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1. Introduction
We consider the two-dimensional nonlinear problem describing steady waves in a horizontal open channel of uniform rectangular cross-section. The motion of an inviscid incompressible heavy fluid, say water, occupying the channel is supposed to be rotational with a prescribed vorticity distribution. This type of motion commonly occurs in nature, as is indicated by observations (see, for example, Thomas (Reference Thomas1990), Swan, Cummins & James (Reference Swan, Cummins and James2001) and references cited therein). The plethora of results obtained for various models describing waves with vorticity is briefly characterised by Kozlov & Kuznetsov (Reference Kozlov and Kuznetsov2014). In that paper, flows with counter-currents as well as unidirectional ones were studied. Further details about the latter flows concerning, in particular, global branches of solutions can be found in § 3 of the survey article by Strauss (Reference Strauss2010).
Here, our aim is to investigate some new properties of the problem of waves on unidirectional flows. In particular, we generalise fundamental bounds for steady irrotational waves found by Kozlov & Kuznetsov (Reference Kozlov and Kuznetsov2009) (see also Keady & Norbury (Reference Keady and Norbury1975) and Amick & Toland (Reference Amick and Toland1981) for the cases of Stokes and solitary waves respectively), thus extending results obtained in Keady & Norbury (Reference Keady and Norbury1978), Constantin & Strauss (Reference Constantin and Strauss2011) and Kozlov & Kuznetsov (Reference Kozlov and Kuznetsov2012) for waves with vorticity. Only periodic waves were considered in the first two papers, whereas some superfluous requirements were imposed in the last paper on the behaviour of the stream function, on the free-surface profile and, most importantly, on the derivative of the vorticity distribution.
1.1. Statement of the problem
Let an open channel of uniform rectangular cross-section be bounded below by a horizontal rigid bottom and let water occupying the channel be bounded above by a free surface not touching the bottom. The water motion is supposed to be two-dimensional and rotational; the surface tension is neglected on the free surface of the water, where the pressure is constant. These assumptions and the fact that water is incompressible allow us to seek the velocity field in the form
$({\it\psi}_{y},-{\it\psi}_{x})$
, where
${\it\psi}(x,y)$
is referred to as the stream function. The vorticity distribution
${\it\omega}$
is supposed to be a prescribed continuous function depending on
${\it\psi}$
.
The non-dimensional variables proposed by Keady & Norbury (Reference Keady and Norbury1978) (see Appendix A in Kozlov & Kuznetsov (Reference Kozlov and Kuznetsov2014) for details of scaling) are used here. In particular, lengths and velocities are scaled to
$(Q^{2}/g)^{1/3}$
(the depth of the critical uniform stream in the irrotational case) and
$(Qg)^{1/3}$
respectively;
$Q$
and
$g$
are the dimensional quantities for the rate of flow and the gravity acceleration respectively.
In appropriate Cartesian coordinates
$(x,y)$
, the bottom coincides with the
$x$
-axis and gravity acts in the negative
$y$
-direction. We choose the frame of reference so that the velocity field is time-independent as well as the unknown free-surface profile. The latter is assumed to be the graph of
$y={\it\eta}(x)$
,
$x\in \mathbb{R}$
, where
${\it\eta}$
is a positive continuous function, and so the longitudinal section of the water domain is
$D=\{x\in \mathbb{R},0<y<{\it\eta}(x)\}$
.
The following free-boundary problem for
${\it\psi}$
and
${\it\eta}$
which describes all kinds of waves has long been known (cf. Keady & Norbury Reference Keady and Norbury1978):
$$\begin{eqnarray}\displaystyle & {\it\psi}_{xx}+{\it\psi}_{yy}+{\it\omega}({\it\psi})=0,\quad (x,y)\in D; & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & {\it\psi}(x,0)=0,\quad x\in \mathbb{R}; & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & {\it\psi}(x,{\it\eta}(x))=1,\quad x\in \mathbb{R}; & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & |\boldsymbol{{\rm\nabla}}{\it\psi}(x,{\it\eta}(x))|^{2}+2{\it\eta}(x)=3r,\quad x\in \mathbb{R}. & \displaystyle\end{eqnarray}$$
$r$
is a constant considered as the problem’s parameter and referred to as Bernoulli’s constant/the total head. In what follows, we suppose that
${\it\psi}$
is a monotonic function of
$y$
for all
$x\in \mathbb{R}$
, which means that the flows we are going to study are unidirectional.
1.2. Main results
Prior to formulating our results we list some auxiliary facts required in what follows.
By a stream (shear-flow) solution of problem (1.1)–(1.4) we mean a pair
$(u(y),d)$
(
$u$
stands for the stream function instead of
${\it\psi}$
and the constant depth of flow
$d$
replaces the wave profile
${\it\eta}$
) such that the following relations hold:
$$\begin{eqnarray}u^{\prime \prime }+{\it\omega}(u)=0\quad \text{on }(0,d),\quad u(0)=0,\quad u(d)=1,\quad |u^{\prime }(d)|^{2}+2d=3r.\end{eqnarray}$$
Here, the prime denotes differentiation with respect to
$y$
. A detailed study of these solutions, in particular those describing flows with counter-currents, is given in Kozlov & Kuznetsov (Reference Kozlov and Kuznetsov2011). The set of unidirectional solutions of (1.5a–c
) is parametrised by
$s=u^{\prime }(0)$
; it is greater than or equal to
$s_{0}=\sqrt{2\max _{0\leqslant {\it\tau}\leqslant 1}{\rm\Omega}({\it\tau})}$
, as follows from the following expressions for
$u$
and
$d$
(implicit and explicit respectively):
$$\begin{eqnarray}y=\int _{0}^{u}\frac{\text{d}{\it\tau}}{\sqrt{s^{2}-2{\rm\Omega}({\it\tau})}}\quad \text{and}\quad d=\int _{0}^{1}\frac{\text{d}{\it\tau}}{\sqrt{s^{2}-2{\rm\Omega}({\it\tau})}},\quad \text{where }{\rm\Omega}({\it\tau})=\int _{0}^{{\it\tau}}{\it\omega}(t)\text{d}t.\end{eqnarray}$$
(It should be noted that
${\rm\Omega}$
belongs to
$C^{1}([0,1])$
.) It is clear that
$d~[=d(s)]$
decreases strictly monotonically and tends to zero as
$s\rightarrow +\infty$
, whereas
$d_{0}=\lim _{s\rightarrow s_{0}+0}d(s)$
can be finite or infinite depending on the behaviour of
${\rm\Omega}$
on
$[0,1]$
(see below).
Equations (1.6a ) and (1.5d ) imply the equation
$$\begin{eqnarray}r=\mathscr{R}(s),\quad \text{where }\mathscr{R}(s)=[s^{2}-2{\rm\Omega}(1)+2d(s)]/3.\end{eqnarray}$$
It is clear that, as it is convex, this function has only one minimum, say
$r_{c}>0$
, attained at some
$s_{c}>s_{0}$
. If
$d_{0}=+\infty$
and
$r>r_{c}$
, then (1.7) has two solutions
$s_{+}$
and
$s_{-}$
such that
$s_{0}<s_{+}<s_{c}<s_{-}$
. By substituting
$s_{+}$
and
$s_{-}$
into (1.6), one obtains the stream solutions
$(u_{+},d_{+})$
and
$(u_{-},d_{-})$
respectively. The shear flows described by these solutions are analogous to the uniform sub- and supercritical flows existing in the irrotational case. If
$d_{0}<+\infty$
, then both
$s_{+}$
and
$s_{-}$
, and, consequently, the corresponding stream solutions exist only for
$r\in (r_{c},r_{0})$
, where
$r_{0}=\mathscr{R}(s_{0})$
. If
$r>r_{0}$
, then only
$s_{-}$
exists and defines
$(u_{-},d_{-})$
.
In order to describe how
$d_{0}$
and the corresponding stream solution depend on the vorticity distribution Kozlov & Kuznetsov (Reference Kozlov and Kuznetsov2011) considered the following three options.
-
(i)
$\max _{0\leqslant {\it\tau}\leqslant 1}{\rm\Omega}({\it\tau})$
is attained either at an inner point of
$(0,1)$
or at one (or both) of the end points. In the latter case, either
${\it\omega}(1)=0$
when
${\rm\Omega}(1)>{\rm\Omega}({\it\tau})$
for
${\it\tau}\in (0,1)$
or
${\it\omega}(0)=0$
when
${\rm\Omega}(0)>{\rm\Omega}({\it\tau})$
for
${\it\tau}\in (0,1)$
(or both of these conditions hold simultaneously). -
(ii)
${\rm\Omega}(0)>{\rm\Omega}({\it\tau})$
for
${\it\tau}\in (0,1]$
and
${\it\omega}(0)<0$
. -
(iii)
${\rm\Omega}({\it\tau})<{\rm\Omega}(1)$
for
${\it\tau}\in (0,1)$
and
${\it\omega}(1)>0$
. Moreover, if
${\rm\Omega}(1)=0$
, then
${\it\omega}(0)<0$
and
${\it\omega}(1)>0$
must hold simultaneously.
Conditions (i)–(iii) define three disjoint sets of vorticity distributions whose union is the whole set of distributions continuous on
$[0,1]$
. It occurs that conditions (i) imply that
$d_{0}=+\infty$
, whereas conditions (ii) and (iii) yield that
$d_{0}$
is finite as well as
$r_{0}$
. The last two conditions have the following consequences for solutions of problem (1.5) when
$s=s_{0}$
:
$u^{\prime }(d_{0})=0$
under conditions (iii), whereas (ii) implies that
$u^{\prime }(0)=0$
and
$u^{\prime }(d_{0})\neq 0$
.
In order to formulate our result on fundamental bounds for
$\hat{{\it\eta}}=\sup _{x\in \mathbb{R}}{\it\eta}(x)$
and
$\check{{\it\eta}}=\inf _{x\in \mathbb{R}}{\it\eta}(x)$
we have to impose some restrictions on
${\it\psi}$
and
${\it\eta}$
.
Conditions (A). The function
${\it\psi}$
belongs to
$C^{1,{\it\alpha}}(\bar{D})$
; that is,
$|{\it\psi}|$
,
$|{\it\psi}_{x}|$
and
$|{\it\psi}_{y}|$
are bounded on
$\bar{D}$
, whereas the derivatives satisfy the Hölder condition there. Moreover,
${\it\psi}_{y}(x,y)\geqslant {\it\delta}$
on
$\bar{D}$
for some
${\it\delta}>0$
. The function
${\it\eta}$
belongs to
$C_{loc}^{0,{\it\alpha}}(\mathbb{R})$
.
In view of conditions (A), problem (1.1)–(1.4) and its reformulation introduced in § 1.3 are considered in certain weak senses (see Constantin & Strauss (Reference Constantin and Strauss2011), where the question has been raised whether different weak formulations of the problem describing steady waves with vorticity are equivalent, and Varvaruca & Zarnescu (Reference Varvaruca and Zarnescu2012) for a detailed study of this question).
Theorem 1. Let for some
$r>0$
problem (1.1)–(1.4) have a non-stream solution
$({\it\psi},{\it\eta})$
satisfying conditions (A). Then the following two assertions hold.
-
(a)
$r>r_{c}$
and
${\it\eta}(x)>d_{-}$
for all
$x\in \mathbb{R}$
. -
(b) If
$r_{0}$
is finite and
$r\in (r_{c},r_{0})$
, then
$\hat{{\it\eta}}\geqslant d_{+}>\check{{\it\eta}}$
. Moreover, the left inequality is also strict provided
$\hat{{\it\eta}}$
is attained at some point.
Furthermore, if
${\it\omega}$
satisfies conditions (iii), then two additional assertions hold.
-
(c) If
$r\in (r_{c},r_{0})$
, then the inequality
$d_{0}>\hat{{\it\eta}}$
complements the above ones for a non-stream solution.
We expect that the left inequality in assertion (b) will always be strict, which is the case for irrotational waves; see Kozlov & Kuznetsov (Reference Kozlov and Kuznetsov2009), where assertions similar to (a) and (b) were obtained. However, there is no analogue of assertion (d) for irrotational waves. On the other hand, it extends the result of Kozlov & Kuznetsov (Reference Kozlov and Kuznetsov2014) on the absence of small-amplitude non-stream solutions (not necessarily unidirectional) when
$r=r_{0}$
,
$s=s_{0}>0$
and a certain restriction is imposed on vorticity. Since Theorem 1 does not cover the case when
$r>r_{0}$
and the vorticity distribution satisfies conditions (ii), we discuss it in § 2.3. It is worth emphasising that the classification (i)–(iii) plays an important role in our proof of Theorem 1.
1.3. The partial hodograph transform
Since we consider unidirectional flows, it is convenient to reformulate problem (1.1)–(1.4) using the partial hodograph transform; that is, by mapping the unknown domain
$D$
onto the strip
$S=\mathbb{R}\times (0,1)$
(cf. Groves & Wahlén Reference Groves and Wahlén2008; Constantin & Strauss Reference Constantin and Strauss2011):
$$\begin{eqnarray}\bar{D}\ni (x,y)\mapsto (q,p)\in \bar{S},\quad \text{where }q=x\text{ and }p={\it\psi}(x,y).\end{eqnarray}$$
These variables are treated as independent, whereas
$y=h(q,p)$
is the new unknown function such that
$h_{p}(q,p)\geqslant {\it\delta}^{\prime }>0$
on
$\bar{S}$
, which follows from conditions (A). A straightforward calculation shows that problem (1.1)–(1.4) takes the following form:
$$\begin{eqnarray}\mathscr{D}h=0,\quad (q,p)\in S;\quad h(q,0)=0,\quad q\in \mathbb{R};\quad (1+h_{q}^{2})h_{p}^{-2}+2h=3r,\quad p=1,q\in \mathbb{R}.\end{eqnarray}$$
Here,
$\mathscr{D}h$
stands for
$$\begin{eqnarray}\left[\frac{h_{q}}{h_{p}}\right]_{q}-\left[\frac{1+h_{q}^{2}}{2h_{p}^{2}}+{\rm\Omega}(\,p)\right]_{p}.\end{eqnarray}$$
On the other hand, a solution of this problem allows us to recover
${\it\eta}$
using the equality
$$\begin{eqnarray}{\it\eta}(x)=h(x,1),\quad x\in \mathbb{R}.\end{eqnarray}$$
Therefore, we write
$\hat{{\it\eta}}$
and
$\check{{\it\eta}}$
for
$\sup _{q\in \mathbb{R}}h(q,1)$
and
$\inf _{q\in \mathbb{R}}h(q,1)$
respectively in what follows. It should be noted that
$h(q,1)>0$
because
$h(q,0)=0$
and
$h_{p}$
is positive in
$S$
.
In order to recover
${\it\psi}$
one has to solve the Dirichlet problem (1.1)–(1.3) in the domain whose upper boundary is defined by (1.11). An alternative way of recovering
${\it\psi}$
is through the implicit relation
$y=h(x,{\it\psi}(x,y))$
.
2. Proof of Theorem 1
To prove Theorem 1 we reformulate it in terms of problem (1.9).
Theorem 1′. Let for some
$r>0$
problem (1.9) have a non-stream solution
$h\in C^{1,{\it\alpha}}(\bar{S})$
such that
$h_{p}\geqslant {\it\delta}^{\prime }$
on
$\bar{S}$
for some
${\it\delta}^{\prime }>0$
. Then the following two assertions hold.
-
(a')
$r>r_{c}$
and
$h(q,1)>d_{-}$
for all
$q\in \mathbb{R}$
. -
(b') If
$r_{0}$
is finite and
$r\in (r_{c},r_{0})$
, then
$\hat{{\it\eta}}\geqslant d_{+}>\check{{\it\eta}}$
. Moreover, the left inequality is also strict provided
$\hat{{\it\eta}}$
is attained at some point.
Furthermore, if
${\it\omega}$
satisfies conditions (iii), then two additional assertions hold.
-
(c') If
$r\in (r_{c},r_{0})$
, then the inequality
$d_{0}>\hat{{\it\eta}}$
complements the above ones for a non-stream solution.
-
(d') If
$r\geqslant r_{0}$
, then every solution of problem (1.9) belonging to
$C^{1,{\it\alpha}}(\bar{S})$
is a stream solution.
2.1. Auxiliary propositions
The proof of Theorem 1′ is based on two auxiliary propositions generalising the ‘Inclusion Theorem’ obtained by Keady & Norbury (Reference Keady and Norbury1982). The first of these is the maximum principle for an elliptic operator that arises when considering
$\mathscr{D}{\it\xi}-\mathscr{D}{\it\zeta}$
, where
${\it\xi}$
and
${\it\zeta}$
are two different solutions of (1.9a
). In Constantin & Strauss (Reference Constantin and Strauss2011) (see pp. 155, 156), it is shown that
$u={\it\xi}-{\it\zeta}$
satisfies the equation
$$\begin{eqnarray}\mathscr{L}u=\big[a^{(qq)}u_{q}+a^{(qp)}u_{p}\big]_{q}+\big[a^{(\,pp)}u_{p}+a^{(\,pq)}u_{q}\big]_{p}=0\quad \text{in }S,\end{eqnarray}$$
which is uniformly elliptic provided
${\it\xi}$
and
${\it\zeta}$
have bounded gradients and
${\it\xi}_{p}$
and
${\it\zeta}_{p}$
are greater than some positive constant.
Proposition 1. Let
$u\in W_{loc}^{1,2}(\bar{S})$
be bounded in
$\bar{S}$
. If
$\mathscr{L}u=0$
on
$S$
, where
$\mathscr{L}$
has the form (2.1) and its coefficients are bounded and such that
$\mathscr{L}$
is uniformly elliptic on
$\bar{S}$
with the ellipticity constant
$C_{\mathscr{L}}>0$
, then the following equalities hold:
$$\begin{eqnarray}\sup _{(q,p)\in S}u(q,p)=\sup _{q\in \mathbb{R}}\{u(q,0),u(q,1)\}\quad \text{and}\quad \inf _{(q,p)\in S}u(q,p)=\inf _{q\in \mathbb{R}}\{u(q,0),u(q,1)\}.\end{eqnarray}$$
Proof. It is clear that (2.2b
) follows from (2.2a
). It should be noted that the assumptions imposed on
$u$
imply its Hölder continuity (see, for example, § 8.9 in Gilbarg & Trudinger (Reference Gilbarg and Trudinger2001)). Moreover, the Hölder norm of
$u$
over
$[t,t+1]\times [0,1]$
is bounded by a constant depending on
$C_{\mathscr{L}}$
, the Hölder exponent and various bounds (for the coefficients of
$\mathscr{L}$
and for
$u$
itself), but independent of
$t$
.
In order to prove (2.2a
) we assume the contrary; that is,
$\hat{u} =\sup _{(q,p)\in S}u$
is strictly greater than
$\sup _{q\in \mathbb{R}}\{u(q,0),u(q,1)\}$
. Then there exist positive numbers
${\it\epsilon}$
and
$\ell$
(the latter depends on
${\it\epsilon}$
, generally speaking) such that
$$\begin{eqnarray}\hat{u} -u(q,p)\geqslant {\it\epsilon}\quad \text{on }S_{\ell }=\{(q,p)\in S:q\in \mathbb{R},~p\in (0,\ell )\cup (1-\ell ,1)\}.\end{eqnarray}$$
There also exists a sequence
$\{(q_{k},p_{k})\}_{k=1}^{\infty }$
such that
$\ell <p_{k}<1-\ell$
and
$u(q_{k},p_{k})\rightarrow \hat{u}$
.
Let
$B_{{\it\rho}}(q,p)$
denote the open circle of radius
${\it\rho}$
centred at
$(q,p)$
. Since
$\hat{u} -u\geqslant 0$
satisfies
$\mathscr{L}(\hat{u} -u)=0$
in
$S$
, Harnack’s inequality (see Corollary 8.21 in Gilbarg & Trudinger (Reference Gilbarg and Trudinger2001)) is applicable in every
$B_{{\it\rho}(\ell )}(q_{k},1/2)$
, where
${\it\rho}(\ell )=(1-\ell )/2$
, and so
$(q_{k},p_{k})\in B_{{\it\rho}(\ell )}(q_{k},1/2)$
. Therefore, we have
$$\begin{eqnarray}\sup _{(q,p)\in B_{{\it\rho}(\ell )}(q_{k},1/2)}\big[\hat{u} -u(q,p)\big]\leqslant C\inf _{(q,p)\in B_{{\it\rho}(\ell )}(q_{k},1/2)}\big[\hat{u} -u(q,p)\big]\leqslant C\big[\hat{u} -u(q_{k},p_{k})\big]\rightarrow 0\end{eqnarray}$$
as
$k\rightarrow \infty$
for some
$C>0$
that depends on bounds for the coefficients of
$\mathscr{L}$
,
$C_{\mathscr{L}}$
and
$\ell$
. Hence, the supremum on the left is arbitrarily small provided
$k$
is sufficiently large, but this is incompatible with (2.3), because
$S_{\ell }$
overlaps with
$B_{{\it\rho}(\ell )}(q_{k},1/2)$
for all
$k=1,\dots ,\infty$
. The obtained contradiction proves the proposition.
The next proposition provides a surrogate of Hopf’s lemma for an elliptic equation in the strip
$\bar{S}$
.
Proposition 2. Let
$\mathscr{L}$
be an elliptic operator of the form (2.1). If
$u\in C^{1,{\it\alpha}}(\bar{S})$
satisfies the equation
$\mathscr{L}u=0$
in
$S$
and
$\sup _{(q,p)\in S}u(q,p)=\sup _{q\in \mathbb{R}}u(q,1)$
, then for any sequence
$\{q_{k}\}_{k=1}^{\infty }$
such that
$$\begin{eqnarray}u(q_{k},1)\rightarrow \sup _{q\in \mathbb{R}}u(q,1)\quad \text{as }k\rightarrow \infty\end{eqnarray}$$
the following relations hold:
$$\begin{eqnarray}u_{q}(q_{k},1)\rightarrow 0\quad \text{as }k\rightarrow \infty \quad and\quad \mathop{\limsup }_{k\rightarrow \infty }\;u_{p}(q_{k},1)\geqslant 0.\end{eqnarray}$$
A similar assertion is true with supremum changed to infimum in the assumption, whereas
$\limsup$
is replaced by
$\liminf$
and the inequality sign is opposite in (2.6).
If supremum is attained at some point
$(q_{\ast },1)$
, then
$u_{p}(q_{\ast },1)>0;$
if infimum is attained, the inequality sign is opposite.
Proof. The last assertion for supremum is a consequence of Proposition 1 and Hopf’s lemma, whereas for infimum it follows by changing
$u$
to
$-u$
. Therefore, we assume that
$\hat{u} =\sup _{(q,p)\in S}u(q,p)=\sup _{q\in \mathbb{R}}u(q,1)$
and take any sequence
$\{q_{k}\}_{k=1}^{\infty }$
satisfying (2.5) (without loss of generality, we take it tending to
$+\infty$
).
In order to prove relations (2.6) we assume the contrary, that is, there exist
${\it\epsilon}>0$
such that either
$|u_{q}(q_{k},1)|\geqslant {\it\epsilon}$
or
$u_{p}(q_{k},1)\leqslant -{\it\epsilon}$
holds for all sufficiently large
$k$
. Since
$u\in C^{1,{\it\alpha}}(\bar{S})$
, we have
$$\begin{eqnarray}u(q_{k}\pm t,1)=u(q_{k},1)\pm u_{q}(q_{k},1)t+O\big(t^{1+{\it\alpha}}\big)\quad \text{as }t\rightarrow +0,\end{eqnarray}$$
which yields that
$\hat{u} <u(q_{k}\pm t,1)$
when
$\pm u_{q}(q_{k},1)>{\it\epsilon}$
provided
$k$
is sufficiently large and
$t>0$
is sufficiently small. However, the last inequality for
$\hat{u}$
contradicts its definition. Similarly, we have
$$\begin{eqnarray}u(q_{k},1-t)=u(q_{k},1)-u_{p}(q_{k},1)t+O\big(t^{1+{\it\alpha}}\big)\quad \text{as }t\rightarrow +0.\end{eqnarray}$$
Then the assumption about
$u_{p}(q_{k},1)$
gives that
$\hat{u} <u(q_{k},1-t)$
provided
$t$
is sufficiently small. This is impossible in view of the definition of
$\hat{u}$
. Thus, relations (2.6) are proved.
2.2. Proof of Theorem 1′
For the reader’s convenience the proof is divided into three parts, each given a title elucidating which assertion(s) is (are) dealt with in it. Prior to turning to the proof, we note that it is straightforward to verify that for any
$s>s_{0}$
such that
$\mathscr{R}(s)=r$
the corresponding stream solution of problem (1.9), say
$H(\,p;s)$
, has the following form:
$$\begin{eqnarray}H(\,p;s)=\int _{0}^{p}\frac{\text{d}{\it\tau}}{\sqrt{s^{2}-2{\rm\Omega}({\it\tau})}}.\end{eqnarray}$$
If
$s=s_{0}$
and either of conditions (ii), (iii) is fulfilled, then this formula gives a continuous function whose derivative is infinite at one of the end points of
$[0,1]$
. In our proof, this function plays a substantial role.
We recall that assertions
$(a^{\prime })$
and
$(b^{\prime })$
are consequences of the assumption that for some
$r>0$
problem (1.9) has a non-stream solution
$h\in C^{1,{\it\alpha}}(\bar{S})$
such that
$h_{p}\geqslant {\it\delta}^{\prime }$
on
$\bar{S}$
for some
${\it\delta}^{\prime }>0$
.
2.2.1. Proof of assertion
$(a^{\prime })$
Here, we show that the following inequalities hold:
$r>r_{c}$
and
$h(q,1)>d_{-}$
for all
$q\in \mathbb{R}$
.
Proof. First we prove that
$r\geqslant r_{c}$
, for which purpose we demonstrate that there exists
$s>s_{0}$
such that
$H(1;s)=\check{{\it\eta}}$
. Let us consider vorticity distributions satisfying conditions (i)–(iii) separately.
If conditions (i) are fulfilled, then
$H(1;s)$
decreases monotonically from positive infinity to zero as
$s$
goes from
$s_{0}$
to
$+\infty$
. Hence,
$H(1;s)$
attains the value
$\check{{\it\eta}}>0$
at some
$s\in (s_{0},+\infty )$
.
Since conditions (ii) imply that
$s_{0}=0$
and
$H_{p}(0;s)\rightarrow +\infty$
as
$s\rightarrow 0$
, we have that the function
$h_{p}(q,0)-H_{p}(0;s_{\ast })$
is negative and separated from zero for some
$s_{\ast }>0$
. Then
$h(q_{\ast },1)-H(1;s_{\ast })\leqslant 0$
for some
$q_{\ast }\in \mathbb{R}$
, because otherwise the function
$h(q,p)-H(\,p;s_{\ast })$
violates Proposition 1 combined with Proposition 2. It follows from the last inequality that
$H(1;s)=\check{{\it\eta}}$
for some
$s>s_{\ast }$
, because
$H(1;s)$
monotonically decreases and tends to zero as
$s$
grows, whereas
$\check{{\it\eta}}\leqslant h(q_{\ast },1)$
.
Since conditions (iii) imply that
$s_{0}>0$
and
$H_{p}(1;s)\rightarrow +\infty$
as
$s\rightarrow s_{0}$
, we have that
$h_{p}(q,1)-H_{p}(1;s_{\ast })$
is negative and separated from zero for all
$q\in \mathbb{R}$
and some
$s_{\ast }>s_{0}$
. The latter fact is similar to that obtained from conditions (ii). This allows us to repeat literally the considerations used in that case, thus arriving at the conclusion that
$H(1;s)=\check{{\it\eta}}$
for some
$s>s_{\ast }$
.
Let us prove the inequalities
$r\geqslant r_{c}$
and
$h(q,1)>\check{{\it\eta}}$
for arbitrary
$q$
. For this purpose we consider
$h(q,p)-H(\,p;s)$
with
$s>s_{0}$
such that
$H(1;s)=\check{{\it\eta}}$
(the existence of that
$s$
was just established). According to Proposition 2, we have
$h_{p}(q_{k},1)-H_{p}(1;s)\leqslant {\it\epsilon}_{k}$
for some sequences
$\{q_{k}\}_{k=1}^{\infty }$
and
$\{{\it\epsilon}_{k}\}_{k=1}^{\infty }$
; the last of these has positive elements and tends to zero. By combining this and Bernoulli’s equation for
$h$
, we obtain
$$\begin{eqnarray}r=\frac{1}{3}\left[\frac{1+h_{q}^{2}}{h_{p}^{2}}+2h\right]\geqslant \frac{1}{3}\left[\frac{1}{(H_{p}(1,s)+{\it\epsilon}_{k})^{2}}+2h(q_{k},1)\right].\end{eqnarray}$$
By letting
$k\rightarrow \infty$
in this inequality and taking into account relation (2.5) and the definition of
$\mathscr{R}(s)$
, we obtain that
$r\geqslant \mathscr{R}(s)$
, and so
$r\geqslant r_{c}$
.
In order to prove the strict inequality
$r>r_{c}$
, let us assume that problem (1.9) with
$r=r_{c}$
has a non-stream solution
$h$
. To show that this is impossible we use the equality
$$\begin{eqnarray}\displaystyle \left[{\rm\Phi}(1;s)-1\right]\int _{-q_{1}}^{q_{2}}w(q,1;s)\text{d}q & + & \displaystyle \int _{-q_{1}}^{q_{2}}\!\int _{0}^{1}\frac{H_{p}^{3}w_{q}^{2}+(2h_{p}+H_{p})w_{p}^{2}}{2h_{p}^{2}}\text{d}p\text{d}q\nonumber\\ \displaystyle & = & \displaystyle -\int _{0}^{1}\left[\frac{h_{q}}{h_{p}}{\rm\Phi}(\,p;s)\right]_{q=-q_{1}}^{q=q_{2}}\text{d}p.\end{eqnarray}$$
Here,
$w(q,p;s)=h(q,p)-H(\,p;s)$
,
${\rm\Phi}(\,p;s)=\int _{0}^{p}H_{t}^{3}(t;s)\text{d}t$
and
$q_{1},q_{2}>0$
, where
$H(\,p;s)$
is defined by (2.9) with
$s$
such that
$\mathscr{R}(s)=r$
, where
$r$
is the same as in problem (1.9) for
$h$
. The derivation of equality (2.11) is implicitly given by Wheeler (Reference Wheeler2014) in the proof of his Lemma 3.1. Indeed, the expression in his formula (3.1) is similar to the left-hand side of (2.11). On the other hand, the right-hand side of (2.11) corresponds to the last term in Wheeler’s identity (3.3).
Taking
$s=s_{c}$
in (2.11), in which case
${\rm\Phi}(1;s_{c})=1$
in view of formula (2.9) for
$H$
, we reduce (2.11) to
$$\begin{eqnarray}\int _{-q_{1}}^{q_{2}}\!\int _{0}^{1}\frac{H_{p}^{2}w_{q}^{2}+(2h_{p}+H_{p})w_{p}^{2}}{2h_{p}^{2}}\text{d}p\text{d}q=-\int _{0}^{1}\left[\frac{w_{q}}{h_{p}}{\rm\Phi}(\,p;s_{c})\right]_{q=-q_{1}}^{q=q_{2}}\text{d}p.\end{eqnarray}$$
Then the positive integral on the left converges as
$q_{1},q_{2}\rightarrow +\infty$
. Since the integrand is positive, there exist two sequences
$\{q_{1}^{(k)}\}_{k=1}^{\infty }$
and
$\{q_{2}^{(k)}\}_{k=1}^{\infty }$
tending to
$+\infty$
and
$-\infty$
respectively as
$k\rightarrow \infty$
and yielding that
$$\begin{eqnarray}\int _{0}^{1}\frac{w_{q}^{2}\big(q_{j}^{(k)},p;s_{c}\big)}{w_{p}^{2}\big(q_{j}^{(k)},p;s_{c}\big)}\text{d}p\rightarrow 0\quad \text{as}~k\rightarrow \infty ,~j=1,2.\end{eqnarray}$$
This implies the following relations:
$$\begin{eqnarray}\int _{0}^{1}\Bigg|\frac{w_{q}\big(q_{j}^{(k)},p;s_{c}\big)}{h_{p}\big(q_{j}^{(k)},p\big)}\Bigg|{\rm\Phi}(\,p;s_{c})\text{d}p\rightarrow 0\quad \text{as }k\rightarrow \infty ,~j=1,2.\end{eqnarray}$$
Therefore, we have that
$$\begin{eqnarray}\int _{-\infty }^{\infty }\!\int _{0}^{1}\frac{H_{p}^{2}w_{q}^{2}+(2h_{p}+H_{p})w_{p}^{2}}{2h_{p}^{2}}\text{d}p\text{d}q=0,\end{eqnarray}$$
according to which the gradient of
$w$
vanishes a.e., and this is incompatible with the assumption that
$h$
is a non-stream solution.
Let us complete the proof of the proposition. Since there exists
$s$
such that
$H(1;s)=\check{{\it\eta}}$
and consequently
$r\geqslant \mathscr{R}(s)$
, we have that
$d_{-}=d(s_{-})$
(it corresponds to
$r$
) is less than or equal to
$\check{{\it\eta}}$
. Indeed, the latter is equal to
$H(1;s)=d(s)$
which decreases monotonically, whereas
$s_{-}\geqslant s$
because
$r\geqslant \mathscr{R}(s)$
. Assuming that there exists
$q^{\circ }$
such that
$d_{-}=h(q^{\circ },1)$
, we apply Hopf’s lemma which implies that
$h_{p}(q^{\circ },1)-H_{p}(1;s_{-})<0$
, and this is incompatible with (2.10), where
$q=q^{\circ }$
instead of
$q_{k}$
. The obtained contradiction proves assertion
$(a^{\prime })$
.
2.2.2. Proof of assertion
$(b^{\prime })$
Here, we show that the following inequalities hold:
$$\begin{eqnarray}\hat{{\it\eta}}\geqslant d_{+}>\check{{\it\eta}}\end{eqnarray}$$
provided
$r\in (r_{c},r_{0})$
. Moreover, the left inequality is strict when
$\hat{{\it\eta}}$
is attained at some point.
Proof. Assuming that
$\check{{\it\eta}}\geqslant d_{+}$
(this is contrary to the right inequality (2.16)), we apply equality (2.11) with
$s=s_{+}$
. In this case, Proposition 1 implies that the function
$w(q,p;s_{+})=h(q,p)-H(\,p,s_{+})\geqslant 0$
and
${\rm\Phi}(1;s_{+})-1\geqslant 0$
. Thus, the first integral on the left-hand side in (2.11) is non-negative. Therefore, repeating literally the considerations based on (2.11) and used in the proof of assertion
$(a^{\prime })$
for demonstrating that
$r\neq r_{c}$
, one again obtains a contradiction with the assumption that
$h$
is a non-stream solution. Thus, it is shown that the right inequality (2.16) holds.
In order to prove the left inequality we again assume the contrary; that is, that
$\hat{{\it\eta}}<d_{+}$
. This implies that there exists
$H(\,p;s)$
such that
$H(1;s)=\hat{{\it\eta}}$
. Since for this
$H$
the function
$w=h-H$
is less than or equal to zero and its maximum vanishes (indeed, we have that
$h(q,0)-H(q,0)=0$
for all
$q\in \mathbb{R}$
), Proposition 2 yields that
$w(q_{k},1)\rightarrow 0$
as
$k\rightarrow \infty$
and
$w_{p}(q_{k},1)\geqslant -{\it\epsilon}_{k}$
for some sequence
$\{q_{k}\}_{k=1}^{\infty }$
(
${\it\epsilon}_{k}>0$
tends to zero as
$k\rightarrow \infty$
). Using this sequence in Bernoulli’s equation, we obtain that
$r\leqslant \mathscr{R}(s)$
, and so either
$\hat{{\it\eta}}\leqslant d_{-}$
or
$\hat{{\it\eta}}\geqslant d_{+}$
. The last inequality contradicts the assumption made, whereas the former inequality is impossible because
$d_{-}\leqslant \check{{\it\eta}}$
(see the proof of assertion
$(a^{\prime })$
) and
$h$
is a non-stream solution.
If
$\hat{{\it\eta}}$
is attained at some point, then the fact that the left inequality (2.16) is strict is a consequence of Hopf’s lemma.
2.2.3. Proof of assertions
$(c^{\prime })$
and
$(d^{\prime })$
We recall that these assertions follow from conditions (iii) for
${\it\omega}$
.
Proof. Let for
$r\in (r_{c},r_{0})$
problem (1.9) have a non-stream solution
$h\in C^{1,{\it\alpha}}(\bar{S})$
such that
$h_{p}\geqslant {\it\delta}^{\prime }$
on
$\bar{S}$
for some
${\it\delta}^{\prime }>0$
. Since conditions (iii) imply the relations
$s_{0}>0$
and
$H_{p}(1;s)\rightarrow +\infty$
as
$s\rightarrow s_{0}$
, we have that
$h_{p}(q,1)-H_{p}(1;s_{\ast })$
is negative and separated from zero for all
$q\in \mathbb{R}$
and some
$s_{\ast }>s_{0}$
. Then
$h(q,1)-H(1;s_{\ast })\leqslant 0$
for all
$q\in \mathbb{R}$
because otherwise the function
$h(q,p)-H(\,p;s_{\ast })$
violates Proposition 2. Therefore, the last inequality yields that
$H(1;s)=\hat{{\it\eta}}$
for some
$s>s_{0}$
and so
$\hat{{\it\eta}}\leqslant d(s)<d_{0}$
, which proves assertion
$(c^{\prime })$
.
Furthermore, we have that
$$\begin{eqnarray}w(q,p)=h(q,p)-H(\,p;s)\leqslant 0\quad \text{on }\bar{S}\quad \text{and}\quad \sup _{q\in \mathbb{R}}h(q,1)-H(1;s)=0.\end{eqnarray}$$
Since Proposition 2 is applicable to this function, we get a sequence
$\{q_{k}\}_{k=1}^{\infty }$
(generally speaking, other than that in the previous paragraph) such that
$w(q_{k},1)\rightarrow 0$
as
$k\rightarrow \infty$
and
$w_{p}(q_{k},1)\geqslant {\it\epsilon}_{k}>0$
(
${\it\epsilon}_{k}$
again tends to zero as
$k\rightarrow \infty$
). As above, by combining this and Bernoulli’s equation, one arrives at the inequality
$r\leqslant \mathscr{R}(s)$
.
If
$r>r_{0}$
, then we arrive at a contradiction because the inequality
$r\leqslant \mathscr{R}(s)$
implies that
$\hat{{\it\eta}}<\check{{\it\eta}}$
. On the other hand, if
$r_{0}=r$
, then
$r=\mathscr{R}(s)$
, and so either
$s=s_{0}$
or
$\hat{{\it\eta}}=d_{-}$
, both of which lead to a contradiction. Thus, only stream solutions exist for
$r\geqslant r_{0}$
; this proves assertion
$(d^{\prime })$
.
The proof of Theorem 1′ is complete. Since this theorem is equivalent to Theorem 1, the latter is also proved.
2.3. Flows with the vorticity distribution satisfying conditions (ii)
Let the vorticity distribution satisfy conditions (ii), that is,
${\it\omega}(0)<0$
and
${\rm\Omega}({\it\tau})<0$
for
$0<{\it\tau}\leqslant 1$
. Then we have that
$$\begin{eqnarray}s_{0}=0,\quad d_{0}=\int _{0}^{1}\frac{\text{d}{\it\tau}}{\sqrt{-2{\rm\Omega}({\it\tau})}}\quad \text{and}\quad r_{0}=2[d_{0}-{\rm\Omega}(1)]/3.\end{eqnarray}$$
Moreover, a solution of problem (1.5) has a non-vanishing derivative at
$d$
.
First, we examine Stokes waves of small amplitude for
$r\geqslant r_{0}$
. Our aim is to show that the stream function changes sign within the corresponding flow, and so the same is true for the horizontal component of velocity. For this purpose we apply the description of these waves obtained by Kozlov & Kuznetsov (Reference Kozlov and Kuznetsov2014) and based on the following two assumptions:
-
(I) a stream solution
$(u(y),d)$
satisfying problem (1.5) with
$r>r_{c}$
is supposed to be such that
$u^{\prime }(d)\neq 0$
; -
(II) the dispersion equation
${\it\sigma}({\it\tau})=0$
corresponding to a stream solution satisfying assumption (I) has at least one positive root, say
${\it\tau}_{0}$
, such that none of the values
$k{\it\tau}_{0}$
(
$k=1,2,\dots$
) are roots of the dispersion equation. Here, (2.19)and
$$\begin{eqnarray}{\it\sigma}({\it\tau})=u^{\prime }(d){\it\gamma}^{\prime }(d,{\it\tau})-[u^{\prime }(d)]^{-1}+{\it\omega}(1)\end{eqnarray}$$
${\it\gamma}(y,{\it\tau})$
solves the boundary value problem (2.20a−c )
$$\begin{eqnarray}-{\it\gamma}^{\prime \prime }+[{\it\tau}^{2}-{\it\omega}^{\prime }(u)]{\it\gamma}=0\quad \text{on }(0,d),\quad {\it\gamma}(0,{\it\tau})=0,\quad {\it\gamma}(d,{\it\tau})=1,\end{eqnarray}$$
If
${\it\omega}$
belongs to
$C^{2,{\it\alpha}}(\mathbb{R})$
with
${\it\alpha}\in (0,1)$
, then assumptions (I) and (II) guarantee that for all sufficiently small values of the parameter
$t$
there exists a solution to problem (1.1)–(1.4) that has the following representation:
$$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}{\it\psi}(x,y;t)=u\displaystyle \left(\frac{dy}{{\it\eta}(x;t)}\right)+t\cos \frac{{\it\tau}_{0}x}{1+{\it\lambda}(t)}W\left(\frac{dy}{{\it\eta}(x;t)}\right)+O(t),\\ {\it\eta}(x;t)=\displaystyle \frac{d}{1+{\it\lambda}(t)}+t\cos \frac{{\it\tau}_{0}x}{1+{\it\lambda}(t)}+O(t).\end{array}\right\} & & \displaystyle\end{eqnarray}$$
Here,
${\it\lambda}(t)\rightarrow 0$
as
$t\rightarrow 0$
and
$W$
solves the following problem (see formulae (38) and (39) in Kozlov & Kuznetsov (Reference Kozlov and Kuznetsov2014)):
$$\begin{eqnarray}-W^{\prime \prime }+[{\it\tau}^{2}-{\it\omega}^{\prime }(u)]W=d^{-1}[\,yu^{\prime }{\it\tau}^{2}+2{\it\omega}(u)]\quad \text{on }(0,d),\quad W(0)=W(d)=0,\end{eqnarray}$$
and
$W^{\prime }(d)=d^{-1}u^{\prime }(d)-[u^{\prime }(d)]^{-1}$
.
According to Proposition 3.5(ii) in Kozlov & Kuznetsov (Reference Kozlov and Kuznetsov2014), for every
$r>r_{c}$
there exists a unidirectional flow for which
$s>s_{c}$
, and so assumption (II) does not hold for the corresponding stream solution. Moreover, if
$r>r_{0}$
, then all stream solutions, other than the one just mentioned, change sign on
$(0,d)$
and have non-vanishing derivatives on the free surface. If assumption (II) is fulfilled, then the same is true for the stream function (2.21) provided
$|t|$
is small.
Let us consider the remaining case, that is,
$r=r_{0}$
,
$s=0$
and
$u^{\prime }(d)\neq 0$
. To demonstrate that the stream function (2.21) changes sign we have to check that
$W^{\prime }(0)\neq 0$
because
$u(0)=u^{\prime }(0)=0$
. In order to verify the last inequality, we take
$w$
that solves the following problem:
$$\begin{eqnarray}-w^{\prime \prime }+[{\it\tau}^{2}-{\it\omega}^{\prime }(u)]w=0\quad \text{on }(0,d),\quad w(0)=1,\quad w(d)=0.\end{eqnarray}$$
Its existence is guaranteed by Lemma 2.2 in Kozlov & Kuznetsov (Reference Kozlov and Kuznetsov2014). Multiplying (2.22a
) by
$w$
, we integrate the result over
$(0,d)$
and then integrate by parts in the same way as in the proof of the cited lemma. Taking into account that
$u^{\prime }(0)=0$
, this leads to the equality
$W^{\prime }(0)=u^{\prime }(d)w^{\prime }(d)$
, which implies the required inequality.
As in the case when
${\it\omega}$
satisfies conditions (iii) we conjecture that only stream solutions exist when
$r\geqslant r_{0}$
and conditions (ii) hold for
${\it\omega}$
(cf. assertion
$(d^{\prime })$
of Theorem 1′). However, if a solution exists for some
$r>r_{0}$
, then we have the following proposition saying that in this case
$d_{0}$
plays the same role as
$d_{+}$
in assertion
$(b^{\prime })$
of Theorem 1′.
Proposition 3. Let
${\it\omega}$
satisfy conditions (ii). If for some
$r>r_{0}$
problem (1.9) has a solution
$h\in C^{1,{\it\alpha}}(\bar{S})$
, then the inequalities
$\hat{{\it\eta}}\geqslant d_{0}>\check{{\it\eta}}$
hold for it.
Proof. To prove the left inequality we assume the contrary, that is,
$d_{0}>\hat{{\it\eta}}$
. Then for some
$s>s_{0}$
there exists a solution of the form (2.9) such that
$H(1;s)=\hat{{\it\eta}}$
. Applying Proposition 2 to the function
$w=h-H$
in the same way as for proving the inequality
$r\geqslant \mathscr{R}(s)$
in the proof of Theorem 1′, we obtain that
$d_{-}\geqslant \hat{{\it\eta}}$
, but according to assertion
$(a^{\prime })$
of this theorem we have
$\check{{\it\eta}}\geqslant d_{-}$
. Hence,
$h$
is identically equal to
$H$
, which is impossible. Thus, the left inequality is proved.
It was shown in the proof of Theorem 1′ that
$s>s_{0}$
can be found from the equation
$H(1;s)=\check{{\it\eta}}$
. This implies the right inequality, thus completing the proof.
3. Concluding remarks
We have considered the problem describing steady rotational water waves in the case when no counter-currents are present in a flow of finite depth (such flows are referred to as unidirectional). It is shown that for the existence of non-stream solutions the parameter of the problem
$r$
(Bernoulli’s constant/the total head) must be strictly greater than the critical value
$r_{c}$
– the unique minimum of the function
$\mathscr{R}$
(see (1.7) for its definition). Stream solutions describing shear flows with horizontal free surfaces are a kind of trivial solutions like those describing irrotational uniform flows. Thus, the requirement on
$r$
obtained here for waves with vorticity generalises that proved by Kozlov & Kuznetsov (Reference Kozlov and Kuznetsov2009) in the irrotational case when two uniform conjugate flows exist for all
$r>r_{c}$
(see (13) in Kozlov & Kuznetsov (Reference Kozlov and Kuznetsov2010)). Furthermore, another result obtained for irrotational waves in Kozlov & Kuznetsov (Reference Kozlov and Kuznetsov2009) is shown to be also valid for waves with vorticity. It states that the depths of the pair of conjugate shear flows serve as bounds for the supremum and infimum of the wave profile on a unidirectional flow.
It is worth mentioning that other results obtained by Kozlov & Kuznetsov (Reference Kozlov and Kuznetsov2010) for irrotational waves are still awaiting generalisation to the case of waves with vorticity; in particular, the following one. Among all waves corresponding to a given near-critical value of Bernoulli’s constant, the solitary wave has the maximal height at the crest; moreover, the whole family of Stokes waves is parametrised by the wave height at the crest, which attains all values between zero and the height of the solitary wave.
According to a unified theory of conjugate flows developed by Benjamin (Reference Benjamin1971), their important feature (apart of being unidirectional) is that they are transcritical. This means that one flow is supercritical, whereas the other one is subcritical; that is, the depth of the former (latter) is less (greater) than that of the critical flow corresponding to
$r_{c}$
. (For a given value of the rate of flow this relates to the opposite relationship between properly defined values of the flow velocity.) In his paper (see also references cited therein), Benjamin (Reference Benjamin1971) characterised the existence of conjugate flows as a common feature for many hydrodynamic models and emphasised that it ‘is crucial to the understanding of observed wave phenomena’.
In this paper the last statement finds the following confirmation. It occurs that along with
$r_{c}$
another critical value
$r_{0}$
exists in
$(r_{c},+\infty )$
provided the vorticity distribution satisfies either of conditions (ii) and (iii) formulated in § 1.2. It is known that for
$r>r_{0}$
only the supercritical shear flow is unidirectional, whereas all other shear flows corresponding to these values of
$r$
(at least one such flow exists for every
$r>r_{0}$
) have counter-currents (see Kozlov & Kuznetsov Reference Kozlov and Kuznetsov2011). Here, we proved that only stream solutions describe unidirectional flows when
$r\geqslant r_{0}$
and conditions (iii) hold for the vorticity distribution. This, in particular, implies that the presence of two conjugate flows is essential for the existence of solitary waves supported by a supercritical shear flow. In the case when the vorticity distribution satisfies conditions (ii) and
$r\geqslant r_{0}$
we demonstrate that all known flows with waves (these are Stokes waves of small amplitude; see Kozlov & Kuznetsov Reference Kozlov and Kuznetsov2014) have counter-currents. Therefore, it is reasonable to conjecture that there are no unidirectional flows with waves for
$r\geqslant r_{0}$
.
There is another reason to refer to
$r_{0}$
as the critical value of the second kind. Indeed, it is shown in Kozlov & Kuznetsov (Reference Kozlov and Kuznetsov2014) that no steady water waves of small amplitude are supported by a shear flow with a still free surface.
Acknowledgements
The authors are grateful to two anonymous referees for their suggestions aimed at improving the presentation. V.K. and E.L. were supported by the Swedish Research Council (VR). N.K. acknowledges the support from G. S. Magnuson’s Foundation of the Royal Swedish Academy of Sciences and Linköping University.
