高調波共鳴定在波の線形安定性 (非線形波動現象の構造と力学)

高調波共鳴定在波の線形安定性

九大応力研 (RIAM, Kyushu Univ.) 岡村誠 (OKAMURA Makoto)

IRD Mansour IOUALALEN IRPHE Christian KHARIF

Stability calculations ofthree-dimensional short-crested

waves

very

near

their tw0-dimensional standing

wave

limit

are

performed

on

water of

uni-form depth. Non-resonant

waves are

stable while resonant

waves are

unsta-ble, which

means

that the resonant interaction contributes to instability.

1Introduction

Linearly, ashort-crested

wave

is defined

as

superposition between

an

incident travelling

wave

with

an

angle $\theta$ to avertical wall and its reflected

wave, where

0is

the angle between the direction of the incident

wave

and

the normal to the wall. The standing

wave

limit corresponds to the angle

$\theta=0^{0}$

.

The properties of short-crested

waves

have been discussed in Marchant

&Roberts

(1987)

on

water of fifinite depth. The authors conjectured that

short-crested

wave

fields may be unstable through harmonic

resonance

phe-nomena.

Later, Ioualalen et al. (1996) showed that harmonic

resonance

is

associated with sporadic and weak superharmonic instability for

short-crested

waves

in finite depth. $\ln$ particular the instability region exhibits

abubble-like shape in the

wave

steepness parameter space. However their

short-crested

wave

solutions

were

not complete because only

one

branc

数理解析研究所講究録 1271 巻 2002 年 32-40

of the multiple-like solutions associated with harmonic

resonance

has been

analysed. Ioualalen&Okamura (2002) calculated nonlinear short-crested

waves

with multiple-like solutions, $i.e.$, two branches linked by

aturn-$\mathrm{i}\mathrm{n}\mathrm{g}$ point and

one

single branch. They found the solutions by Ioualalen

et al. (1996) incomplete when harmonic

resonance occurs.

They also

ob-tained their stability diagram in the vicinity of harmonic

resonance

and

found that harmonic

resonance

is associated with two bubbles of

instabil-$\mathrm{i}\mathrm{t}\mathrm{y}$ that

are

not anymore sporadic.

In thepresent study,

we

examine the relation between short-crested

waves

with

a

small angle $\theta$ and standing

waves near

the critical depth.

Then

we

perform

a

superharmonic stability analysis of resonant short-crested

waves

very

near

the standing

wave

limit. Our stability scheme does not apply

directly to standing waves in order to use the stability analysis for steady

waves.

2

Formulation

We consider standing gravity

waves on an

inviscid, incompressible fluid

of fifinite depth where the flow is assumed irrotational. The governing

equa-tions

are

given in

a

dimensionless form with respect to the reference length

$1/k$ and the reference time $(gk)^{-1/2}$, where $g$ is the gravitational

accelera-tion and $k$ the wavenumber of the incident

wave

train.

Let

us

defifine aframe of reference $(x^{*}, y^{*}, z^{*}, t^{*}, \phi^{*})$

so

that $x^{*}=x-ct$,

$y^{*}=y$, $z^{*}=z$, $t^{*}=t$ and $\phi^{*}=\phi-cx^{*}$, where $c$ represents the propagation

velocity of the short-crested

wave

train and is equal to $\omega/\alpha$, $\omega$ being the

frequency of the

wave

and $\alpha=\sin\theta$ is $\mathrm{t}^{-}\mathrm{h}\mathrm{e}$

$x$-direction

wave

number, the

$y$-direction

wave

number being $\beta=\cos$$\theta$. $1\mathrm{f}$

we

omit the asterisks for sake

of simplicity, the governing equations

are:

$\Delta\phi=0$, $\mathrm{f}\mathrm{o}\mathrm{r}-d<z<\eta$, (1)

$\phi_{z}=0$,

on

$z=-d$, (2)

$\phi_{t}+\eta+\frac{1}{2}(\phi_{x}^{2}+\phi_{y}^{2}+\phi_{z}^{2}-c^{2})=0$,

on

_$z=\eta$, (3) $\eta_{t}+\phi_{x}\eta_{x}+\phi_{y}\eta_{y}-\phi_{z}=0$,

on

_$z=\eta$, (4)

where $d$ is the depth of the fluid, $\phi(x, y, z, t)$ the velocity potential and

$z=\eta(x, y, t)$ the equation of the free surface.

We

introduce the following functions to construct

a

stability problem:

$\eta(x, y, t)=\overline{\eta}(x, y)+\eta’(x, y, t)$ , (5)

$\phi(x, y, z, t)=\overline{\phi}(x, y, z)+\phi’(x, y, z, t)$, (6)

where we

assume

that the surface elevation and the velocity potential

are

superposition of

a

steady unperturbed

wave

$(\overline{\eta},\overline{\phi})$ and infifinitesimal

pertur-bations $(\eta’, \phi’)$ where $\eta’<<\overline{\eta}$ and $\phi’<<\overline{\phi}$

.

After substituting expressions

(5) and (6) into equations (1)$-(4)$ and linearizing,

we

obtain the zeroth

order system ofequations for which permanent short-crested

waves are

so-lutions and the fifirst order perturbation equations representing the stability

problem.

In order to solve the zeroth order system of equations,

we

look for the

following form of the velocity potential:

$\overline{\phi}=-cx+\sum_{k=0j=2}^{N}\sum_{-(k\mathrm{m}\mathrm{o}\mathrm{d} 2)}^{N}\phi_{jk}\sin(j\alpha x)\cos(k\beta y)\frac{\cosh[\kappa_{jk}(z+d)]}{\cosh(\kappa_{jk}d)}$, (7)

where $\kappa_{JK}=[(J\alpha)^{2}+(K\beta)^{2}]^{1/2}$ and $N$ is the maximum order of expansion

and is chosen tobe 19 $\mathrm{i}_{11}$ this paper. Further details about the computations

of the short-crested

waves can

be found in Okamura ₍₁₉₉₆₎

The first order system of equations is

$\triangle\phi’=0$, $\mathrm{f}\mathrm{o}\mathrm{r}-d<z<\overline{\eta}$, (8)

$\phi_{z}’=0$,

on

$z=-d$, (9)

$\phi_{t}’=-\overline{\phi}_{x}\phi_{x}’-\overline{\phi}_{y}\phi_{y}’-\overline{\phi}_{z}\phi_{z}’-\eta’(1+\overline{\phi}_{x}\overline{\phi}_{xz}+\overline{\phi}_{y}\overline{\phi}_{yz}+\overline{\phi}_{z}\overline{\phi}_{zz})$,

on

$z=\overline{\eta}$, (10)

$\eta_{t}’=\eta’(\overline{\phi}_{zz}-\overline{\eta}_{x}\overline{\phi}_{xz}-\overline{\eta}_{y}\overline{\phi}_{yz})-\overline{\eta}_{x}\phi_{x}’-\overline{\phi}_{x}\eta_{x}’-\overline{\eta}_{y}\phi_{y}’-\overline{\phi}_{y}\eta_{y}’+\phi_{z}’$,

on

$z=\overline{\eta}$. (11)

We look for non-trivial solutions of the following form:

$\eta’=e^{-i\sigma t}\sum_{J=-\infty}^{\infty}\sum_{K=-\infty}^{\infty}a_{JK}e^{i(J\alpha x+K\beta y)}$, (12)

$\phi’=e^{-i\sigma t}\sum_{J=-\infty}^{\infty}\sum_{K=-\infty}^{\infty}b_{JK}e^{i(J\alpha x+K\beta y)_{\frac{\cosh[\kappa_{JK}(z+d)]}{\cosh(\kappa_{JK}d)}}}$ , (13)

which is reduced to the eigenvalue problem determining the eigenvalues $\sigma$

and their eigenvectors consisting of $a_{JK}$ and $b_{JK}$.

3Relation

between

standing

and

short-crested

waves

Marchant&Roberts (1987) showed that harmonic

resonance occurs

for

standing

waves

of fifinite depth when

a

harmonic $(m, n)$ is

a

solution of

the homogeneous differential equation derived from the surface conditions.

Such

case occurs

at critical depths $d$ which satisfy the relation,

$n\tanh(nd)=m^{2}\tanh d$. (14)

The lowest order harmonic

resonance occurs

at depth $d_{\mathrm{h}\mathrm{r}}\approx 0.624$ which is

related to harmonic

resonance

$(3, 5)$.

We analyse the $(3, 5)$

resonance

because it is the strongest harmonic

resonance.

Figure 1exhibits the multiple-like solution structure of the

coefficient $\phi_{35}$

as a

function of the coefficient $\phi_{11}$ of the fundamental mode

$\phi_{J\mathit{5}}$ $\theta=\mathit{0}^{\mathrm{o}}$ $\phi_{JS}$ $\theta=5^{\cdot}$

Figure 1: Coefficient $\phi_{35}$ versus coefficient $\_{11}$ for depth $d=0.58$ and angles $0=0^{0}$ (left)

and $\theta=5^{\mathrm{o}}$ (right). Circle-signs $(\circ)$ and plus-signs $(+)$ denote the unstable and stable

solutions, respectively (displayed only for $\theta=0^{\mathrm{o}}$).

for depth $d=0.58$ at angles$\theta=0^{0}$ and $\theta=5^{\mathrm{O}}$

.

The solutions

are

composed

of three branches: branches (1) and (2) linked by

a

turning point (TP)

and branch (3). The figure shows that the solutions for $\theta=0^{\mathrm{o}}$

are very

similar to those for $\theta=5^{0}$ and thus

we can use

the short-crested

waves

for

$\theta=0.001^{0}$ to obtain the results for the stability of standing

waves.

Figure 1also indicates that the resonant harmonic mode $\phi_{35}$ is relatively

dominant both

on

branch (2) and

on

branch (3) for $\phi_{11}$ smaller than the

turning point (TP). We call it resonant

wave.

However the fundamental

mode $\phi_{11}$ is relatively dominant both

on

branch (1) and

on

branch (3) for

$\phi_{11}$ larger than the turning point(TP). We call it non-resonant

wave

4

Superharmonic instability of

short-crested

waves

near

their standing

wave

limit:

$\mathit{0}--0.001^{\mathrm{O}}$

We perform here the superharmonic instabilities of short-crested

waves

that

are

very close to standing waves; that is, angle $\theta=0.001^{0}$. The aim of

this study is to characterize the superharmonic instability associated with

harmonic

resonance

appearing in standing

waves as

Ioualalen&Okamura

(2002) clarifified the relation between the superharmonic instability and

harmonic

resonance

for short-crested

waves.

The time scale of the strongest

instability tells

us

whether the multiple-like solution related to harmonic

resonance

is observable

or

not.

A superharmonic instability associated with

a

harmonic

resonance

$(m, n)$

can

arise only if the two eigenvalues with opposite signature

are

equal,

$\sigma_{m,n}^{s}(h)=\sigma_{-m,n}^{-s}(h)$, (15)

for

some wave

steepness $h$. For standing

waves

the condition of harmonic

resonance

is equivalent to condition (14). Such superharmonic instability

is described

as an

interaction between the two eigenmodes $(\pm m, n)$ and the

$2m$-modes $(1, \pm 1)$ of the basic unperturbed standing wave, that is,

$\Omega_{1}=-\Omega_{2}+m\Omega_{01}+m\Omega_{02}$, (16)

$k_{1}=k_{2}+mk_{01}+mk_{02}$, (17)

where $\Omega_{i}=[|k_{i}|\tanh(\kappa_{mn}d)]^{1/2}$, $\Omega_{0i}=\tanh^{1/2}d$ for $i=1,2$ and $k_{1}=$

$(\alpha m, \beta n)$, $k_{2}=( \mathrm{m}, \beta n)$, $k_{01}=(\alpha, \beta)$, and $k_{02}=(\alpha, -\beta)$.

In Figures 2 and 3

are

plotted the frequencies and growth rates of the

eigenvalues $\sigma_{\pm 3,5}$ for all branches of the

wave

solutions for depths $d=0.58$ and $d=0.62$ in the vicinity of the critical depth $d_{\mathrm{h}\mathrm{r}}\approx 0.624$

.

For both

$0.\cdot.100.1\mathrm{R}^{\bullet}\mathrm{r}.\mathrm{o}\mathrm{e}_{0.00.10_{\theta_{\mathit{1}\mathrm{J}}}}\mathrm{o}.\mathrm{u}\mathrm{n}.\alpha \mathrm{l}4\mathrm{I}.020.020.1-.\cdot\ldots\ldots\ldots\cdot\cdot..\cdot.\ovalbox{\tt\small REJECT}^{\bullet_{\bullet_{\bullet}}}000.\alpha\cdots\cdots\cdots\cdot\cdot.\cdot.\cdot\cdot.\bullet\cdot.\cdot.\cdot\cdot.\cdot\circ\{\mathrm{n}^{\bullet}10s_{3S}\bullet\bullet_{\bullet}\bullet\bullet^{\bullet^{\bullet^{\bullet^{\bullet^{\bullet}}}}}(1)\epsilon_{l_{l_{\mathrm{o}}}}\dot{(}\mathrm{i}_{)8}\epsilon^{8^{l}}\bullet_{\bullet}\bullet$

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)

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$(1)...\mathrm{o}\mathrm{Q}\mathrm{o}(2..)$ 3 $(3)^{\mathrm{O}}l....\circ\cdot..mathrm{t}_{\mathrm{o}}^{\mathrm{O}}$ $0$ $\mathrm{Q}$ $(3)0$ $0$ _$0$ $\mathrm{Q}$ $0.\cdot(.3.)\cdot$

.

$\mathrm{t}4\theta$ $J_{0}^{9}$ $0$ $0$ $0$ $0$ $0$ $0$ $\mathrm{Q}^{\cdot}.$

.

$\mathrm{Q}(3\mathrm{k}....0...P$ (3)

.

(3)

.

$\circ$

...

..

$(1f.$

.

$\mathrm{Q}$ $\mathrm{o}$ $(2)$

Figure 2: Frequency $[-\Re(\sigma\pm 35)]$ $(\bullet)$ and growth rate $[-\Im(\sigma\pm 35)](0)$ as a function of

coefficient $\phi_{11}$ for angle $\theta=0.001^{0}$ and depth$d=0.58$. The right panel is an enlargement

of the left panel.

$-0.\alpha \mathrm{I}44.\mathfrak{W}^{\bullet^{\bullet}}p\}.\alpha \mathrm{I}2-0.\mathrm{m}_{0.\mathrm{m}\mathrm{o}.0}- 0.010^{\cdot}..\cdot.\cdot..\cdot.\cdot.\cdot...\cdot..\cdot.\cdot..\cdot.\cdot.\cdot.\cdot$

.

$\cdot..- 0\ovalbox{\tt\small REJECT}_{\bullet}^{\bullet}(\bullet\bullet \mathfrak{l}_{\bullet\bullet}^{2\succ 13)}.\infty 20.\mathrm{r}^{(3(3S}0.\alpha \mathrm{I}20.\mathfrak{m}0.M0.\mathrm{m}0.0100.\mathrm{m}..\cdot.\cdot.\cdot..\cdot.\cdot.\nwarrow s^{\bullet}\mathrm{Q}\circ(2\succ(3)\bullet\bullet 3s_{10.020.030.u\mathrm{o}.\mathrm{o}s\mathrm{o}.oe\mathrm{o}.\sigma 70.1-0.w0.100.\alpha 00.\mathrm{M}20.\mathfrak{U}0.\mathrm{m}\mathrm{o}.\mathfrak{m}0.\mathrm{o}n\mathrm{o}.\mathrm{o}n\mathrm{o}.0740.0760.1n\epsilon 0.\mathrm{r}}(\cdot 3\theta(1)_{\bullet(1)}\bullet(\mathrm{i})\bullet^{\bullet^{\bullet^{\bullet^{\bullet^{\bullet}}\omega.\alpha 11}}}\bullet(2)-(3)(2)\prec 3)\bullet \mathrm{O}\bullet\bullet\bullet\bullet\bullet\bullet\bullet trr\ell rr\bullet\bullet 0.01.\tilde{\circ}_{\theta}\cdot.\cdot.\cdot..\cdot.(3)\bullet\bullet 13)\bullet(3)\cdot(1)^{\theta^{\mathrm{o}}(3)}\mathrm{o}^{l}e^{9}\mathrm{o}_{d^{0}}.\cdot.\cdot$

.

$\mathrm{o}_{\mathrm{o}^{(2\succ(3)}}$

.

_.

(1).

.

$.\tilde{l}$ (3). $(3S$

. .

$\mathrm{Q}.\nwarrow \mathrm{Q}$

.

$\cdot$

.

$\cdot$

.

$(\cdot 3\theta$ $.0\swarrow 0^{\cdot}$

.

. .

.

$\cdot \mathrm{Q}$

.

.

$\cdot$ $\theta^{\mathrm{o}}$ (1). (3).

.

.

$\mathrm{o}_{\mathrm{t}2)\prec 3)}^{l}$

Figure 3: The same as Figure 2 except for depth $d=0.62$

depths, branch (1) is stable

on

its whole region, from $\phi_{11}=0$ to the turning

point ($\phi_{11}\approx 0.2305605$ for $d=0.58$ and $\phi_{11}\approx 0.0705$ for $d=0.62$), while

branch (2) is unstable

on

its whole region. The transition from stable to

unstable

occurs

when the frequency reaches the zero-axis, then the growth

rate value leaves it. For both depths the dominant instability appears

for $\phi_{11}=0$ and the instability

on

branch (2) weakens with increasing

$\phi_{11}$ to disappear at the turning point (here at the zer0-axis). Branch(3)

is unstable from $\phi_{11}=0$ to the turning point ahead ($\phi_{11}\approx 0.2591$ for

$d=0.58$ and $\phi_{11}\approx 0.0709$ for $d=0.62$). The maximum of instability

also appears for $\phi_{11}=0$. The instability

occurs

when eigenvalues $\sigma_{3,5}$ and

$\sigma_{-3,5}$ coalesce at zero-frequency ($\mathrm{p}\mathrm{h}_{\mathrm{f}\mathrm{f}\mathrm{i}}\mathrm{e}$-locked with the unperturbed wave).

Such instability is physically associated witha resonant interaction: the

coalescence ofthe two eigenmodes at zero-frequency simply

means

that the

harmonics $(\pm 3,5)$ propagate at the

same

phase speed

as

the basic wave,

bearing in mind that the stability problem has been computed in the frame

of reference moving with the basic

wave.

Ioualalen&Okamura (2002) showed that for resonant short-crested

waves

the instability region is a small range of

₀

like

a

bubble. In the present

case

the instability region is

a

wide range of $\phi_{11}$, which is much different from

that in the short-crested

waves.

The instability is strong for resonant wave,

$i.e.$,

on

branch (2) and the left part of branch (3). The instability weakens

as

$\phi_{11}$ becomes larger. Beyond the turning point the solution

on

branch (3)

remains weakly unstable within

a

certain range of the parameter regime

then it turns stable

5

Conclusion

This study deals with the stability of the two-dimensional standing

waves

with multiple-like solutions for the strongest harmonic

resonance

$(3,5)$

oc-curs.

Since

our

numerical procedure calculating the stability of

three-dimensional short-crested

waves

does not apply to

two-dimensional

stand-$\mathrm{i}\mathrm{n}\mathrm{g}$

waves

because the

waves are

not anymore stationary,

we

fifirst show

that short-crested

waves

and standing

waves

match each other at the limit

$(\thetaarrow 0^{\mathrm{o}})$ in order to extend the stability results here to standing

waves.

Then

we

perform

a

superharmonic stability analysis of short-crested

waves

very

near

$\mathrm{t}\mathrm{h}\mathrm{e},\mathrm{i}\mathrm{r}$

standing

wave

limit. The stability analysis shows that $\mathrm{r}\mathrm{e}\mathrm{s}-$

onant

waves are

strongly unstable. By contrast, non-resonant

waves are

almost stable andweakly unstable within

a

sporadic range of the parameter

region then non-resonant

waves are

therefore only solutions to exist.

参考文献

[1] M.

_Ioualalen&M.

Okamura, ”Structure of the instability associated

with harmonic

resonance

of short-crested waves.” J. Phys. Oceanogr. 32

(2002) 1331 1337.

[2] M. Ioualalen, A. J. Roberts&C. Kharif, _{“On the observability of finite}

depth short-crested water waves.” J. Fluid Mech. 322 (1996)

1-19.

[3] T. R. Marchant&A. J. Roberts, ”Properties of short-crested

waves

in

water offinite depth.” J. Austral Math. Soc. $\mathrm{B}29$ (1987)

103-125.

[4] M. Okamura, ”Notes

on

short-crested

waves

in deep water.”J. Phys.

Soc. $Jpn$. 65 (1996)

2841-2845