The Korteweg–de Vries Equation from the Free-Surface Problem

Contents

The exact free-surface problem
Two small parameters, and a choice
Eliminating the depth
Unidirectional reduction
Two checks

This appendix derives Equation (35.81) of Fluid Dynamics from the exact equations of an ideal free-surface layer. The chapter, in the derivation of Phenomenon 35.96, takes the Korteweg–de Vries equation as given, integrates it for a travelling wave, and obtains the profile Equation (35.82) with Russell's speed Equation (35.80); what it owes, and what is supplied here, is the equation itself [Korteweg:1895]. The travelling-wave integration is not repeated below — Equation (35.82) stands where it is.

The derivation is an expansion in two small parameters, and the single most important thing about it is that the relation between them is chosen, not deduced. That choice is stated up front in Two small parameters, and a choice and revisited at Remark A64.1; a reader who takes it for a consequence has misunderstood the whole calculation.

The exact free-surface problem

Take an ideal, incompressible, irrotational layer of undisturbed depth \(h\) over a flat rigid bottom at \(y=0\), with the free surface at \(y=h+\eta(x,t)\) and gravity \(g\) acting downwards. Surface tension is set to zero here and restored in Remark A64.6. By Proposition 35.28 the velocity is \(\vect{v}=\vect{\nabla}\phi\) with

\begin{equation}\tag{A64.1} \pp_{x}^{2}\phi+\pp_{y}^{2}\phi=0 \quad\text{for }0<y<h+\eta\ec \qquad \pp_{y}\phi=0\ \text{at }y=0\ec \end{equation}

the second being impermeability of the bottom. At the surface two conditions hold, both nonlinear and both imposed on a boundary whose position is itself unknown:

\begin{align} \pp_{t}\eta+\pp_{x}\phi\,\pp_{x}\eta&=\pp_{y}\phi &&\text{at }y=h+\eta\ec \tag{A64.2}\\ \pp_{t}\phi +\tfrac{1}{2}\left[\left(\pp_{x}\phi\right)^{2} +\left(\pp_{y}\phi\right)^{2}\right]+g\eta&=0 &&\text{at }y=h+\eta\ep \tag{A64.3} \end{align}

Equation (A64.2) says the surface is material — a particle on it stays on it — and Equation (A64.3) is the unsteady Bernoulli equation Equation (35.21) evaluated at the surface, where the pressure equals the constant atmospheric value, absorbed into \(\phi\).

Two small parameters, and a choice

Let \(a\) be a typical wave amplitude and \(\ell\) a typical horizontal length, and set

\begin{equation}\tag{A64.4} \epsilon:=\frac{a}{h}\ec \qquad \delta^{2}:=\frac{h^{2}}{\ell^{2}}\ec \qquad c_{0}:=\sqrt{gh}\ep \end{equation}

Nondimensionalize by

\begin{equation}\tag{A64.5} x=\ell\tilde{x}\ec\quad y=h\tilde{y}\ec\quad t=\frac{\ell}{c_{0}}\tilde{t}\ec\quad \eta=a\tilde{\eta}\ec\quad \phi=\frac{ga\ell}{c_{0}}\tilde{\phi}\ec \end{equation}

the scale for \(\phi\) being the one that makes Equation (A64.3) balance at leading order. Substituting and dropping tildes, Equations (A64.1), (A64.2) and (A64.3) become exactly

\begin{align} \delta^{2}\pp_{x}^{2}\phi+\pp_{y}^{2}\phi&=0\ec \qquad \pp_{y}\phi=0\ \text{at }y=0\ec \tag{A64.6}\\ \pp_{t}\eta+\epsilon\,\pp_{x}\phi\,\pp_{x}\eta &=\frac{1}{\delta^{2}}\,\pp_{y}\phi &&\text{at }y=1+\epsilon\eta\ec \tag{A64.7}\\ \pp_{t}\phi+\frac{\epsilon}{2}\left[ \left(\pp_{x}\phi\right)^{2} +\frac{1}{\delta^{2}}\left(\pp_{y}\phi\right)^{2}\right]+\eta&=0 &&\text{at }y=1+\epsilon\eta\ep \tag{A64.8} \end{align}

No approximation has yet been made: the three coefficients \(\epsilon\), \(1/\delta^{2}\) and \(\epsilon/\delta^{2}\) are what the substitution produces, using \(ga/c_{0}^{2}=a/h=\epsilon\) and \(\ell^{2}/h^{2}=1/\delta^{2}\).

Remark A64.1 (The distinguished limit is a choice).

Nothing so far relates \(\epsilon\) to \(\delta^{2}\), and different relations give different equations. Taking \(\epsilon\ll\delta^{2}\) and keeping only \(\delta^{2}\) gives the linear dispersive equation whose relation is Equation (A64.21) below; taking \(\delta^{2}\ll\epsilon\) and keeping only \(\epsilon\) gives the nonlinear non-dispersive equation, whose solutions steepen and break in finite time. The Korteweg–de Vries equation is the distinguished limit

\begin{equation}\tag{A64.9} \epsilon=O\!\left(\delta^{2}\right)\ec \qquad\text{equivalently}\qquad \frac{a\ell^{2}}{h^{3}}=O(1)\ec \end{equation}

in which the two effects enter at the same order and can balance. This is the entire physical content of the solitary wave — steepening against spreading, as Phenomenon 35.96 says — and it is a hypothesis about the class of initial data being described, not a consequence of the equations. Data not satisfying Equation (A64.9) are described by one of the other two limits and do not produce solitary waves. The dimensionless group in Equation (A64.9) is the Ursell number; the solitary wave of Equation (35.82) has \(\ell^{2}=4h^{3}/3a\), i.e. Ursell number \(4/3\) exactly, which is the self-consistency of the whole construction.

Both parameters are treated as first order below: terms of order \(\epsilon\) and \(\delta^{2}\) are retained, and terms of order \(\epsilon^{2}\), \(\epsilon\delta^{2}\) and \(\delta^{4}\) are discarded.

Eliminating the depth

Lemma A64.2 (Exact depth expansion).

Every solution of Equation (A64.6) analytic in \(y\) is

\begin{equation}\tag{A64.10} \phi(x,y,t)=\sum_{n=0}^{\infty}\frac{\left(-1\right)^{n} \delta^{2n}y^{2n}}{\left(2n\right)!}\, \pp_{x}^{2n}F(x,t)\ec \qquad F(x,t):=\phi(x,0,t)\ec \end{equation}

which for the retained orders reads \(\phi=F-\tfrac{1}{2}\delta^{2}y^{2}\pp_{x}^{2}F +\tfrac{1}{24}\delta^{4}y^{4}\pp_{x}^{4}F+O(\delta^{6})\). Rests on Equation (A64.6) and Theorem 11.38.

Proof.

Derives Lemma A64.2. Write \(\phi=\sum_{m\ge0}y^{m}\phi_{m}(x,t)/m!\). The bottom condition kills \(\phi_{1}\), and Equation (A64.6) gives the recursion \(\phi_{m+2}=-\delta^{2}\pp_{x}^{2}\phi_{m}\); hence every odd coefficient vanishes and \(\phi_{2n}=\left(-\delta^{2}\right)^{n}\pp_{x}^{2n}\phi_{0}\), which is Equation (A64.10) with \(\phi_{0}=F\).

∎

The lemma is exact and is what makes the problem one-dimensional: the whole vertical structure is slaved to the single function \(F\), and the long-wave parameter \(\delta^{2}\) is precisely the bookkeeping parameter of that slaving. Introduce the horizontal velocity at the bottom,

\begin{equation}\tag{A64.11} u(x,t):=\pp_{x}F\ec \end{equation}

and read off from Equation (A64.10)

\begin{equation}\tag{A64.12} \frac{1}{\delta^{2}}\pp_{y}\phi =-y\,\pp_{x}u+\frac{\delta^{2}y^{3}}{6}\,\pp_{x}^{3}u +O\!\left(\delta^{4}\right)\ec \qquad \pp_{x}\phi=u+O\!\left(\delta^{2}\right)\ec \qquad \pp_{t}\phi=\pp_{t}F-\frac{\delta^{2}y^{2}}{2}\,\pp_{x}\pp_{t}u +O\!\left(\delta^{4}\right)\ep \end{equation}

Proposition A64.3 (The Boussinesq pair).

To first order in \(\epsilon\) and in \(\delta^{2}\), the free-surface conditions Equations (A64.7) and (A64.8) reduce to the closed system

\begin{align} \pp_{t}\eta+\pp_{x}u +\epsilon\,\pp_{x}\left(\eta u\right) -\frac{\delta^{2}}{6}\,\pp_{x}^{3}u&=0\ec \tag{A64.13}\\ \pp_{t}u+\pp_{x}\eta+\epsilon\,u\,\pp_{x}u -\frac{\delta^{2}}{2}\,\pp_{x}^{2}\pp_{t}u&=0\ep \tag{A64.14} \end{align}

Rests on Lemma A64.2 and Equation (A64.7).

Proof.

Derives Proposition A64.3. Evaluate Equation (A64.12) at \(y=1+\epsilon\eta\) and substitute in Equation (A64.7). The right-hand side is

\[ -\left(1+\epsilon\eta\right)\pp_{x}u +\frac{\delta^{2}}{6}\left(1+\epsilon\eta\right)^{3}\pp_{x}^{3}u +O\!\left(\epsilon\delta^{2},\delta^{4}\right) =-\pp_{x}u-\epsilon\eta\,\pp_{x}u +\frac{\delta^{2}}{6}\pp_{x}^{3}u+\cdots\ec \]

while the left-hand side is \(\pp_{t}\eta+\epsilon u\,\pp_{x}\eta\). Collecting and using \(\epsilon u\,\pp_{x}\eta+\epsilon\eta\,\pp_{x}u =\epsilon\,\pp_{x}\left(\eta u\right)\) gives Equation (A64.13).

In Equation (A64.8) the term \(\epsilon\left(\pp_{y}\phi\right)^{2}/2\delta^{2}\) is \(O\left(\epsilon\delta^{2}\right)\) by Equation (A64.12) and is discarded, while \(\epsilon\left(\pp_{x}\phi\right)^{2}/2=\epsilon u^{2}/2+O(\epsilon \delta^{2})\). Evaluating \(\pp_{t}\phi\) at \(y=1+\epsilon\eta\) to the retained order,

\[ \pp_{t}F-\frac{\delta^{2}}{2}\,\pp_{x}\pp_{t}u +\frac{\epsilon}{2}u^{2}+\eta=0\ec \]

and differentiating once in \(x\), with \(\pp_{x}F=u\), gives Equation (A64.14).

∎

Unidirectional reduction

At \(\epsilon=\delta^{2}=0\) the pair Equations (A64.13) and (A64.14) is \(\pp_{t}\eta+\pp_{x}u=0\), \(\pp_{t}u+\pp_{x}\eta=0\), whence \(\pp_{t}^{2}\eta=\pp_{x}^{2}\eta\): two waves, one running each way, with \(u=\eta\) on the right-running one. Restrict to that branch and correct it.

Theorem A64.4 (The Korteweg–de Vries equation).

A right-running solution of Equations (A64.13) and (A64.14), correct to first order in \(\epsilon\) and \(\delta^{2}\), has

\begin{equation}\tag{A64.15} u=\eta-\frac{\epsilon}{4}\eta^{2} +\frac{\delta^{2}}{3}\,\pp_{x}^{2}\eta +O\!\left(\epsilon^{2},\epsilon\delta^{2},\delta^{4}\right)\ec \end{equation}

and \(\eta\) obeys

\begin{equation}\tag{A64.16} \pp_{t}\eta+\pp_{x}\eta +\tfrac{3}{2}\epsilon\,\eta\,\pp_{x}\eta +\tfrac{1}{6}\delta^{2}\,\pp_{x}^{3}\eta=0\ep \end{equation}

In dimensional variables this is

\begin{equation}\tag{A64.17} \pdv{\eta}{t} +c_{0}\left(1+\frac{3\eta}{2h}\right)\pdv{\eta}{x} +\frac{c_{0}h^{2}}{6}\,\frac{\pp^{3}\eta}{\pp x^{3}}=0\ec \qquad c_{0}=\sqrt{gh}\ec \end{equation}

which is Equation (35.81). Rests on Proposition A64.3 and Equation (A64.9).

Proof.

Derives Theorem A64.4. Write \(u=\eta+\epsilon A+\delta^{2}B\) with \(A\) and \(B\) functions of \(\eta\) and its \(x\) derivatives, to be determined. Substituting into Equations (A64.13) and (A64.14) and discarding second-order terms,

\begin{align} \pp_{t}\eta+\pp_{x}\eta +\epsilon\left(\pp_{x}A+2\eta\,\pp_{x}\eta\right) +\delta^{2}\left(\pp_{x}B-\tfrac{1}{6}\pp_{x}^{3}\eta\right)&=0\ec \tag{A64.18}\\ \pp_{t}\eta+\pp_{x}\eta +\epsilon\left(\pp_{t}A+\eta\,\pp_{x}\eta\right) +\delta^{2}\left(\pp_{t}B -\tfrac{1}{2}\pp_{x}^{2}\pp_{t}\eta\right)&=0\ep \tag{A64.19} \end{align}

Inside a term already carrying a factor \(\epsilon\) or \(\delta^{2}\) the leading-order relation \(\pp_{t}=-\pp_{x}\) may be used, since the error is of second order; so \(\pp_{t}A=-\pp_{x}A\), \(\pp_{t}B=-\pp_{x}B\) and \(\pp_{x}^{2}\pp_{t}\eta=-\pp_{x}^{3}\eta\), and Equation (A64.19) becomes

\[ \pp_{t}\eta+\pp_{x}\eta +\epsilon\left(-\pp_{x}A+\eta\,\pp_{x}\eta\right) +\delta^{2}\left(-\pp_{x}B +\tfrac{1}{2}\pp_{x}^{3}\eta\right)=0\ep \]

The two equations must be consistent: a single function \(\eta\) cannot obey two different evolution laws. Subtracting them removes the leading part and leaves

\[ \epsilon\left(2\pp_{x}A+\eta\,\pp_{x}\eta\right) +\delta^{2}\left(2\pp_{x}B-\tfrac{2}{3}\pp_{x}^{3}\eta\right)=0\ec \]

and since \(\epsilon\) and \(\delta^{2}\) are independent parameters each bracket must vanish. Integrating, and taking the constants of integration to be zero so that \(u\) vanishes with \(\eta\),

\begin{equation}\tag{A64.20} A=-\tfrac{1}{4}\eta^{2}\ec \qquad B=\tfrac{1}{3}\pp_{x}^{2}\eta\ec \end{equation}

which is Equation (A64.15). Adding the two equations instead, the terms in \(A\) and \(B\) cancel identically and what survives is

\[ 2\left(\pp_{t}\eta+\pp_{x}\eta\right) +3\epsilon\,\eta\,\pp_{x}\eta +\tfrac{1}{3}\delta^{2}\,\pp_{x}^{3}\eta=0\ec \]

which is Equation (A64.16).

For Equation (A64.17), undo Equation (A64.5): multiplying Equation (A64.16) by \(ac_{0}/\ell\) and using \(\pp_{\tilde{t}}\tilde{\eta} =\left(\ell/ac_{0}\right)\pp_{t}\eta\), \(\pp_{\tilde{x}}\tilde{\eta}=\left(\ell/a\right)\pp_{x}\eta\), \(\epsilon\tilde{\eta}\pp_{\tilde{x}}\tilde{\eta} =\left(\ell/ah\right)\eta\,\pp_{x}\eta\) and \(\delta^{2}\pp_{\tilde{x}}^{3}\tilde{\eta} =\left(h^{2}\ell/a\right)\pp_{x}^{3}\eta\), every occurrence of \(a\) and \(\ell\) cancels and Equation (A64.17) results — as it must, since \(a\) and \(\ell\) were bookkeeping devices and no physical statement may depend on them.

∎

Remark A64.5 (No general perturbation theory was needed).

The step just carried out is what is usually called the removal of secular terms, and it is often presented as an application of the method of multiple scales. Part II carries no such method — its asymptotic material is the WKB expansion of Section 13.6 and the matched expansions named at Definition 13.86 — and none is needed. The reason is visible in the proof: because Equations (A64.13) and (A64.14) are two equations for the same pair of unknowns, consistency alone fixes \(A\) and \(B\), and the freedom that a general theory would have to organize into slow and fast variables has already been used up. Had the reduction been attempted on a single equation the situation would be different, and the apparatus would be owed.

Two checks

The chapter asserts, without demonstration, that the two nontrivial terms of Equation (35.81) are exactly the steepening and the dispersion. Both halves can be verified against results already proved.

Drop the third derivative.

Equation (A64.17) becomes \(\pp_{t}\eta+c_{0}\left(1+3\eta/2h\right)\pp_{x}\eta=0\), a quasilinear first-order equation whose solution is constant along the characteristics \(\dd x/\dd t=c_{0}\left(1+3\eta/2h\right)\). Higher parts of the profile therefore travel faster and overtake lower ones, the front steepens, and after a finite time the solution becomes multivalued: this is the simple-wave steepening of Remark 35.90, which in a compressible gas ends in a shock. Note that the amplitude-dependent speed \(c_{0}\left(1+3\eta/2h\right)\) is consistent with Russell's measured \(c^{2}=g\left(h+a\right)\) only through the factor \(3/2\) that the expansion supplies and dimensional analysis cannot.

Drop the nonlinear term.

Setting \(\eta\propto\ee^{\ii\left(kx-\omega t\right)}\) in Equation (A64.17) without the \(\eta\,\pp_{x}\eta\) term gives \(-\ii\omega+\ii c_{0}k+\left(c_{0}h^{2}/6\right)\left(\ii k\right)^{3} =0\), that is

\begin{equation}\tag{A64.21} \omega=c_{0}k\left(1-\tfrac{1}{6}\left(kh\right)^{2}\right)\ep \end{equation}

Expanding the exact gravity-wave relation Equation (35.78) at \(\gamma=0\) with \(\tanh\left(kh\right)=kh-\tfrac{1}{3}\left(kh\right)^{3}+\cdots\),

\[ \omega^{2}=gk\tanh\left(kh\right) =ghk^{2}\left(1-\tfrac{1}{3}\left(kh\right)^{2}+\cdots\right) \quad\Longrightarrow\quad \omega=c_{0}k\left(1-\tfrac{1}{6}\left(kh\right)^{2}+\cdots\right)\ec \]

which is Equation (A64.21) exactly, coefficient included. The third-derivative term of Equation (35.81) is therefore the first dispersive correction to the shallow-water speed and nothing else, as the chapter says.

Remark A64.6 (What is not in the equation).

Four idealizations were made and each has an observable consequence.

No viscosity. Equation (A64.1) is the ideal-fluid problem, so the solitary wave of Equation (35.82) propagates without loss. Real waves in a channel decay, chiefly through the boundary layer on the bed and walls (Section 35.7.2), and Russell's own observation ended when his wave died away.

No surface tension. Restoring it means replacing the constant surface pressure in Equation (A64.3) by the Young–Laplace jump Equation (35.18), and the expansion then delivers the same equation with the dispersive coefficient \(\left(c_{0}h^{2}/2\right)\left(\tfrac{1}{3}-\tau\right)\), \(\tau:=\gamma/\rho gh^{2}\) the Bond parameter. The check above confirms the coefficient independently: expanding Equation (35.78) with \(\gamma\) retained gives \(\omega=c_{0}k\bigl[1-\tfrac{1}{2}\left(kh\right)^{2} \left(\tfrac{1}{3}-\tau\right)\bigr]\). The sign of the dispersion therefore reverses at \(\tau=1/3\), that is for \(h<\sqrt{3\gamma/\rho g}\), about \(4.7\,\mathrm{mm}\) for clean water with \(\gamma=7.3\times 10^{-2}\,\mathrm{N}/\mathrm{m}\) and \(\rho=998\,\mathrm{kg}/\mathrm{m}^{3}\). In a layer thinner than that there are no elevation solitary waves, and Equation (35.82) does not apply — a limitation invisible in the chapter's statement.

Irrotational and one-directional. The reduction of Unidirectional reduction discards the left-running wave entirely. An initial disturbance at rest splits into two, and Equation (A64.17) describes one of them.

Small amplitude. The expansion is in \(\epsilon=a/h\) with \(\epsilon=O\left(\delta^{2}\right)\), so it is valid only for \(a\ll h\) — Russell's regime, as the chapter notes in the derivation of Phenomenon 35.96. Solitary waves of large amplitude exist but are not described by Equation (35.81): they have a limiting height of about \(0.8h\) and a corner at the crest, neither of which this equation knows about.

Remark A64.7.

The Korteweg–de Vries Equation from the Free-Surface Problem discharges Equation (35.81) in the derivation of Phenomenon 35.96 in Fluid Dynamics, which takes the equation as given and proceeds from it to the profile Equation (35.82) and to Russell's speed Equation (35.80) [Russell:1845]. The reader returning there now knows what the two nontrivial terms are and where each comes from — the first from the \(\epsilon\) ordering of the surface conditions, the second from the \(\delta^{2}\) ordering of the depth expansion — and, more importantly, that their coexistence was imposed by the choice Equation (A64.9) rather than discovered. The reading of Equation (35.82) as a soliton, with the elastic collisions and the infinite hierarchy of conserved quantities that follow, belongs to Nonlinear Dynamics and Chaos; nothing of that is visible in the derivation above, which is why Korteweg and de Vries did not see it [Korteweg:1895].