2. Preliminaries

As is quite common in the theory of two-dimensional water waves, it is more convenient to work with the streamfunction instead of the velocity field. Here, the streamfunction $\bar \psi$ and the relative velocity field $(\bar u,\bar v)$ are related by

(2.1) \begin{align} \bar u=-\bar \psi _{\bar y}\lt 0,\qquad \bar v=\bar \psi _{\bar x}. \end{align}

It turns out that for our purposes it is most convenient to work with non-dimensional variables as in Keady & Norbury (Reference Keady and Norbury1975), where the mass flux and the gravitational constant are scaled to unity. More precisely, the (dimensional) streamfunction is measured in units of the mass flux $\bar M$ ( $=\bar c\bar d$ for a solitary wave), lengths in units of $\bar M^{2/3}\bar g^{-1/3}$ and velocities in units of $\bar M^{1/3}\bar g^{1/3}$ . That is,

(2.2) \begin{align} \bar \psi =\bar M\psi ,\quad (\bar x,\bar y,\bar \eta ,\bar d)=\bar M^{2/3}\bar g^{-1/3}(x,y,\eta ,d),\quad (\bar u,\bar v,\bar c)=\bar M^{1/3}\bar g^{1/3}(u,v,c), \end{align}

where the variables without bars are non-dimensional. Note that still

(2.3) \begin{align} u=-\psi _y\lt 0,\qquad v=\psi _x. \end{align}

Then, the non-dimensionalised version of (1.1) reads

(2.4a) \begin{align} \Delta \psi &= 0 & \text{in } 0 \lt y \lt \eta (x), \\[-10pt] \nonumber \end{align}

(2.4b) \begin{align} \frac {|\boldsymbol{\nabla }\psi |^2}{2} + y &= r & \text{on } y = \eta (x), \\[-10pt] \nonumber \end{align}

(2.4c) \begin{align} \psi &= 1 & \text{on } y = \eta (x), \\[-10pt] \nonumber \end{align}

(2.4d) \begin{align} \psi &= 0 & \text{on } y = 0. \\[9pt] \nonumber \end{align}

Above, $r$ is referred to as the Bernoulli constant and is a parameter of the problem. In what follows we only require $u=-\psi _y$ to be continuous within the fluid domain up to the boundary, while the surface profile is just continuous. In particular, all our results will be valid for extreme waves.

Throughout the rest of this paper, we shall only work with the system (2.4), but state the results also in terms of the relative fluid velocity $(u,v)$ , which is the physically more relevant quantity. In particular, the reader should always keep the relation (2.3) in mind.

We are concerned with solitary wave solutions of (2.4) that can be characterised by surface profiles $\eta$ satisfying the non-dimensional counterpart of (1.2), that is,

(2.5) \begin{align} \lim _{x\to \pm \infty }\eta (x)=d. \end{align}

This means that

(2.6) \begin{align} \psi \to \frac yd,\quad u\to -\frac 1d,\quad v\to 0,\quad \textrm {for }x\to \pm \infty . \end{align}

In particular,

(2.7) \begin{align} c=\frac 1d. \end{align}

In fact, any solitary wave is a wave of elevation with $\eta (x) \gt d$ , while the symmetric surface profile $\eta$ is strictly monotonically decreasing for $x \gt 0$ (and increasing for $x\lt 0$ ); for more details see Craig & Sternberg (Reference Craig and Sternberg1988) and Kozlov et al. (Reference Kozlov, Lokharu and Wheeler2021). The crest is located at the vertical $x = 0$ , and we denote

(2.8) \begin{align} \hat \eta := \max \eta = \eta (0). \end{align}

Given any Bernoulli constant $r \gt 3/2$ one finds two conjugate streams with depths $d_-(r)$ and $d_+(r)$ that are the unique roots of the equation

(2.9) \begin{align} \frac {1}{2d^2} + d = r. \end{align}

Note that the assumption $r \gt 3/2$ is valid for all water waves; check Keady & Norbury (Reference Keady and Norbury1975), Amick & Toland (Reference Amick and Toland1981) and Kozlov & Kuznetsov (Reference Kozlov and Kuznetsov2009). Thus, any solitary water wave is subject to the bounds

(2.10) \begin{align} d = d_-(r) \lt 1 \lt d_+(r) \lt \hat {\eta } \leq r. \end{align}

See Keady & Norbury (Reference Keady and Norbury1975) and Kozlov & Kuznetsov (Reference Kozlov and Kuznetsov2009) for more details. Finally, we note that the Froude number $ \textit{Fr}$ can be computed as

(2.11) \begin{align} \textit{Fr} = d^{-3/2} \gt 1 \end{align}

in view of (1.6) and (2.7) together with our normalisation $g=1$ .

Another important quantity, introduced by Benjamin & Lighthill (Reference Benjamin and Lighthill1954), is the flow force constant given by

(2.12) \begin{align} S := \int _0^\eta \left (\frac {u^2}{2}-\frac {v^2}{2}+r-y\right )\,{\mathrm{d}} y = \frac {d^2}{2} + \frac {1}{d}; \end{align}

here, the first expression is easily seen to be independent of $x$ , just by means of (2.4), while the second identity follows by taking the limits $x\to \pm \infty$ , using (2.5) and (2.6). As for an upper bound for the Froude number, let us present the following simple argument which involves $S$ and leads to the inequality $ \textit{Fr} \lt \sqrt {2}$ , and is in fact very similar to an argument in Keady & Pritchard (Reference Keady and Pritchard1974).

We apply (2.12) right below the crest (i.e. for $x=0$ ) and use the Cauchy–Schwarz inequality (noting that $u$ is not constant)

(2.13) \begin{align} 1=-\int _0^{\hat \eta }u\,{\mathrm{d}} y\lt \sqrt {\hat \eta }\left (\int _0^{\hat \eta }u^2\,{\mathrm{d}} y\right )^{1/2} \end{align}

to estimate

(2.14) \begin{align} S = r\hat \eta - \frac {\hat \eta ^2}{2} + \frac 12 \int _0^{\hat \eta } u^2\,{\mathrm{d}} y \gt r \hat {\eta } - \frac {\hat {\eta }^2}{2} + \frac {1}{2 \hat {\eta }} =: \mathcal{S}(\hat {\eta }). \end{align}

(The Cauchy–Schwarz inequality asserts that $|\int _I fg|\le (\int _I|f|^2)^{1/2}(\int _I|g|^2)^{1/2}$ for any square-integrable functions $f,g$ on an interval $I\subset {\mathbb{R}}$ . Moreover, equality holds if and only if $f$ and $g$ are linearly dependent. In (2.13), the inequality is applied for $f\equiv1$ , $g=u$ , $I=(0,\hat\eta)$ .)

A direct computation reveals that $\mathcal{S}'(t) = 0$ only for $t = d$ and $t = d_+$ . Thus, by (2.10) and since $\mathcal{S}$ is decreasing for $t \gt d_+$ , we obtain

(2.15) \begin{align} S \gt \mathcal{S}(r) = \frac {r^2}{2} + \frac {1}{2r}. \end{align}

In combination with the identities (2.9) and (2.12), one immediately finds that this inequality is only true if $ \textit{Fr} \lt \sqrt {2}$ . Furthermore, one can plug (2.9), (2.12) and $\hat {\eta } = d + a$ into (2.14) to obtain Starr’s inequality (1.8), and that is completely avoiding any integral identities. Let us also remark that the bound $ \textit{Fr}\lt \sqrt 2$ yields the upper bound $r\lt 4^{1/3}\approx 1.5874$ via (2.9) and (2.11); therefore, throughout this paper it is sufficient to consider

(2.16) \begin{align} r\in \left(\frac 32, 4^{1/3} \right). \end{align}

Our proof of the main theorem is based on improving the Cauchy–Schwarz inequality (2.13) or, equivalently, on estimating the integral

(2.17) \begin{align} \frac 12 \int _0^{\hat {\eta }} \big ( u + \hat {\eta }^{-1} \big )^2 \, {\mathrm{d}} y \end{align}

from below, where $-\hat \eta ^{-1}$ is the average of $u$ .

3. Estimates below the crest

Since the classical works of Starr (Reference Starr1947), or Long (Reference Long1956) and Keady & Pritchard (Reference Keady and Pritchard1974), it has been known that a key ingredient is a careful estimate of $u$ at the crest line. A basic property is the following: by a maximum principle argument for $v=\psi _x$ on $x\gt 0$ , using the monotonicity of the surface profile, we have

(3.1) \begin{align} \psi _{yy}(0,y) = -\psi _{xx}(0,y)\lt 0,\text{ that is, }u_y(0,y)\gt 0\text{ for }y\gt 0. \end{align}

This means that the horizontal velocity is strictly increasing from bottom to top at the crest line. Little, however, does it say how fast this increase is. We shall make this much more precise in the following completely new result.

Lemma 1. Given an arbitrary solitary wave solution, consider the harmonic function

(3.2) \begin{align} f := \left(\psi _y^2-\psi _x^2 \right)\psi _{xy}-2\psi _x\psi _y\psi _{xx}+\frac 35\psi _x = -\big(u^2-v^2 \big)u_x+2u u_y v+\frac 35v. \end{align}

Then, $f$ is negative everywhere in $\varOmega :=\{(x,y)\in {\mathbb{R}}^2:0\lt y\lt \eta (x),-\infty \lt x\lt 0\}$ , which is the left open half of the fluid domain, where $\eta ' \gt 0$ .

We formulate the claim for solitary waves, while it is also true for any Stokes wave solution.

Proof. First, note that $f$ is indeed harmonic, which follows from a straightforward computation. Moreover, $f \lt 0$ along the bottom of $\varOmega$ because $\psi _{xy} \lt 0$ there. At the crest line $f = 0$ , so we assume that there is a global positive maximum, attained somewhere at the surface. Let us compute the normal derivative there. In the following, all quantities are understood to be evaluated at this maximum. Introducing the tangential derivative

(3.3) \begin{align} T:= \partial _x + \eta '\partial _y, \end{align}

we know that

(3.4) \begin{align} \textit{Tf} = 0. \end{align}

Moreover, by (2.4) we have

(3.5) \begin{align} \Delta \psi _x = \Delta \psi _y = 0, \quad T^2\left (\frac {|\boldsymbol{\nabla }\psi |^2}{2}+y\right ) = 0, \quad T^3\psi = 0. \end{align}

These, in total five equations form an inhomogeneous system of linear equations for the third derivatives $\psi _{xxx}$ , $\psi _{xxy}$ , $\psi _{xyy}$ , $\psi _{yyy}$ and $\eta '''$ , with lower-order derivatives as coefficients. Thus, these third derivatives can be uniquely expressed in terms of lower-order derivatives. Similarly, with the same line of reasoning, by

(3.6) \begin{align} & \Delta \psi = 0,\quad T\left (\frac {|\boldsymbol{\nabla }\psi |^2}{2}+y\right ) = 0, \quad T^2\psi = 0, \nonumber \\& \left(\psi _y^2-\psi _x^2 \right)\psi _{xy}-2\psi _x\psi _y\psi _{xx}+\frac 35\psi _x = f, \end{align}

the second-order derivatives $\psi _{xx}$ , $\psi _{xy}$ , $\psi _{yy}$ and $\eta ''$ can be expressed in terms of first-order derivatives and $f$ . Finally, through $T\psi =0$ , $\psi _x$ can be written in terms of $\psi _y$ and $\eta '$ . Altogether, we see that the normal derivative of $f$ can be written in terms of $\psi _y$ , $\eta '$ and $f$ . Omitting the lengthy details of this computation, we arrive at

(3.7) \begin{align} -\eta ' f_x + f_y = A + B f + C f^2, \end{align}

where

(3.8) \begin{align} A &= -\eta ' \frac {3-24\eta^{\prime 2}+57\eta ^{\prime 4}-32\eta ^{\prime 6}+4\eta ^{\prime 8}}{25 \big(1-\eta^{\prime 2} \big)^3\psi _y}, \\[-10pt] \nonumber \end{align}

(3.9) \begin{align} B &= -\frac {\big(1+\eta^{\prime 2}\big)^2 \big(2-5\eta^{\prime 2}-9\eta ^{\prime 4}+4\eta ^{\prime 6}\big)}{20\eta^{\prime 2} \big(1-\eta^{\prime 2}\big)^3\psi _y^2}, \\[-10pt] \nonumber \end{align}

(3.10) \begin{align} C &= -\frac {\big(1+\eta^{\prime 2}\big)^5}{16 \eta {\prime }^3 \big(1-\eta^{\prime 2} \big)^3\psi _y^3}. \\[9pt] \nonumber \end{align}

Now notice

(3.11) \begin{align} |\eta '| \lt 0.60443, \end{align}

which corresponds to the upper bound $31.15^\circ$ for the slope obtained in Amick (Reference Amick1987). Thus, it is easy to see that $A\lt 0$ . Note that the coefficient $3/5$ in the definition of $f$ is the minimal value of this coefficient such that $A\lt 0$ for all possible values of $\eta '$ .

Thus, $A,C \lt 0$ . If $B \leq 0$ , then we immediately arrive at a contradiction with the Hopf lemma. Assume now $B\gt 0$ , which forces

(3.12) \begin{align} \eta ' \gt 0.52729. \end{align}

Then we complete the expression in (3.7) to a full square

(3.13) \begin{align} -\eta ' f_x + f_y = A + D^2 + \big(B f + C f^2-D^2 \big), \end{align}

so that the expression in brackets is a negative full square, and compute

(3.14) \begin{align} A + D^2 = \frac {4-24\eta^{\prime 2}+21\eta ^{\prime 4}+40\eta ^{\prime 6}}{100 \eta '\big(1-\eta ^{\prime 4} \big)\psi _y}. \end{align}

This quantity is strictly negative for $0.49028 \lt \eta ' \lt 0.60605$ , which is satisfied in view of (3.11) and (3.12). Thus, the normal derivative of $f$ is negative, leading to a contradiction. Therefore, $f \lt 0$ everywhere in $\varOmega$ .

Corollary 1. Everywhere below the crest we have

(3.15) \begin{align} \left(\psi _y^2 \psi _{yy}+\frac 35 \psi _y \right)_y = - \left(u^2 u_y+\frac 35 u \right)_y \lt 0. \end{align}

Moreover, at the crest line,

(3.16) \begin{align} -u = \psi _y \le \frac {\sqrt 2\sqrt {80r-27y-53\hat \eta }}{3\sqrt 5} - \frac {\sqrt {2(r-\hat \eta )}}{3}. \end{align}

In particular,

(3.17) \begin{align} u^2 \leq \frac 65 (r-y) \end{align}

at the crest line for an extreme wave.

Proof. First, (3.15) follows directly from the Hopf lemma, since $f = 0$ below the crest, while $0$ is the global maximum of $f$ on the closure of $\varOmega$ .

Let us now consider an extreme wave for which $\psi _y(0,\hat \eta )=0$ . Then, by (3.15), $\psi _y^2 \psi _{yy}+3 \psi _y/5\ge 0$ at the crest line. Dividing by $\psi _y$ and integrating from $y$ to $\hat \eta$ yields (3.17).

Consider now a non-extreme wave. At the crest, we have $\psi _{yy}=\eta ''\psi _y$ in view of $T^2\psi =0$ . Similarly as in the previous proof, $\eta ''$ can be expressed in terms of $\psi _y$ , $\eta '$ , and $f$

(3.18) \begin{align} \eta '' = -\frac {5\big(1+\eta^{\prime 2}\big)^2 f + 8\eta ' \big(1-3\eta^{\prime 2}+\eta ^{\prime 4}\big)\psi _y}{20\eta ' \big(1-\eta^{\prime 2}\big) \psi _y^3}. \end{align}

Since $f/\eta '\le 0$ for $x\ne 0$ by Lemma 1, we get

(3.19) \begin{align} \eta '' \ge -\frac {2}{5\psi _y^2} \end{align}

at the crest. Therefore, at the crest line,

(3.20) \begin{align} \psi _y^2 \psi _{yy}+\frac 35 \psi _y \ge \frac {\psi _y(0,\hat \eta )}{5} \gt 0. \end{align}

Thus, we have a differential inequality of the form

(3.21) \begin{align} h' + ah^{1/3} \ge b, \end{align}

with $h=\psi _y^3(0,\boldsymbol{\cdot })$ , $a=9/5$ and $b=3\psi _y(0,\hat \eta )/5=3\sqrt {2(r-\hat \eta )}/5$ , such that $ah^{1/3}\gt b$ . This inequality can be rearranged and integrated from $y$ to $\hat \eta$ to yield

(3.22) \begin{align} y-\hat \eta \le \left [\frac {3(2b+ah^{1/3})h^{1/3}}{2a^2} + \frac {3b^2\ln (ah^{1/3}-b)}{a^3}\right ]\Bigg |_{h=h(y)}^{h(\hat \eta )}. \end{align}

Therefore,

(3.23) \begin{align} \psi _y^2 + \frac 23 \sqrt {2(r-\hat \eta )}\psi _y &\le \frac 65 (\hat \eta -y) + \frac 29 (r-\hat \eta ) \left (15 + \ln \frac {8(r-\hat \eta )}{\left (3\psi _y-\sqrt {2(r-\hat \eta )}\right )^2}\right ) \nonumber \\[-5pt] &\le \frac 65 (\hat \eta -y) + \frac {10}{3} (r-\hat \eta ), \\[9pt] \nonumber \end{align}

where in the last step we simply used $\psi _y\ge \sqrt {2(r-\hat \eta )}$ . Completing the square, we arrive at (3.16).

Now we are ready to combine the previous results, presenting a proof of the main theorem.

4. A proof of Theorem 1

Throughout the proof, we only estimate quantities below the crest line, that is, all quantities are understood to be evaluated at $x=0$ .

Let us recall that the crucial task is to estimate the integral (2.17) from below. That is, we have to estimate carefully how much $u$ deviates from its average with the help of the findings of § 3. Let us explain the main idea on how to do this: we split the integral (2.17) into three pieces: close to bottom, close to crest and in between. We know that $u$ is strictly increasing with height, so $u+\hat \eta ^{-1}$ is positive sufficiently close to the crest. What ‘sufficiently’ means can be made precise with the help of our new estimate (3.16). Now, this inequality also provides an explicit lower bound for the ‘close to crest’ piece. About the ‘in between’ piece we cannot say much, unfortunately, and therefore, we simply throw it away. Finally, close to the bottom $u+\hat \eta ^{-1}$ is negative, and there the integral of its square can be estimated from below by (minus) its average by means of the Cauchy–Schwarz inequality. But since the integral of $u+\hat \eta ^{-1}$ along the whole crest line vanishes, that average can be translated again to an integral close to the crest, for which we can use the same techniques as before.

Put together, we then get an explicit lower bound for the integral (2.17), with which $S$ can be estimated from below, similarly to (2.14). The resulting inequality can be written only in terms of $S$ , $r$ and $d$ , but not of $\hat \eta$ , after establishing some monotonicity property and therefore being allowed to replace $\hat \eta$ by $r$ . Next, this inequality can be written only in terms of $d$ by means of (2.9) and (2.12), then consisting only of explicit analytic expressions in $d$ . It turns out that the resulting inequality can be viewed as a lower bound for $d$ (in much the same way as we recovered (1.8) in § 2). This lower bound, being the root of a complicated, but explicit analytic function, we cannot state exactly, but rather numerically up to arbitrary precision. Finally, such a lower bound for $d$ immediately provides an upper bound for the Froude number in view of (2.11).

More concretely, the first step is to estimate

(4.1) \begin{align} S & = \frac {1}{2 \hat {\eta }} + r \hat {\eta } - \frac {\hat {\eta }^2}{2} + \frac 12\int _0^{\hat {\eta }} \big(u + \hat {\eta }^{-1} \big)^2 \, {\mathrm{d}} y \nonumber \\& \ge \frac {1}{2 \hat {\eta }} + r \hat {\eta } - \frac {\hat {\eta }^2}{2} + \frac 12\int _0^{y_0} \big(u + \hat {\eta }^{-1}\big)^2 \, {\mathrm{d}} y + \frac 12\int _{y_1}^{\hat {\eta }} \big(u + \hat {\eta }^{-1} \big)^2 \, {\mathrm{d}} y \nonumber \\& =: \frac {1}{2 \hat {\eta }} + r \hat {\eta } - \frac {\hat {\eta }^2}{2} + I_1 + I_2. \end{align}

Here, $y_0 \lt y_1$ are defined from the respective relations

(4.2) \begin{align} u(0,y_0) = -\hat \eta ^{-1}, \quad \frac {\sqrt 2\sqrt {80r-27y_1-53\hat \eta }}{3\sqrt 5} - \frac {\sqrt {2(r-\hat \eta )}}{3} = \hat \eta ^{-1}; \end{align}

recall (3.16) and that $u(0,y)$ is strictly increasing in $y$ . For the first integral we use

(4.3) \begin{align} \int _0^{\hat \eta } \big(u + \hat \eta ^{-1}\big)\, {\mathrm{d}} y = 0 \end{align}

and the Cauchy–Schwarz inequality to find

(4.4) \begin{align} I_1 & \ge \frac {1}{2y_0} \left (\int _0^{y_0} |u + \hat {\eta }^{-1}| \, {\mathrm{d}} y\right )^2 \ge \frac {1}{2y_1} \left (\int _0^{y_0} |u + \hat {\eta }^{-1}| \, {\mathrm{d}} y\right )^2 \nonumber \\ & = \frac {1}{2y_1} \left (\int _{y_0}^{\hat {\eta }} |u + \hat {\eta }^{-1}| \, {\mathrm{d}} y\right )^2 \nonumber \\ &\ge \frac {1}{2y_1} \left (\int _{y_1}^{\hat {\eta }} \left ( \hat {\eta }^{-1} - \frac {\sqrt 2\sqrt {80r-27y-53\hat \eta }}{3\sqrt 5} + \frac {\sqrt {2(r-\hat \eta )}}{3} \right ) \,{\mathrm{d}} y\right )^2 \nonumber \\ & =: \tilde I_1(\hat {\eta },r) \end{align}

with the help of (3.16). The second integral can be estimated directly by means of (3.16) as

(4.5) \begin{align} I_2 \ge \frac {1}{2} \int _{y_1}^{\hat {\eta }} \left (\hat {\eta }^{-1} - \frac {\sqrt 2\sqrt {80r-27y-53\hat \eta }}{3\sqrt 5} + \frac {\sqrt {2(r-\hat \eta )}}{3}\right )^2\, {\mathrm{d}} y =: \tilde I_2(\hat {\eta },r). \end{align}

Both $\tilde I_1$ and $\tilde I_2$ can be computed explicitly in terms of $\hat \eta$ and $r$ , but we omit the resulting complicated formulas. Combining the two results, we conclude

(4.6) \begin{align} S \ge \frac {1}{2 \hat {\eta }} + r \hat {\eta } - \frac {\hat {\eta }^2}{2} + \tilde I_1(\hat {\eta },r) + \tilde I_2(\hat {\eta },r) =: I(\hat {\eta },r). \end{align}

To study $I(\hat{\eta} ,r)$ , it is beneficial to introduce $s=\sqrt {r-\hat \eta }$ and $J(s,r)=I(\hat{\eta},r)-I(r,r)$ because $J$ is analytic in $(s,r)\in [0,1/\sqrt 2)\times (3/2,4^{1/3})=: D$ . Here, recall (2.16) and $r-\hat \eta \lt r-d_+ = (2d_+^2)^{-1} \lt 1/2$ . We find that $J\ge 0$ in $D$ . We have validated this inequality rigorously with interval arithmetic up to an error of $10^{-6}$ which is sufficient to ensure the level of precision for the bounds later on. To convince the reader, we include a plot of $J(s,r)/s$ (we are dividing by $s$ since $J(0,r)=0$ ) in figure 1.

Figure 1.

Plot of $J(s,r)/s$ .

Thus, $I(\hat{\eta},r) \ge I(r,r)$ and hence

(4.7) \begin{align} S\ge I(r,r). \end{align}

This, as mentioned before, can be viewed as an explicit inequality only in terms of $d$ through (2.9) and (2.12), and we find that necessarily

(4.8) \begin{align} d \gt 0.82066 \end{align}

and

(4.9) \begin{align} \textit{Fr} = d^{-3/2} \lt 1.3451. \end{align}

This finishes the proof of Theorem 1.

5. Further novel estimates of the horizontal velocity

While we were previously concerned with lower bounds on $u$ , we shall now also derive a novel upper bound. Combined with Theorem 1, this then implies a novel estimate for the velocity at the bottom below the crest. We will formulate this estimate in dimensional variables, which is most relevant for practical applications.

To this end, let us look at the bottom. A basic property is the following: comparing with a uniform laminar flow with depth $\hat \eta$ , it is clear that

(5.1) \begin{align} \frac 12 u^2(x,0) \ge \frac {1}{2\hat \eta ^2}. \end{align}

This inequality, however, is not very good for near-extreme waves, and we shall improve it in Lemma 2.

There, we make use of the flow force function

(5.2) \begin{align} F:=\int _0^y\left (\frac {u^2}{2}-\frac {v^2}{2}+r-y'\right )\,{\mathrm{d}} y', \end{align}

defined in the fluid domain and satisfying

(5.3) \begin{align} \Delta F = -1 \text{ in } 0\lt y\lt \eta (x);\quad F=S\text{ on } y=\eta (x);\quad F=0\text{ on } y=0. \end{align}

The following lemma follows from maximum principles applied to a carefully chosen quantity based on $F$ .

Lemma 2. For an arbitrary solution, we have

(5.4) \begin{align} \frac 12 u^2(x,0) \ge \frac {S}{r} - \frac {r}{2} \end{align}

everywhere at the bottom.

Proof. We consider the harmonic function

(5.5) \begin{align} h = F - \left (-\frac 12 y^2 + \mu y\right ), \ \ \mu = \frac {S}{\hat {\eta }} + \frac {\hat {\eta }}{2}. \end{align}

Note that $-y^2/2 + \mu y$ is increasing on the interval $[0, \hat {\eta }]$ , since the derivative is zero only for $y = \mu$ but

(5.6) \begin{align} \mu \ge \frac {S}{r} + \frac {r}{2} \gt r; \end{align}

this inequality follows from the definition of $\mu$ and its monotonicity in $\hat {\eta }$ on the interval $[0, \sqrt {2S}]$ , which contains $r$ by (2.15).

Thus, we see that $h$ is non-negative at the surface by (5.3) and the definition of $\mu$ , so it attains its minimum everywhere at the bottom, where $h = 0$ . Now, the weak maximum principle and (5.6) give the desired inequality.

Using Lemma 2, we can estimate from above the velocity at the bottom right below the crest. Indeed, (5.4) expressed in terms of $d$ by means of (2.9) and (2.12) give, together with (2.7) and our novel lower bound (4.8) for $d$ obtained in the proof of Theorem 1

(5.7) \begin{align} -\frac {u(0,0)}{c} \ge d\sqrt {2\left (\frac Sr-\frac r2\right )} \gt 0.53624. \end{align}

Clearly, this inequality then also holds for the ratio $-\bar u(0,0)/\bar c$ of dimensional quantities. In particular,

(5.8) \begin{align} \bar u(0,0)+\bar c \lt 0.46376\, \bar c. \end{align}

Thus, for any solitary wave the velocity at the bottom below the crest is restricted to at most $46.376\,\%$ of the propagation speed.