsen & Dysthe (Reference Trulsen and Dysthe1997) inspired by the effect of cubic nonlinearities in the nonlinear Schrödinger equation. The last four columns of the table are explained in the next section.
First column run number. Next six columns nominal parameter values: period, wave height, dimensionless depth (for the shallower depth
$h = 5\,$
m), steepness, width parameter, largest fetch. Final four columns downshift at largest fetch: peak across tank, peak at centreline, mean across tank, mean at centreline.
The change of depth at
$80\,$
m from the wave maker does not have any impact on the results because we are in a deep-water regime. The results in Appendix A show that the interesting downshift happens after a longer propagation distance, which was the reason why we moved most of the probes to greater fetch for runs 8018–8026.
Runs 4–6 are test cases with large transverse parameter
$\Delta \mu$
and small fetch
$\epsilon ^2k_cx_4$
,. They are not interesting and therefore not included in the following discussion.
3. Data analysis
The
$x$
-axis is along the tank, the
$y$
-axis is across and the
$z$
-axis is vertical pointing up. The surface elevation is
$z = \eta$
with quiescent surface
$z=0$
. The velocity field is
$\boldsymbol{v} = \boldsymbol{\nabla }\phi$
according to an assumption of potential flow.
At the lateral vertical walls there is no flow through the walls
$v_y = 0$
. In the case of gravity waves and potential flow, the normal derivative of the surface elevation vanishes at the walls,
$ {\partial \eta }/{\partial y} = 0$
. This can be established from the exact dynamic surface condition
\begin{equation} \frac {\partial \phi }{\partial t} + g \eta + \frac {1}{2} (\boldsymbol{\nabla }\phi )^2 = 0 \quad \mbox{at $z = \eta $} .\end{equation}
Upon taking the
$y$
-derivative and evaluating at the lateral wall we get the desired result.
In the following we outline a Fourier analysis in the transverse (
$y$
) and time (
$t$
) domains, with no Fourier decomposition in the longitudinal (
$x$
) direction.
Having established that
$ {\partial \eta }/{\partial y} = 0$
at the lateral walls, we can represent the transverse behaviour of the surface elevation by an infinite cosine series
\begin{equation} \eta (x, y, t) = \sum _{n=0}^\infty \tilde {\eta }_n(x,t) \cos \left(\frac {\pi n y}{b}\right)\!. \end{equation}
We proceed as in Trulsen et al. (Reference Trulsen, Stansberg and Velarde1999), but now truncating (3.2) to the first five terms, insisting on exact reproduction at the collocation points
\begin{equation} \eta (x,y_l,t) = \sum _{n=0}^4 \check {\eta }_n(x,t) \cos \left(\frac {\pi n y_l}{b}\right) \qquad \mbox{for} \quad l = 1,2,3,4,5. \end{equation}
With
$y_l/b = l/6$
the coefficients are obtained by the transform
\begin{equation} \left ( \begin{array}[c]{c} \check {\eta }_0 \\ \check {\eta }_1 \\ \check {\eta }_2 \\ \check {\eta }_3 \\ \check {\eta }_4 \end{array} \right ) = \frac {1}{6} \left ( \begin{array}[c]{ccccc} 2 & 0 & 2 & 0 & 2 \\ 2\sqrt {3} & 0 & 0 & 0 & -2\sqrt {3} \\ 3 & -3 & 0 & -3 & 3 \\ \sqrt {3} & -3 & 0 & 3 & - \sqrt {3} \\ 1 & -3 & 4 & -3 & 1 \end{array} \right ) \left ( \begin{array}[c]{c} \eta (y_1) \\ \eta (y_2) \\ \eta (y_3) \\ \eta (y_4) \\ \eta (y_5) \end{array} \right)\!.\end{equation}
In the following, we use the truncated (3.3) as the basis for interpolation across the tank
\begin{equation} \eta (x,y,t) = \sum _{n=0}^4 \check {\eta }_n(x,t) \cos \left(\frac {\pi n y}{b}\right)\!.\end{equation}
We take the temporal Fourier transform over a time interval
$\tau$
starting at an offset time depending on the position
$\tau _0(x)$
\begin{equation} \check {\eta }_n(x,t) = \sum _{\!j} \hat {\eta }_{n,j}(x) {\mathrm e}^{-{\mathrm i} \omega _{\!j} t} \qquad \mbox{for} \quad \tau _0(x) \leq t \lt \tau _0(x)+\tau ,\end{equation}
where
$\omega _{\!j} = 2\pi j/\tau$
. The offset starting time
$\tau _0(x)$
serves two purposes. First,
$\tau _0(x)$
is chosen large enough that all startup effects have disappeared. Second,
$ {{\mathrm d} \tau _0}/{{\mathrm d} x} = c_g$
in order to recognise that energy is transported with the group velocity. Between two positions
$x_1$
and
$x_2$
we have
$x_1 - x_2 = c_g (\tau _0(x_1) - \tau _0(x_2))$
, where
$c_g$
is the group velocity corresponding to the nominal wave period
$T$
.
With the advantage of the orthogonality of the interpolating functions, we can now compute the mean power in the combined time and transverse domain
\begin{equation} I(x) = \frac {1}{\tau } \int _{\tau _0(x)}^{\tau _0(x) + \tau } \frac {1}{b} \int _0^b \eta ^2(x,y,t) \, {\mathrm d} y {\mathrm d} t = \sum _{\!j} \sum _{n=0}^4 \nu _n |\hat {\eta }_{n,j}(x)|^2 ,\end{equation}
where we have introduced the coefficient
$\nu _n$
such that
$\nu _0 = 1$
and
$\nu _n = 1/2$
for
$n \gt 0$
.
Introducing the power of each mode
\begin{equation} J_n(x,\omega _{\!j}) = \nu _n |\hat {\eta }_{n,j}(x)|^2 ,\end{equation}
the transversally integrated power
\begin{equation} F(x,\omega _{\!j}) = \sum _{n=0}^4 J_n(x, \omega _{\!j}) ,\end{equation}
and the temporally integrated power
\begin{equation} G_n(x) = \sum _{\!j} J_n(x, \omega _{\!j}) ,\end{equation}
we have that the total mean power is
\begin{equation} I(x) = \sum _{n=0}^4 G_n(x) = \sum _{\!j} F(x,\omega _{\!j}) = \sum _{\!j} \sum _{n=0}^4 J_n(x,\omega _{\!j}) .\end{equation}
Sometimes experiments are carried out with only one probe across the tank for each longitudinal position
$x$
; typically, such probes are located along the centreline of the tank. In our case this corresponds to transverse position
$y_3 = b/2$
and we define
$\zeta (x,t) \equiv \eta (x,y_3,t) = \check {\eta }_0(x,t) - \check {\eta }_2(x,t) + \check {\eta }_4(x,t)$
, where we have employed the interpolation formula (3.5). Again we take the temporal Fourier transform over a time interval
$\tau$
starting at an offset time depending on position
$\tau _0(x)$
\begin{equation} \zeta (x,t) \equiv \eta (x,y_3,t) = \sum _{\!j} \hat {\zeta }_{\!j}(x) {\mathrm e}^{-{ i} \omega _{\!j} t} \qquad \mbox{for} \quad \tau _0(x) \leq t \lt \tau _0(x)+\tau ,\end{equation}
from which we identify the mean power in the time domain
\begin{equation} C(x) = \frac {1}{\tau } \int _{\tau _0(x)}^{\tau _0(x) + \tau } \eta ^2(x,y_3,t) \, {\mathrm d} t = \sum _{\!j} |\hat {\zeta }_{\!j}(x)|^2 = \sum _{\!j} D(x, \omega _{\!j}) ,\end{equation}
where
$D(x, \omega _{\!j}) = |\hat {\zeta }_{\!j}(x)|^2$
.
We have computed three different kinds of peak frequencies of the power spectrum: The peak frequency of each transverse mode
$\omega _{{\textit{peak}}, n}(x)$
is the frequency
$\omega _{\!j}$
that maximises
$J_n(x,\omega _{\!j})$
for each
$n\in \{0,1,2,3,4\}$
. The peak frequency for the combined measurements across the tank
$\omega _{{\textit{peak}}}(x)$
is the frequency
$\omega _{\!j}$
that maximises
$F(x,\omega _{\!j})$
. The peak frequency at the centreline of the tank
$\omega _{{\textit{peak}},c}(x)$
is the frequency
$\omega _{\!j}$
that maximises
$D(x,\omega _{\!j})$
.
For all of the peak frequencies, we limit the search to a frequency range strictly within the first harmonic domain
$\delta \lt \omega _{{\textit{peak}}}/\omega _c \lt 2 - \delta$
, where
$\delta$
is a small number.
Similarly, we have computed three different kinds of mean frequencies of the power spectrum: the mean frequency of each transverse mode is
\begin{equation} \omega _{{\textit{mean}},n}(x) = \sum _{\!j} |\omega _{\!j}| J_n (x, \omega _{\!j}) / G_n (x) .\end{equation}
The mean frequency for the combined measurements across the tank is
\begin{equation} \omega _{{\textit{mean}}}(x) = \sum _{\!j} |\omega _{\!j}| F(x, \omega _{\!j}) / I (x) .\end{equation}
The mean frequency at the centreline of the tank is
\begin{equation} \omega _{{\textit{mean}},c}(x) = \sum _{\!j} |\omega _{\!j}| D(x, \omega _{\!j}) / C (x) .\end{equation}
The evolution of these quantities along the tank is expected to be dominated by cubic nonlinear interactions, e.g. described by the nonlinear Schrödinger equation, therefore we have chosen to normalise the evolution
$x$
-axis by the square of the nominal steepness
$\epsilon$
(Trulsen & Dysthe Reference Trulsen and Dysthe1997).
4. Results
Here, we give detailed results for run 8019. Detailed results for all runs can be found in Appendix A. Pictures from the runs can be found in Appendix B.
In order to help the interpretation of the following figures we offer table 2 with a summary of the quantities that are being presented.
Summary of quantities presented.
Figure 3 shows the frequency power spectrum
$J_n(x,\omega )$
for each transverse mode number
$n$
at each station
$x$
, normalised such that the peak value is 1 at each station
$x$
. Two different views are shown: to the left, spectral curves as a function of
$\omega$
for each value of
$n$
; to the right, intensity plots in the
$(\omega ,n)$
-plane. Notice that
$\omega$
increases to the right in the left column while it increases downward in the right column. We observe significant downshift of the peak frequency, and a certain amount of energy transfer to transverse modes
$n\gt 0$
. The long-crested mode
$n=0$
and the first transverse mode
$n=1$
have significantly downshifted peaks, while all modes have significant energy leakage to higher frequencies. All figures are limited to the first harmonic frequency domain
$0 \lt \omega \lt 2\omega _c$
.
Frequency power spectrum
$J_n(x,\omega )$
for each transverse mode number
$n$
at each station
$x$
, normalised such that the peak value is 1 at each station
$x$
.
Figure 4 shows the mean power of each transverse mode
$G_n(x)$
, the total mean power
$I(x)$
and the mean power along the centreline of the tank
$C(x)$
. If we limit consideration to only the centreline, or only the long-crested part of the wave
$n=0$
, then we may overestimate the amount of dissipation. A significant portion of the energy is transferred to transverse modes
$n \gt 0$
.
Mean power along the tank for each transverse mode
$G_n(x)$
, the total mean power
$I(x)$
and the mean power along the centreline of the tank
$C(x)$
. Linear and logarithmic power axes. Each set of four discrete points is connected with straight lines (linear interpolation).
Peak and mean frequencies along the tank,
$\omega _{{\textit{peak}}, n}(x)$
and
$\omega _{{\textit{mean}}, n}(x)$
for each transverse mode
$J_n(x,\omega )$
,
$\omega _{{\textit{peak}}}(x)$
and
$\omega _{{\textit{mean}}}(x)$
for the combined measurements across the tank
$F(x,\omega )$
,
$\omega _{{\textit{peak}},c}(x)$
and
$\omega _{{\textit{mean}},c}(x)$
for the centreline of the tank
$D(x,\omega )$
. Each set of four discrete points is connected with straight lines (linear interpolation).
Both figures 3 and 4 imply that a significant amount of energy can be transferred to transverse modes, and thus dissipation can be overestimated by just observing the long-crested mode or just measuring at the centreline. These two figures also suggest that the main contributions to the energy budget can be captured by just considering the long-crested mode
$n=0$
and the first transverse mode
$n=1$
.
Figure 5 shows the peak and mean frequencies. To the left we see the peak frequency
$\omega _{{\textit{peak}}, n}(x)$
for each transverse mode
$J_n(x,\omega )$
, the peak frequency
$\omega _{{\textit{peak}}}(x)$
for the combined measurements across the tank
$F(x,\omega )$
and the peak frequency
$\omega _{{\textit{peak}},c}(x)$
at the centreline of the tank
$D(x,\omega )$
. To the right we see the mean frequency
$\omega _{{\textit{mean}}, n}(x)$
for each transverse mode
$J_n(x,\omega )$
, the mean frequency
$\omega _{{\textit{mean}}}(x)$
for the combined measurements across the tank
$F(x,\omega )$
and the mean frequency
$\omega _{{\textit{mean}},c}(x)$
at the centreline of the tank
$D(x,\omega )$
. Several of the transverse modes
$n\gt 0$
may actually experience significant upshift, but they do not carry significant amount of energy, therefore there is a clear tendency of downshift for the wave system as a whole and along the centreline.
5. Discussion
The results can be summarised in parametric
$(\epsilon ,\Delta \mu )$
-plots in order to distinguish which parametric regimes that give downshift. Figure 6 shows results for the downshift of the peak frequency
$\omega _{{\textit{peak}}}(x)$
, while figure 7 shows results for the downshift of the mean frequency
$\omega _{{\textit{mean}}}(x)$
, in both cases for the combined measurements across the tank
$F(x,\omega )$
. Each run corresponds to a cross,
$\times$
, or a cross in a circle,
$\otimes$
.
Parametric plot for peak frequency downshift. Runs without,
$\times$
, and with,
$\otimes$
, clear visual observation of type II horseshoe patterns. Run numbers reduced by 8000. Colours: black, no downshift
$\Delta \omega _{{\textit{peak}}}/\omega _c \gt 0.9$
; red, small downshift
$0.9 \geq \Delta \omega _{{\textit{peak}}}/\omega _c \gt 0.76$
; blue, clear downshift
$\Delta \omega _{{\textit{peak}}}/\omega _c \leq 0.76$
. Solid red contour at value 0.9, solid blue contour at value 0.76. Dotted contours attempted parameterisation explained in text.
Parametric plot for mean frequency downshift. Runs without,
$\times$
, and with,
$\otimes$
, clear visual observation of type II horseshoe patterns. Run numbers reduced by 8000. Colours: black, no downshift
$\Delta \omega _{{\textit{mean}}}/\omega _c \gt 0.95$
; red, small downshift
$0.95 \geq \Delta \omega _{{\textit{mean}}}/\omega _c \gt 0.89$
; blue, clear downshift
$\Delta \omega _{{\textit{mean}}}/\omega _c \leq 0.89$
. Solid red contour at value 0.95, solid blue contour at value 0.89. Dotted contours attempted parameterisation explained in text.
Colours indicate the amount of downshift at the last probe location
$x_4$
. For the duplicate runs with identical parameter values
$(\epsilon ,\Delta \mu )$
, only the highest run numbers with the largest value of
$x_4$
were employed in order to compute the contour levels. Runs with the largest amount of downshift are shown in blue. Runs with smaller amounts of downshift are shown in red. Runs without downshift or with little downshift are shown in black. The chosen contour levels are different for the peak frequency in figure 6 and for the mean frequency in figure 7, and are given in the figure captions, because the peak frequency downshift is larger than the mean frequency downshift.
In these plots the solid contours are obtained from the irregular grid of data points using the ‘griddata’ command in Octave/Matlab with linear interpolation. We have attempted to parameterise the experimental result by the formulas
$\Delta \mu = 0.2 (\epsilon - 0.15)^{0.25}$
shown in red dots and
$\Delta \mu = 0.1 (\epsilon - 0.22)^{0.1}$
shown in blue dots.
Run numbers 2 and 26 both correspond to the single experiment reported in Trulsen et al. (Reference Trulsen, Stansberg and Velarde1999). We confirm that this choice of
$(\epsilon ,\Delta \mu )$
indeed produces a downshift, as previously reported. The fact that run number 10, with equal steepness
$\epsilon$
and greater
$\Delta \mu$
, does not produce a downshift suggests that transverse modulations are essential for this downshift to occur.
Run 11 has downshift while run 10 does not. They have the same value of
$\Delta \mu$
, but run 11 has greater steepness
$\epsilon$
than run 10. This is likely associated with the greater amount of energy loss in the steeper run 11, possibly giving support to previous findings that damping associated with heavy modulations can give downshift for long-crested waves (e.g. Trulsen & Dysthe Reference Trulsen and Dysthe1990)
Run 19 has downshift while run 1 does not, for the same values of
$(\epsilon , \Delta \mu )$
. This is due to the probes being too close to the wave maker for run 1. The probes were moved farther down the tank for runs 8018–8026. We employed the result from run 8019 in order to draw the solid contour curves.
Experimental runs with clear visual observation of type II horseshoe patterns are indicated by circles with crosses
$\otimes$
, this happens for the largest values of steepness
$\epsilon \gt 0.3$
. The results clearly show that the downshift does not depend on type II modulational instability or the horseshoe modulation pattern. It is interesting to note that run 12 does not have a clear downshift of the peak frequency, while run 11 does, suggesting that the downshift depends on type I rather than type II modulational instability.
The downshift clearly depends on the combination of the tank being sufficiently wide and the wave being sufficiently steep. The thresholds are approximately
$\Delta \mu \approx 0.1$
and
$\epsilon \approx 0.15$
. Downshift can happen in wide wave tanks
$\Delta \mu \lt 0.1$
for steepness
$\epsilon \gt 0.15$
. For narrower wave tanks
$0.1 \lt \Delta \mu \lt 0.12$
, downshift can happen for larger steepness
$\epsilon \approx 0.25$
, but it is not clear that an even greater steepness
$\epsilon \gt 0.3$
will guarantee downshift for these values of
$\Delta \mu$
. For even narrower wave tanks
$\Delta \mu \gt 0.12$
it is not clear that a downshift can happen regardless of the steepness.
Those runs with the largest steepness denoted by circle and cross,
$\otimes$
, correspond to clear observations of type II broad-band horseshoe patterns along the wave crests. Not all of the cases with horseshoe patterns did show downshift, only those with sufficiently small
$\Delta \mu$
showed downshift. We therefore do not expect that type II modulations play any role in the frequency downshift.
The detailed results in Appendix A clearly show that if measurements were limited to only the centreline along the tank, with only one probe across the tank, the conclusions would have been different from those discussed above. This could have resulted in an overestimation of the energy dissipation and an incorrect estimation of the frequency downshift. This is because energy and frequency content can be ‘hidden’ in the transverse modes. Our results suggest that it may be sufficient to include only the first transverse mode
$n=1$
in addition to the long-crested mode
$n=0$
, this should be investigated with appropriate models in the future.
Pictures from all the experiments are shown in Appendix B.
The most significant limitation of these experiments is the rather limited maximum normalised evolution distance
$\epsilon ^2 k_c x_4$
, possibly concealing downshift at greater distances for some of the runs.
Comparing these results with previous literature, we should keep in mind that many previous investigations on downshift were done with seeded waves, while our experiments employ unseeded waves.
While the most accepted paradigms for downshifting argue that dissipation is required for downshifting to occur. Our experiments contain both dissipation and transverse modulation at the same time. The full explanation for downshifting likely depends on both.
6. Conclusions
We have presented experimental evidence for the numerical prediction of Trulsen & Dysthe (Reference Trulsen and Dysthe1997), confirming and extending the experimental observation of Trulsen et al. (Reference Trulsen, Stansberg and Velarde1999), that transverse narrow-banded modulations can play an essential role in the evolution of initially uniform long-crested waves, and can facilitate a frequency downshift. Insufficient measurement resolution across the tank can result in overestimating dissipation and incorrect estimation of the amount of downshifting, because energy and frequency content can be ‘hidden’ in transverse modes. We observe clear downshift only in cases that allow significant interactions with unstable narrow-banded transverse modes, corresponding to transverse type I instability. We do not necessarily observe downshift with modulations of unstable wide-banded transverse modes, corresponding to type II instability. A criterion for downshift appears to be that the wave tank should accommodate crest lengths of at least 10 wavelengths.
Funding
This work was initiated in 1997 as a collaboration between Kristian B. Dysthe at the University of Bergen, Carl Trygve Stansberg at Marintek and Karsten Trulsen first at the University of Bergen and then at Universidad Complutense de Madrid. The experiments were carried out at Marintek in 1998. Kristian B. Dysthe died 30 July 2023, before the paper was finished. The analysis and writing was completed by Karsten Trulsen at the University of Oslo. The experiments were funded by Norsk Hydro. We thank Gunnar Furnes, then at Norsk Hydro, for his support securing that funding. The early stage of this work was supported by the Research Council of Norway grant RCN 109382 and the European Union grant MAS3–CT96–5016.
Declaration of interests
The authors report no conflict of interest.
Appendix A. Detailed analysis for each run
Figures 8–31 show detailed analysis of the various experimental runs. For explanation please see the legends of figures 3–5.
Run 8000.
Run 8001.
Run 8002.
Run 8003.
Run 8007.
Run 8008.
Run 8009.
Run 8010.
Run 8011.
Run 8012.
Run 8013.
Run 8014.
Run 8015.
Run 8016.
Run 8017.
Run 8018.
Run 8019.
Run 8020.
Run 8021.
Run 8022.
Run 8023.
Run 8024.
Run 8025.
Run 8026.
Appendix B. Pictures from various runs
Pictures from run 8008.
Pictures from run 8009.
Pictures from run 8010.
Pictures from run 8011.
Pictures from run 8012.
Pictures from run 8013.
Pictures from run 8014.
Pictures from run 8015.
Pictures from run 8016.
Pictures from run 8017.
Pictures from run 8019.
Pictures from run 8025.
Open access
