2. Two-dimensional Dirichlet–Neumann operator

Let $\psi$ be the streamfunction harmonic conjugate of the velocity potential $\phi$ (Milne-Thomson Reference Milne-Thomson2011). These two functions are related by the Cauchy–Riemann relations $\phi _x =\psi _y=u$ and $\phi _y=-\psi _x=v$. Thus, the complex potential $f\stackrel {\textrm def}{=}\phi +\mathrm {i}\psi$ is a holomorphic function of $z\stackrel {\textrm def}{=} x+\mathrm {i} y$, with $z={{z_{s}}}\stackrel {\textrm def}{=} x+\mathrm {i}\eta$ at the free surface and $z={{z_{b}}}\stackrel {\textrm def}{=} x-\mathrm {i} d$ at the bottom. (As general notation, subscripts ‘$s$’ and ‘$b$’ denote quantities written, respectively, at the free surface and at the bottom.) The seabed being impermeable and static, it is a streamline where $\psi ={\psi _{b}}$ is constant. Without loss of generality, we then choose ${\psi _{b}}=0$ for simplicity.

For any complex abscissa $z_0$, the Taylor expansion around $z_0=0$ is (omitting temporal dependences for brevity)

(2.1)\begin{equation} f(z-z_0)=\exp[{-}z_0\partial_z]f(z)\ \stackrel{\textrm def}{=} \sum_{n=0}^{\infty}\,\frac{({-}1)^{n}z_0^{n}}{n!}\,\frac{\partial^{n}f(z)}{\partial z^{n}}. \end{equation}

For instance, taking $z_0=\mathrm {i} h=\mathrm {i}(d+\eta )$, the relation (2.1) written at the free surface becomes

(2.2)\begin{equation} f({{z_{s}}}-\mathrm{i} h)=\exp[-\mathrm{i} h\partial_{{{z_{s}}}}]f({{z_{s}}}), \end{equation}

with the formal operator $\exp [-\mathrm {i} h \partial _{{{z_{s}}}}]\stackrel {\textrm def}{=}\sum _{n=0}^{\infty } (n!)^{-1}(-\mathrm {i} h)^{n}\partial _{{{z_{s}}}}^{n}$ together with $\partial _{{{z_{s}}}} \stackrel {\textrm def}{=}(1+\mathrm {i}\eta _x)^{-1}\partial _x$, $\partial _{{{z_{s}}}}^{2}=(1+\mathrm {i}\eta _x)^{-1} \partial _x(1+\mathrm {i}\eta _x)^{-1}\partial _x$, etc. (Throughout this paper, we use the classical convention that any operator acts on everything it multiplies on its right, unless parentheses enforce otherwise.)

It should be noticed that exponents denote differential compositions, so $h^{n}$ is the $n$th power of the function $h=d+\eta$, while $\partial _{{{z_{s}}}}^{n}$ is the $n$th iteration of the differential operator $\partial _{{{z_{s}}}}$. Therefore, for example, if $h$ is not constant then $h^{2}=h(x)^{2}\neq h(h(x))$, $h^{2}\partial _{{{z_{s}}}}^{2} \neq (h\partial _{{{z_{s}}}})^{2}=h\partial _{{{z_{s}}}}h\partial _{{{z_{s}}}}$, $\exp [-\mathrm {i} h\partial _{{{z_{s}}}}]\neq \sum _{n=0}^{\infty }(n!)^{-1}(-\mathrm {i} h\partial _{ {{z_{s}}}})^{n}$ and the operator inverse of $\exp [-\mathrm {i} h\partial _{{{z_{s}}}}]$ is not $\exp [\mathrm {i} h\partial _{{{z_{s}}}}]$ (but $\exp [\mathrm {i} h\partial _{{{z_{b}}}} ]$ as shown in § 3).

Since ${{z_{s}}}-\mathrm {i} h=x-\mathrm {i} d={{z_{b}}}$ then $f({{z_{s}}}-\mathrm {i} h)=f({{z_{b}}}) ={{\phi _{b}}}$ is real (recall that ${\psi _{b}}=\mathrm {Im}\,{f_{b}}=0$ by definition), while $f({{z_{s}}})={{\phi _{s}}}+\mathrm {i}{{\psi _{s}}}$ is complex. Therefore, the imaginary part of (2.2), i.e.

(2.3)\begin{equation} 0 = \mathrm{Re} \{\exp [-\mathrm{i} h\partial_{{{z_{s}}}}] \} {{\psi_{s}}} + \mathrm{Im} \{\exp [-\mathrm{i} h\partial_{{{z_{s}}}}] \} {{\phi_{s}}}, \end{equation}

yields at once

(2.4)\begin{equation} {{\psi_{s}}} ={-} \left(\mathrm{Re} \{ \exp [-\mathrm{i} h\partial_{{{z_{s}}}}] \}\right)^{{-}1} \mathrm{Im} \{\exp [-\mathrm{i} h\partial_{{{z_{s}}}}] \} {{\phi_{s}}}. \end{equation}

The equation for the free-surface impermeability being $\partial _t\eta =\mathscr {G}{{\phi _{s}}}=-\partial _x{{\psi _{s}}}={{v_{s}}}-{{u_{s}}}\partial _x\eta$, an explicit definition of the DNO is obtained directly from (2.4) as

(2.5)\begin{equation} \mathscr{G} = \partial_x\left(\mathrm{Re} \{ \exp [-\mathrm{i} h \partial_{{{z_{s}}}} ] \}\right)^{{-}1} \mathrm{Im} \{\exp [-\mathrm{i} h\partial_{{{z_{s}}}} ] \}. \end{equation}

Formula (2.5) provides an explicit expression for the DNO, i.e. $\mathscr {G}$ appears only on the left-hand side. It is the main result of this paper, which can be generalised to higher dimensions and for moving bottoms (see below). It is also suitable to derive various approximations, in particular high-order shallow-water approximations without assuming small amplitudes (see § 4.3 below; in fact, this goal was the original motivation for deriving (2.5)).

For applications, it is convenient to introduce an operator $\mathscr {J}$ such that $\mathscr {G}=-\partial _x\mathscr {J}\partial _x$, so

(2.6)\begin{equation} \mathscr{J} ={-}\left(\mathrm{Re}\{\exp[- \mathrm{i} h \partial_{{{z_{s}}}}]\}\right)^{{-}1} \mathrm{Im}\{ \exp[-\mathrm{i} h\partial_{{{z_{s}}}}]\}\partial_x^{{-}1}. \end{equation}

Since $\mathscr {G}$ is a self-adjoint positive-definite operator (Lannes Reference Lannes2013), so is $\mathscr {J}$. Further, it is also convenient to introduce the operators $\mathscr {R}$ and $\mathscr {I}$ defined by

(2.7a,b)\begin{equation} \mathscr{R}\stackrel{\textrm def}{=} \mathrm{Re} \{\exp[-\mathrm{i} h \partial_{{{z_{s}}}}]\}, \quad \mathscr{I}\stackrel{\textrm def}{=}-\mathrm{Im} \{\exp[-\mathrm{i} h \partial_{{{z_{s}}}}]\}\partial_x^{{-}1}, \end{equation}

so $\mathscr {J}=\mathscr {R}^{-1}\mathscr {I}$.

Although explicit, the formulae (2.5) and (2.6) are not quite in closed form since they involve series (via the definition of the exponential operator) and operator inversion. Additional relations, suitable for practical applications, are thus derived below.

3. Auxiliary relations

With different choices of $z$ and $z_0$, the Taylor expansion (2.1) provides various relations of practical interest. Several variants of (2.5) can then be derived, their convenience depending on the problem at hand.

With the choice $z={{z_{b}}}$ and $z_0=-\mathrm {i} h=-\mathrm {i}(d+\eta )$, the relation (2.1) becomes

(3.1)\begin{equation} f({{z_{b}}}+\mathrm{i} h) = f({{z_{s}}}) = \exp [\mathrm{i} h \partial_{{{z_{b}}}}]f({{z_{b}}}), \end{equation}

so a comparison with (2.2) yields at once

(3.2)\begin{equation} \left(\exp[-\mathrm{i} h \partial_{{{z_{s}}}}]\right)^{{-}1} = \exp[\mathrm{i} h\partial_{{{z_{b}}}}] \quad \Longleftrightarrow \quad \left(\exp[\mathrm{i} h\partial_{{{z_{b}}}}]\right)^{{-}1} = \exp[-\mathrm{i} h\partial_{{{z_{s}}}}]. \end{equation}

With this relation, the operator involving $\partial _{{{z_{s}}}}$ in the DNO (2.5) can be replaced by one involving $\partial _{{{z_{b}}}}$. This is somewhat convenient in constant depth because, then, $\partial _{{{z_{b}}}}=\partial _x$. However, two operators then need to be inverted instead of one with (2.5), so further simplifications are desirable.

Taking $z=x$ together with $z_0=-\mathrm {i}\eta$ and $z_0=\mathrm {i} d$, (2.1) yields

(3.3a,b)\begin{equation} f(x+\mathrm{i}\eta) = f({{z_{s}}}) = \exp\!\left[\mathrm{i}\eta\partial_x\right]f(x), \quad f(x-\mathrm{i} d) = f({{z_{b}}}) = \exp\!\left[-\mathrm{i} d\partial_x\right]f(x), \end{equation}

and the elimination of $f(x)$ between these two relations, together with (2.2), yields

(3.4)\begin{equation} \exp[-\mathrm{i} h\partial_{{{z_{s}}}}]=\exp\!\left[-\mathrm{i} d\partial_x\right] \left(\exp\!\left[\mathrm{i}\eta \partial_x\right]\right)^{{-}1} =\exp\!\left[-\mathrm{i} d\partial_x\right] \left(\exp\!\left[-\mathrm{i}\eta\partial_x\right]\,\right)^{{\dagger}}\left(1+ \mathrm{i}\eta_x\right), \end{equation}

where a ${\dagger}$ denotes the adjoint operator. (For any complex function $\gamma$ of a single real variable $x$, the operator $\exp [\gamma \partial _x]\stackrel {\textrm def}{=}\sum _{n=0} ^{\infty }(n!)^{-1}\gamma ^{n}\partial _x^{\,n}$ has for Hermitian adjoint $(\exp [\gamma \partial _x])^{{\dagger}}\stackrel {\textrm def}{=}\sum _{n=0}^{\infty }(n!)^{-1}\partial _x^{n} (-\gamma ^{\ast })^{n}$, a star denoting the complex conjugate. We then have $(\exp [\gamma \partial _x])^{-1}=(\exp [\gamma ^{\ast }\partial _x])^{{\dagger}}(1+\gamma _x)$.) We thus have relations that allow us to avoid the computation of the $\partial _z$ operators, moreover without inversions. The operator $\mathscr {R}$ remains to be inverted, however.

For a real or complex function $\gamma$ depending on a single real variable $x$, let the operators and their Hermitian adjoints be

(3.5a,b)\begin{gather} \mathscr{C}_\gamma \stackrel{\textrm def}{=} \sum_{n=0}^{\infty}\frac{({-}1)^{n}}{(2n)!}\,\gamma^{2n} \partial_x^{2n},\quad \mathscr{S}_\gamma \stackrel{\textrm def}{=} \sum_{n=0}^{\infty}\frac{({-}1)^{n}}{(2n+1)!}\, \gamma^{2n+1}\,\partial_x^{2n+1}, \end{gather}

(3.6a,b)\begin{gather}\mathscr{C}_\gamma^{{\dagger}}=\sum_{n=0}^{\infty}\frac{({-}1)^{n}}{(2n)!}\,\partial_x^{2n} {\gamma^{{\ast}}}^{2n},\quad \mathscr{S}_\gamma^{{\dagger}}=\sum_{n=0}^{\infty}\frac{({-}1)^{n+1}}{(2n+1)!}\, \partial_x^{2n+1}\,{\gamma^{{\ast}}}^{2n+1}. \end{gather}

We then have

(3.7a,b)\begin{equation} \exp\!\left[-\mathrm{i} d\partial_x\right]=\mathscr{C}_d - \mathrm{i}\,\mathscr{S}_d, \quad \left(\exp\!\left[-\mathrm{i}\eta\,\partial_x\right]\,\right)^{{\dagger}}= \mathscr{C}_\eta^{{\dagger}}+\mathrm{i}\,\mathscr{S}_\eta^{{\dagger}}, \end{equation}

and the relation (3.4) is split into real and imaginary parts as

(3.8)\begin{gather} \mathrm{Re}\{\exp[-\mathrm{i} h\partial_{{{z_{s}}}}]\}= \mathscr{C}_d\,\mathscr{C}_\eta^{{\dagger}}+\mathscr{S}_d\,\mathscr{S}_\eta^{{\dagger}}- \mathscr{C}_d\,\mathscr{S}_\eta^{{\dagger}}\eta_x + \mathscr{S}_d\,\mathscr{C}_\eta^{{\dagger}} \eta_x, \end{gather}

(3.9)\begin{gather}\mathrm{Im}\{\exp[-\mathrm{i} h\partial_{{{z_{s}}}}]\}= \mathscr{C}_d\,\mathscr{S}_\eta^{{\dagger}}-\mathscr{S}_d\,\mathscr{C}_\eta^{{\dagger}}+ \mathscr{C}_d\,\mathscr{C}_\eta^{{\dagger}}\eta_x + \mathscr{S}_d\,\mathscr{S}_\eta^{{\dagger}} \eta_x. \end{gather}

With the operator relation $\eta _x=\partial _x\eta -\eta \partial _x$ (resulting from the Leibniz rule), we have

(3.10a,b)\begin{equation} \mathscr{C}_\eta^{{\dagger}}\eta_x=\partial_x^{{-}1}\mathscr{S}_\eta^{{\dagger}}\partial_x - \mathscr{S}_\eta^{{\dagger}}, \quad \mathscr{S}_\eta^{{\dagger}}\eta_x= \mathscr{C}_\eta^{{\dagger}} -\partial_x^{{-}1}\,\mathscr{C}_\eta^{{\dagger}}\partial_x, \end{equation}

so the relations (3.8) and (3.9) yield

(3.11a,b)\begin{equation} \mathscr{R}=\mathscr{C}_d\partial_x^{{-}1}\,\mathscr{C}_\eta^{{\dagger}}\partial_x + \mathscr{S}_d\partial_x^{{-}1}\,\mathscr{S}_\eta^{{\dagger}}\partial_x,\quad \mathscr{I}=\mathscr{S}_d\partial_x^{{-}1}\,\mathscr{C}_\eta^{{\dagger}}- \mathscr{C}_d\partial_x^{{-}1}\,\mathscr{S}_\eta^{{\dagger}}. \end{equation}

The latter relations are particularly convenient to derive analytic approximations and to extrapolate the DNO to higher dimensions, as shown below.

4. Approximate Dirichlet–Neumann operators

From the explicit DNO (2.5) and the relations derived in the previous section, several approximations of practical interest can be easily obtained. We consider here only two special cases.

4.1. Infinitesimal waves in arbitrary depth

Assuming that the free surface $\eta$ remains close to zero, one can formally expand the DNO in increasing order of nonlinearities in $\eta$ (Craig & Sulem Reference Craig and Sulem1993; Craig et al. Reference Craig, Guyenne, Nicholls and Sulem2005). Thus, writing $\mathscr {G}=\mathscr {G}_0+\mathscr {G}_1+\mathscr {G}_2+\cdots$ and similarly for $\mathscr {R}$, $\mathscr {I}$ and $\mathscr {J}$, one obtains at once from (3.11) the following:

(4.1a–c)\begin{gather} \mathscr{R}_0 = \mathscr{C}_d, \quad \mathscr{R}_1 ={-}\mathscr{S}_d\eta\partial_x, \quad \mathscr{R}_2 ={-}{{\textstyle{1\over 2}}}\,\mathscr{C}_d\partial_x\eta^{2}\partial_x, \quad \mathrm{etc.}, \end{gather}

(4.2a–c)\begin{gather}\mathscr{I}_0 = \mathscr{S}_d\partial_x^{{-}1}, \quad \mathscr{I}_1 = \mathscr{C}_d\eta, \quad \mathscr{I}_2 ={-}{{\textstyle{1\over 2}}}\,\mathscr{S}_d\partial_x\eta^{2}, \quad \mathrm{etc}. \end{gather}

The relation

(4.3)\begin{align} \mathscr{R}^{{-}1}&=[\mathscr{R}_0+\mathscr{R}_1+\mathscr{R}_2+\cdots]^{{-}1} =[1+\mathscr{R}_0^{{-}1}\,\mathscr{R}_1+\mathscr{R}_0^{{-}1}\,\mathscr{R}_2+ \cdots]^{{-}1}\mathscr{R}_0^{{-}1}\nonumber\\ &=[1-\mathscr{R}_0^{{-}1}\,\mathscr{R}_1- \mathscr{R}_0^{{-}1}\,\mathscr{R}_2+\mathscr{R}_0^{{-}1}\,\mathscr{R}_1\, \mathscr{R}_0^{{-}1}\,\mathscr{R}_1+\cdots]\mathscr{R}_0^{{-}1} \end{align}

then yields, after some algebra,

(4.4)\begin{equation} \left.\begin{gathered} \mathscr{J}_0=\mathscr{C}_d^{{-}1}\,\mathscr{S}_d\,\partial_x^{{-}1}, \quad \mathscr{J}_1 = \eta + \mathscr{J}_0\partial_x\eta\partial_x\mathscr{J}_0, \\ \mathscr{J}_2={{\textstyle{1\over 2}}}\,\partial_x\eta^{2}\partial_x\mathscr{J}_0 + \mathscr{J}_0\partial_x\eta\partial_x\,\mathscr{J}_1-{{\textstyle{1\over 2}}}\,\mathscr{J}_0 \partial_x^{2}\eta^{2}, \quad \mathrm{etc.}, \end{gathered}\right\} \end{equation}

hence

(4.5)\begin{equation} \left.\begin{gathered} \mathscr{G}_0={-}\partial_x\,\mathscr{C}_d^{{-}1}\,\mathscr{S}_d, \quad \mathscr{G}_1={-}\partial_x\eta\partial_x-\mathscr{G}_0\eta\,\mathscr{G}_0, \\ \mathscr{G}_2={{\textstyle{1\over 2}}}\,\partial_x^{2}\eta^{2}\,\mathscr{G}_0-\mathscr{G}_0\eta\,\mathscr{G}_1 -{{\textstyle{1\over 2}}}\,\mathscr{G}_0\partial_x\eta^{2}\partial_x, \quad\mathrm{etc}. \end{gathered}\right\}\end{equation}

In constant depth, the expansion of Craig & Sulem (Reference Craig and Sulem1993) is, as expected, recovered by introducing the operator $\mathscr {D}\stackrel {\textrm def}{=}\mathrm {i}\partial _x$, i.e. replacing $\partial _x$ by $-\mathrm {i}\mathscr {D}$. With a variable bottom, the expansion of Craig et al. (Reference Craig, Guyenne, Nicholls and Sulem2005) is also recovered, except for the definition of $\mathscr {G}_0$. Indeed, Craig et al. (Reference Craig, Guyenne, Nicholls and Sulem2005) define $\mathscr {G}_0$ with an expansion for small amplitudes of the bottom corrugation (i.e. ${\max }|d(x)-\bar {d}|$ is small, where $\bar {d}$ is the mean depth), and they provide a recursion formula for computing this series. In (4.5), $\mathscr {G}_0$ is defined explicitly for arbitrary (non-overturning) bottom and no additional expansions are required.

4.3. Long waves in shallow water

For long waves in shallow water, the characteristic wavelength $L_c$ is much larger than the characteristic depth $d_c$, so $\sigma \stackrel {\textrm def}{=} d_c/L_c\ll 1$ is a ‘shallowness’ dimensionless small parameter. The horizontal derivative $\partial _x$ is then of first order in shallowness and the DNO can be expanded is power series of $\sigma$, without assuming small amplitude for the waves and/or for the bottom corrugation. Thus, we do not need to explicitly introduce scalings to assess the order of terms – it is sufficient to count the number of derivatives. For instance, $\partial _x^{3} \eta$, $(\partial _x^{2}\eta )(\partial _x\eta )$ and $(\partial _x\eta )^{3}$ are all of third order in shallowness, as well as $\partial _x^{3}d$, $(\partial _x^{2} d)(\partial _xd)$ and $(\partial _xd)^{3}$.

We then have the shallow-water even-terms expansions $\mathscr {R}=\mathscr {R}_0+\mathscr {R}_2 +\mathscr {R}_4+\cdots$ (and similarly for $\mathscr {I}$ and $\mathscr {J}$), so, from (3.11),

(4.6a,b)\begin{gather} \mathscr{R}_0=1,\quad \mathscr{R}_2={-}{{\textstyle{1\over 2}}}\,d^{2}\partial_x^{2}- d\partial_x\eta\partial_x-{{\textstyle{1\over 2}}}\,\partial_x\eta^{2}\partial_x,\quad \mathrm{etc.}, \end{gather}

(4.7a,b)\begin{gather}\mathscr{I}_0=h,\quad \mathscr{I}_2={-}{{\textstyle{1\over 6}}}\,d^{3}\partial_x^{2}- {{\textstyle{1\over 2}}}\,d^{2}\partial_x^{2}\eta-{{\textstyle{1\over 2}}}\,d\partial_x^{2}\eta^{2}- {{\textstyle{1\over 6}}}\,\partial_x^{2}\eta^{3},\quad \mathrm{etc.}, \end{gather}

hence, after some algebra,

(4.8a,b)\begin{equation} \mathscr{J}_0=h,\quad \mathscr{J}_2={{\textstyle{1\over 2}}}\,h^{2}d_{xx}+ hh_xd_x-hd_x^{2}+{{\textstyle{1\over 3}}}\,\partial_xh^{3}\partial_x, \quad \mathrm{etc.} \end{equation}

Note that $\mathscr {J}_0$ and $\mathscr {J}_2$ are obviously self-adjoint, as they should be.

It should be emphasised that these approximations were obtained directly from the explicit DNO, considering weak variations in $x$ (i.e. long waves in shallow water) but without assuming small amplitudes of the free surface and of the seabed (i.e. there are no restrictions on the magnitude of $|\eta |$ and $|d(x)-\bar {d}|$, $\bar {d}$ being the mean depth).

5. Dirichlet–Neumann operator in higher dimensions

It is rather straightforward to extrapolate the DNO given by (2.5) to three (and more) spatial dimensions. In higher dimensions, the holomorphic functions cannot be used but series representations remain. This feature is exploited here to obtain an explicit expression for the DNO in an arbitrary number of dimensions.

With $\boldsymbol {x}=(x_1,x_2,\ldots,x_N)\in \mathbb {R}^{N}$ referring to the ‘horizontal’ coordinates, the mathematical problem is then posed in the $(N+1)$-dimensional Cartesian $(\boldsymbol {x},y)$-space, with $y$ the ‘upward vertical’ coordinate. Obviously, only the two-dimensional (i.e. $N=1$) and three-dimensional (i.e. $N=2$) cases are of physical interest for water waves. Let $\boldsymbol {\nabla }\stackrel {\textrm def}{=}(\partial _{x_1},\ldots,\partial _{x_N})$, $\varDelta \stackrel {\textrm def}{=} \boldsymbol {\nabla }\boldsymbol {\cdot }\boldsymbol {\nabla }$ and $\mathscr {D}\stackrel {\textrm def}{=}(-\varDelta )^{1/2}$ denote, respectively, the horizontal gradient, Laplacian and semi-Laplacian operators.

The DNO is naturally extended in higher dimensions by extrapolating the relation $\mathscr {G}=-\partial _x\mathscr {R}^{-1}\mathscr {I}\partial _x$, the operators $\mathscr {R}$ and $\mathscr {I}$ having to be redefined. In the two-dimensional case, these operators are defined via complex expressions in § 2. In order to extend these operators to higher dimensions, one must consider their real form (3.11), so their extrapolation is natural.

One-dimensional operators involving only even-order derivatives have straightforward extensions in higher dimensions by replacing the second-order horizontal derivative $\partial _x^{2}$ by the horizontal Laplacian $\varDelta$. For instance,

(5.1)\begin{gather} \mathscr{C}_d \mapsto \sum_{n=0}^{\infty}\frac{({-}1)^{n}}{(2n)!}\,d^{2n} \varDelta^{n} \stackrel{\textrm def}{=} \cosh (d\,\mathscr{D}), \end{gather}

(5.2)\begin{gather}\mathscr{C}_\eta^{{\dagger}} \mapsto \sum_{n=0}^{\infty}\frac{({-}1)^{n}}{(2n)!}\, \varDelta^{n}\eta^{2n}\stackrel{\textrm def}{=} \cosh(\mathscr{D}\,\eta), \end{gather}

(5.3)\begin{gather}\mathscr{S}_d\partial_x^{{-}1}\mapsto \sum_{n=0}^{\infty}\frac{({-}1)^{n}}{(2n+1)!} \,d^{2n+1}\varDelta^{n} \stackrel{\textrm def}{=} \sinh(d\,\mathscr{D})\mathscr{D}^{{-}1}, \end{gather}

(5.4)\begin{gather}\partial_x^{{-}1}\mathscr{S}_\eta^{{\dagger}} \mapsto \sum_{n=0}^{\infty}\frac{({-}1)^{n+1}}{(2n+1)!}\, \varDelta^{n}\eta^{2n+1} \stackrel{\textrm def}{=} -\mathscr{D}^{{-}1}\sinh(\mathscr{D}\,\eta). \end{gather}

It should be emphasised that, as in the one-dimensional case, the operators do not commute, so, for example, $\cosh (d\mathscr {D})\neq \cosh ( \mathscr {D}d)$ and $\cosh (d\mathscr {D})^{-1}\neq \operatorname {sech}(d\mathscr {D})$, the equalities holding only in constant depth because then $d\mathscr {D}=\mathscr {D}d$.

A natural extension of $\mathscr {I}\partial _x$ is thus $\mathscr {I}\boldsymbol {\nabla }$ with

(5.5)\begin{equation} \mathscr{I} \mapsto \sinh(d\,\mathscr{D})\mathscr{D}^{{-}1} \cosh(\mathscr{D}\,\eta)+\cosh(d\,\mathscr{D}) \mathscr{D}^{{-}1}\sinh(\mathscr{D}\,\eta). \end{equation}

In order to find the extension of $\partial _x\mathscr {R}^{-1}$, the operator $\mathscr {R}$ given by (3.11a) is rewritten as

(5.6)\begin{equation} \mathscr{R}=\mathscr{C}_d\partial_x{}^{{-}1}[\mathscr{C}_\eta^{{\dagger}} +\partial_x\,\mathscr{C}_d^{{-}1}(\mathscr{S}_d\,\partial_x^{{-}1}) \partial_x (\partial_x^{{-}1}\,\mathscr{S}_\eta^{{\dagger}})]\partial_x. \end{equation}

Thus, we have the natural extension

(5.7)\begin{equation} \partial_x\,\mathscr{R}^{{-}1}\mapsto[\cosh(\mathscr{D}\eta) +\mathscr{G}_0\,\mathscr{D}^{{-}1}\sinh(\mathscr{D}\eta)]^{{-}1} \boldsymbol{\nabla}\boldsymbol{\cdot}\cosh(d\mathscr{D})^{{-}1}, \end{equation}

where $\cosh (d\mathscr {D})^{-1}$ is the inverse operator of $\cosh (d\mathscr {D})$, and

(5.8)\begin{equation} \mathscr{G}_0 \stackrel{\textrm def}{=} -\boldsymbol{\nabla}\boldsymbol{\cdot}\cosh(d\mathscr{D})^{{-}1}\sinh (d\mathscr{D})\mathscr{D}^{{-}1}\,\boldsymbol{\nabla}.\end{equation}

Therefore, the DNO becomes at once

(5.9)\begin{equation} \mathscr{G}={-}[\cosh(\mathscr{D}\eta)+\mathscr{G}_0 \,\mathscr{D}^{{-}1}\sinh(\mathscr{D}\eta)]^{{-}1} \boldsymbol{\nabla}\boldsymbol{\cdot}\cosh(d\mathscr{D})^{{-}1}\mathscr{I}\,\boldsymbol{\nabla}. \end{equation}

In order to avoid misinterpretations of the formula (5.9), it is worth re-emphasising here that: (i) any operator acts on everything it multiplies on its right, so (5.9) should be applied successively leftward starting from the furthest right; and (ii) exponents denote operator compositions, so an exponent $-1$ means an operator inversion.

Note that $\mathscr {G}\to \mathscr {G}_0$ as $\eta \to 0$. Moreover, processing as in § 4.1 for infinitesimal waves, one finds the expansion of Craig et al. (Reference Craig, Guyenne, Nicholls and Sulem2005), except for $\mathscr {G}_0$ that is defined implicitly by Craig et al. (Reference Craig, Guyenne, Nicholls and Sulem2005) but explicitly here.

5.1. Constant depth

In constant depth, $d$ commuting then with both $\mathscr {D}$ and $\boldsymbol {\nabla }$, we have the simplified relation $\mathscr {G}_0=\mathscr {D}\tanh (d \mathscr {D})$, while $\mathscr {I}$ and $\mathscr {G}$ become (see Appendix A for details)

(5.10)\begin{align} \mathscr{I}&=\mathscr{D}{}^{{-}1}[\sinh(d\mathscr{D}) \cosh(\mathscr{D}\eta)+\cosh(d\mathscr{D})\sinh(\mathscr{D}\eta)] =\mathscr{D}^{{-}1}\sinh(\mathscr{D}h),\end{align}

(5.11)\begin{align} \mathscr{G}&={-}[\cosh(d\mathscr{D})\cosh(\mathscr{D}\eta) +\sinh(d\mathscr{D})\sinh(\mathscr{D}\eta)]^{{-}1} \boldsymbol{\nabla}\boldsymbol{\cdot}\mathscr{I}\,\boldsymbol{\nabla}\nonumber\\ &={-}\cosh(\mathscr{D}h)^{{-}1}\,\mathscr{D}^{{-}1}\,\boldsymbol{\nabla}\boldsymbol{\cdot} \sinh(\mathscr{D}h)\boldsymbol{\nabla}. \end{align}

A better-conditioned formulation, avoiding the computation of $\mathscr {D}$, is

(5.12)\begin{equation} \mathscr{G}={-}[\operatorname{sech}(d\mathscr{D}) \cosh(\mathscr{D}h)]^{{-}1}\,\boldsymbol{\nabla}\boldsymbol{\cdot}[ \operatorname{sech}(d\mathscr{D})\operatorname{sinhc} (\mathscr{D}h)h]\boldsymbol{\nabla}, \end{equation}

with

(5.13a,b)\begin{equation} \operatorname{sech}(d\mathscr{D})\stackrel{\textrm def}{=}\sum_{n=0}^{\infty} \frac{E_{2n}}{(2n)!}\,d^{2n}\,\mathscr{D}^{2n}, \quad \operatorname{sinhc}(\mathscr{D}h)\stackrel{\textrm def}{=}\sum_{n=0}^{\infty} \frac{1}{(2n+1)!}\,\mathscr{D}^{2n}h^{2n}, \end{equation}

where $E_n$ are the Euler numbers (Abramowitz & Stegun Reference Abramowitz and Stegun1965) (since $d$ and $\mathscr {D}$ commute, we have $\operatorname {sech}(d\mathscr {D})=\cosh (d \mathscr {D})^{-1}$). The relation (5.12) involving only even powers of $\mathscr {D}$, only Laplacian and gradient operators need to be evaluated, i.e. the computation of the non-local operator $\mathscr {D}$ can be avoided.

As the DNO appears in Hamiltonian formulations of water waves, its functional variations are crucial to derive the equations of motion and to investigate stability (Fazioli & Nicholls Reference Fazioli and Nicholls2010). Thanks to the explicit DNO (5.11), these variations can be obtained quite effortlessly. Indeed, with the relations (cf. Appendix A)

(5.14)\begin{gather} \cosh(\mathscr{D}(h+\delta h))= \cosh(\mathscr{D}h)+ \mathscr{D}\sinh(\mathscr{D} h)\delta h + O(\delta h^{2}), \end{gather}

(5.15)\begin{gather}\sinh(\mathscr{D}(h+\delta h))= \sinh(\mathscr{D}h)+\mathscr{D}\cosh(\mathscr{D} h)\delta h+O(\delta h^{2}), \end{gather}

the first variation of the DNO is obtained at once as

(5.16)\begin{align} \mathscr{G}(h+\delta h)&=\mathscr{G}(h)-\cosh(\mathscr{D}h )^{{-}1}\,\boldsymbol{\nabla}\boldsymbol{\cdot}\cosh(\mathscr{D}h)\delta h\,\boldsymbol{\nabla}\nonumber\\ &\quad+\cosh(\mathscr{D}h)^{{-}1}\mathscr{D}\sinh(\mathscr{D} h)\delta h\,\mathscr{G}(h)+O(\delta h^{2}), \end{align}

or

(5.17)\begin{align} \mathscr{G}(h+\delta h)&=\mathscr{G}(h) -\cosh(\mathscr{D}h )^{{-}1}\,\boldsymbol{\nabla}\boldsymbol{\cdot}\cosh(\mathscr{D} h)\delta h\boldsymbol{\nabla} + \mathscr{G}(h)\delta h\,\mathscr{G}(h)\nonumber\\ &\quad- \cosh(\mathscr{D}h)^{{-}1}\boldsymbol{\nabla}\boldsymbol{\cdot} \cosh(\mathscr{D}h)(\boldsymbol{\nabla} h)\delta h\,\mathscr{G}(h) + O(\delta h^{2}). \end{align}

Similarly, higher-order functional variations of $\mathscr {G}$ can be easily obtained. This is one illustration of the advantage of dealing with an explicit DNO.

6. Moving bottom

We consider finally the generalisation of a moving bottom, i.e. $d=d(x,t)$. Of course, for simplicity, we begin with the two-dimensional case, the generalisation to higher dimensions being straightforward.

When $\partial _td\neq 0$, the bottom is no longer a streamline, so the streamfunction is not zero at the seabed, i.e. ${\psi _{b}}={\psi _{b}}(x,t)\neq 0$. The lower boundary condition (1.2) becomes $\partial _td=\partial _x{{\psi _{b}}}=-{{v_{b}}}- {{u_{b}}}\partial _xd$. With a moving bottom, the relations (2.1), (2.2) and (3.1) still hold, but (2.3) becomes

(6.1)\begin{equation} {{\psi_{b}}} = \mathrm{Re}\{\exp[-\mathrm{i} h\partial_{{{z_{s}}}}]\}{{\psi_{s}}} + \mathrm{Im}\{\exp[-\mathrm{i} h\partial_{{{z_{s}}}}]\}{{\phi_{s}}}. \end{equation}

The condition for the bottom impermeability yielding ${{\psi _{b}}}=\partial _x^{-1} \partial _td$, the relation (6.1) gives

(6.2)\begin{equation} {{\psi_{s}}}=\mathrm{Re}\{\exp[-\mathrm{i} h\partial_{{{z_{s}}}}] \}{}^{{-}1}(\partial_x^{{-}1}\partial_td-\mathrm{Im}\{\exp[-\mathrm{i} h\partial_{{{z_{s}}}}]\}{{\phi_{s}}}). \end{equation}

The relation (6.2) shows that the Dirichlet-to-Neumann transformation at the free surface is no longer a homogeneous linear function of ${{\phi _{s}}}$. The impermeability of the free surface is then $\partial _t\eta =G({{\phi _{s}}})$, the generalised DNO $G$ being

(6.3)\begin{equation} G\!\left({{\phi_{s}}}\right) = \mathscr{G}\,{{\phi_{s}}} - \partial_x\, \mathrm{Re}\{\exp[-\mathrm{i} h\,\partial_{{{z_{s}}}}]\}^{{-}1}\partial_x^{{-}1}\,\partial_td, \end{equation}

where $\mathscr {G}$ is given by (2.5). Note that $\partial _x^{-1}\partial _td$ is not uniquely defined due to the antiderivative, uniqueness being enforced by the definitions of the mean water level and of the frame of reference.

In higher dimensions, with $\partial _x^{-1}=\partial _x\partial _x^{-2}\mapsto \boldsymbol {\nabla }\varDelta ^{-1}=-\mathscr {D}^{-2}\boldsymbol {\nabla }$, the DNO obviously becomes

(6.4)\begin{equation} G\!\left({{\phi_{s}}}\right) =[\cosh(\mathscr{D}\,\eta)+\mathscr{G}_0 \,\mathscr{D}^{{-}1}\sinh(\mathscr{D}\,\eta)]^{{-}1} \boldsymbol{\nabla}\boldsymbol{\cdot}\cosh(d\,\mathscr{D})^{{-}1}(\mathscr{D}^{{-}2}\boldsymbol{\nabla} \partial_td-\mathscr{I}\boldsymbol{\nabla}{{\phi_{s}}}), \end{equation}

with $\mathscr {I}$ and $\mathscr {G}_0$ being defined, respectively, by (5.5) and (5.8).