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Water Waves

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Initial Stage of the Finite-Amplitude Cauchy–Poisson Problem

  • Peder A. TyvandEmail author
Original Article
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Abstract

The nonlinear Cauchy–Poisson problem for an incompressible inviscid fluid to start flowing under gravity is investigated analytically. The general nonlinear initial/boundary-value problem is formulated, including both an initial surface deflection and an initial velocity generated by a pressure impulse on the surface. Two subproblems are: (1) a finite-amplitude surface deflection released from rest; (2) the fluid is forced into motion by a pressure impulse on the initially horizontal surface. Solutions for these two subproblems are given to the leading order. One exact solution is given for the fully nonlinear initial-value problem, where a surface pressure impulse is applied on a surface with finite initial deflection. The concept of the highest non-breaking wave is illustrated by dipole acceleration fields at a state of gravitational release from rest. This is done for two phenomena: run-up of a non-breaking solitary wave on a sloping beach, and free nonlinear sloshing in an open container.

Keywords

Cauchy–Poisson problem Free surface Gravitational flow Initial value problem 

1 Introduction

The Cauchy–Poisson (CP) problem for water waves is classical in fluid mechanics. It is named after Augustin Louis Cauchy (1789–1857) and Siméon Denis Poisson (1781–1840). Their pioneering work on transient waves was documented by Poisson [14] and Cauchy [2]. This linearized theory is now well-developed; see Lamb [8] and Wehausen and Laitone [18].

The basic CP problem is concerned with deep-water waves, since the initial value problem for shallow-water waves can be classified as a different category. The key property of deep-water waves is that their evolution process does not possess any given physical scales, when the fluid (liquid) is inviscid and incompressible, and the air motion above the surface is neglected. The gravitational acceleration g offers the only scale. Any theoretical deep-water wave model must postulate at least one additional physical scale, to be combined with gravity. Since a flow problem cannot be specified without some assumptions concerning geometry, a length scale L is the most obvious choice of a physical scale, and this choice will induce a gravitational time scale \(\sqrt{L/g}\). A time scale T for the wave motion could alternatively be postulated, inducing a length scale \(g T^2\) for the waves.

The density \(\rho \) of the fluid enters the picture only if there is an exterior dynamical causing of the flow. Density does not enter the dynamic free-surface condition, as a contrast to internal waves at the interface between two fluids, where the density of each fluid is essential. In a constant-density liquid, the acceleration of surface particles is dictated by gravity alone. The dynamic condition is therefore a kinematic constraint of each particle slides tangentially along the instantaneous surface with a tangential acceleration equal to the tangential component of the gravitational acceleration vector. The dynamic condition of surface waves thus serves as a kinematic converter of potential energy for the vertical direction into kinetic energy for the horizontal direction. Philosophically it means that the Cauchy–Poisson problem has a time arrow. This time arrow is of algorithmic nature, and it exists in spite of the fact that water waves represent a conservative process without entropy production. A practical implication of this time arrow, is that a water wave can never be stopped as long as the surface is free. Any curved surface shape will induce a free-surface acceleration field that is incompatible with bringing the wave motion to rest.

There exists a considerable amount of formal research on the mathematical properties of the CP problem; see e.g. Shinbrot [15] and Alazard et al. [1]. Not much work has addressed the nonlinear dynamics of the finite-amplitude CP problem, in the context of physical causation. Debnath [5] studied weakly nonlinear surface waves by the Lagrangian description of motion, but he considered developed oscillatory flows and not the early stages of the CP flow. In the present paper we will concentrate on the nonlinearities that are present already initially as the flow starts its free-surface evolution. Early nonlinearities are crucial because dispersion tends to induce gradual reduction of the nonlinearities in a wave packet, as long as the waves do not break during the early stage where the flow has not yet become oscillatory.

We will discuss the finite-amplitude CP problem and its physical causation. Before treating the general problem where a deformed surface is put into motion, we will investigate two basic subproblems: (i) gravitational release from rest of an initially deformed free surface; (ii) an initially horizontal surface put impulsively into motion by an instantaneous pressure impulse.

These two subproblems do not obey superposition when we add up their causes in a full nonlinear problem: this full problem consists of an initially deformed surface where an instantaneous pressure impulse on the surface forces an initial motion which is released to evolve under gravity with a free surface. We will investigate one example of a slightly deformed free surface being put into motion by a moderately strong pressure impulse, where the full nonlinear effects for the early flow will be calculated and compared with an asymptotic expansion.

2 Formulation of the Causal CP Problem

We consider an inviscid and incompressible fluid (liquid) with a free surface subject to constant atmospheric pressure \(p_\mathrm{atm}\). Time is denoted by t. Cartesian coordinates xyz are introduced, where the z axis is directed upwards in the gravity field and the horizontal xy plane represents a reference level corresponding to an undisturbed free surface. The gravitational acceleration is g, and \(\rho \) denotes the constant fluid density. The fluid pressure is denoted by p. The velocity vector is \(\vec {v}\). The surface elevation is \(\eta (x,y,t)\).

We will take a causal viewpoint, considering flows that follow as effects of specified physical causes. Before the initiation of motion, time is negative (\(t<0\)), and the fluid is in a motionless state. We include two simultaneous physical causes for the flow: (i) a deformed surface shape at \(t=0\), to be released for gravitational free-surface flow at \(t=0^+\); (ii) a finite pressure impulse of infinitesimal duration \((0<t<0^+)\), distributed over the surface to force the fluid into initial motion without changing the initial elevation that was already present at \(t=0\). This instantaneous pressure impulse is applied on the deformed surface of the fluid, working in the normal direction at each surface point.

The CP problem is of second order in time, and allows two different causes for the flow to start. These two causes can be combined, with the surface already given a deflection before the surface pressure impulse sets in. The first cause alone gives the CP subproblem of an initially deflected free surface released from rest. The second cause alone gives the CP subproblem of a pressure impulse on an initially horizontal surface, delivering momentum to the fluid for subsequent free-surface flow.

The initial state at \(t=0^+\) is irrotational, which follows from Lord Kelvin’s circulation theorem, since the fluid is motionless for \(t<0\). Irrotational flow implies the existence of a velocity potential \(\Phi (x,y,z,t)\) so that \(\vec {v} = \nabla \Phi \). The incompressible flow of the homogeneous fluid implies Laplace’s equation
$$\begin{aligned} \nabla ^2 \Phi = 0, \end{aligned}$$
(1)
to be valid in the fluid domain. From the equation of motion Bernoulli’s equation follows
$$\begin{aligned} \frac{p-p_\mathrm{atm}}{\rho } + \frac{\partial \Phi }{\partial t} + \frac{1}{2} \vert \nabla \Phi \vert ^2 + g z = 0. \end{aligned}$$
(2)
The nonlinear kinematic free-surface condition is
$$\begin{aligned} \frac{\partial \eta }{\partial t}+ \nabla \Phi \cdot \nabla \eta = \frac{\partial \Phi }{\partial z},~~z=\eta (x,y,t). \end{aligned}$$
(3)
The nonlinear dynamic free-surface condition is given by
$$\begin{aligned} \frac{\partial \Phi }{\partial t} + \frac{1}{2} \vert \nabla \Phi \vert ^2 + g \eta = 0,~~z=\eta (x,y,t), \end{aligned}$$
(4)
where surface tension is neglected. Both these nonlinear conditions are relevant for \(t>0^+\), after the forcing of the flow has been finished, so that the surface is free for the fluid to flow under gravity.
The initial-value problem is a nonlinear Cauchy–Poisson problem where the initial free surface is given as a function \(\eta _0(x,y)\) at \(t=0\)
$$\begin{aligned} \eta (x,y,0) = \eta _0(x,y). \end{aligned}$$
(5)
Figure 1 gives a 2D sketch of the initial forcing of the flow. Generally in 3D we assume an initial forcing stage of infinitesimal duration \(0<t<0^+\), during which a surface pressure impulse P(xy) is forcing the surface into a finite-amplitude flow described by the velocity potential \(\Phi (x,y,\eta _0,0^+)\). The pressure impulse P(xy) has the dimension of pressure multiplied by time.
Fig. 1

Sketch of the finite amplitude Cauchy–Poisson problem in 2D. An initial state at \(t=0^{+}\) is shown. The initial surface elevation is \(z = \eta (x,0) = \eta _{0}(x)\). A pressure-impulse distribution \(P(x) = p_{-1} (x, \eta _{0}) = - \rho \varphi _{0}(x, \eta _{0})\), assembled with infinitesimal duration, is applied on this deformed surface. This pressure impulse forces the liquid into initial motion with the velocity potential \(\varphi (x,z,0^{+}) = \varphi _{0}(x,z)\)

We introduce the following small-time expansion:
$$\begin{aligned} (p, \Phi , \eta ) = (p_{-1},0 ,0) \delta (t) + H(t)((p_0, \phi _0, \eta _0) + t (p_1, \phi _1, \eta _1) + t^2(p_2, \phi _2,\eta _2) + \cdots ), \end{aligned}$$
(6)
where \(\delta (t)\) is the Dirac delta function and H(t) is the Heaviside unit step function. The consecutive Taylor series in time may only exist if the leading order solution is regular (without singularities), but we will avoid these difficulties by considering only leading-order solutions in the present paper.
From the Bernoulli equation (2), we have the relationship
$$\begin{aligned} p_{-1} = -\rho \phi _0. \end{aligned}$$
(7)
The surface pressure impulse P(xy) received by the surface during the infinitesimal time interval is generally defined by
$$\begin{aligned} P(x,y) = p_{-1}(x,y,\eta _0) = -\rho \phi _0 (x, y, \eta _0), \end{aligned}$$
(8)
which follows from integrating Bernoulli’s equation (2) over the time interval \(0<t<0^+\). During this integration in time, the other terms in Bernoulli’s equation are negligible in comparison with the pressure impulse being assembled at \(t=0^+\). The surface is forced into finite velocity, thus remaining at its initial position \(\eta _0(x,y)\) during the infinitesimal time of surface forcing. The surface pressure impulse creates horizontal forces on the surface particles, and they will not have a purely vertical motion, as they do when the surface remains free during an impulsive start [16].
The initial condition for the pressure is
$$\begin{aligned} p(x,y,\eta _0,0^+) = 0, \end{aligned}$$
(9)
as the surface is again free after the surface forcing has been finished.

3 Initial Elevations Released from Rest

We will study some exact acceleration fields of initial flow for finite amplitude deflections of a free surface. Even though the elevation is recognized as the physical cause of the potential flow, it is mathematically legal to choose the entire initial acceleration flow field and compute its corresponding initial elevation, keeping in mind that it is the initial elevation that generates the flow field. The only force is gravity, and there is no initial velocity. An exact acceleration field that is valid at \(t=0^+\) will also be valid for a viscous fluid [11], even though the later time evolution with viscosity will be very different from inviscid flows.

The dynamic free-surface condition (4) is reduced to
$$\begin{aligned} \phi _1 = - \rho \eta _0,~~z=\eta _0(x,y), \end{aligned}$$
(10)
while the kinematic condition does not play any role for this initial flow released from rest. The task is to link a flow field \(\phi _1\) to its associated initial surface elevation \(\eta _0\).

The mathematical solutions can be given in dimensionless form, simply by putting \(g=1\) and \(\rho =1\), with a unit of length chosen for each particular case. If there is a horizontal bottom, it will be located at unit depth.

3.1 Fourier Component Potentials

We first consider the single Fourier component with constant unit depth
$$\begin{aligned} \phi _1 = A \cos (k x) \cosh k(z+1) + \mathrm{constant}, \end{aligned}$$
(11)
where A is the amplitude for the potential, k is the wave number with the associate wavelength \(2 \pi /k\). In Fig. 2 we have chosen \(k=1\) so that the wavelength is \(2 \pi \), and we select the case with the steepest possible free surface, where \(\vert A \vert = 0.2661\), so that the flow is downward at \(x=0\), where there is a right-angle surface peak of approximately unit height above the undisturbed free surface. The choice \(k=1\) has the role of setting a length scale for the peaked surface.
Fig. 2

The initial surface shape with associated isobars and initial streamlines for a single 2D Fourier-component acceleration potential, with dimensionless wavelength \(2\pi \). Impermeable bottom at unit depth \(z=-1\). This finite-depth potential is \(\varphi _{1} =A ~cos(x) cosh (z+1)\). The given initial shape that has the maximum possible height with a pointed apex has amplitude \(A = - 0.2661\). One wavelength is shown

Any isobar can be reinterpreted as an initial surface of fluid released from rest. The initial dam-break type of potential flow considered by Penney and Thornhill [13] is therefore implicitly included in this example, when we consider the isobars near the bottom. The bottom itself (\(z=-1\)) is not an isobar, since a pressure gradient is needed to accelerate the fluid along the bottom.

The related 2D wedge problem of dam breaking has been studied by Tyvand et al. [17], who showed the mathematical equivalence between initial dam breaking and constant-acceleration impact of a liquid body on a plane wall. The impact problem for the 2D liquid wedge had earlier been studied by Cooker [3].

Figure 3 shows the deep-water version of the horizontally periodic acceleration field
$$\begin{aligned} \phi _1 = A \cos (x)~ e^{z+1} + \mathrm{constant}. \end{aligned}$$
(12)
The wave number is \(k=1\) is chosen without loss of generality, setting the length scale in terms of the wavelength \(2 \pi \). Selecting the amplitude value as \(\vert A \vert = 1/e = 0.3678\) we achieve the peaked surface with maximal possible surface slope. This surface peak happens to have unit height. This steepest possible free-surface shape is practically the same with and without a constant-depth bottom.
Fig. 3

The initial surface shape with associated isobars and initial streamlines of a single 2D Fourier-mode acceleration potential with wavelength \(2\pi \). The infinite-depth potential is \(\varphi _{1} = A \cos (x) \exp (z)\). One wavelength is shown. The given initial shape with flow amplitude \(A = - 1/e = - 0.3679\) gives the maximum possible height with a pointed apex of unit height. Maximum initial surface slope angle is \(\pi /4\)

The peaked surface shape in Fig. 3 is related to the highest deep-water wave [4]. Remarkably, the highest standing wave reported in the nonlinear simulations by Longuet-Higgins and Dommermuth [10] fits quite closely with the smooth isobars slightly below the peaked surface in our Fig. 3. These authors started a nonlinear CP problem with a flat initial surface (zero potential energy) and sinusoidal velocity distribution. The surface evolution approached a situation with dominating potential energy and small kinetic energy, similar to the situation that we start from in Fig. 3.

3.2 Dipole-Type Potentials

We will consider a basic class of non-periodic solutions: isolated surface peaks corresponding to dipole-type potentials. We will only discuss the 2D case. The single vertical-dipole potential for the acceleration field is
$$\begin{aligned} \phi _1 = A \frac{z - 1}{x^2 + (z - 1)^2}, \end{aligned}$$
(13)
and we choose the same amplitude \(\vert A \vert = 0.25\) for the two dipole potentials represented in Fig. 4. This amplitude for a downward dipole gives a central surface peak of approximate height 0.5 measured from the the undisturbed free surface level, shown the left. The same amplitude with opposite sign is chosen for the rounded trough shape shown to the right. The great contrast in steepness as well as curvature for these two flows of equal but opposite amplitudes in Fig. 4 has a simple explanation: the same singularity that generates these identical but oppositely directed flows is located much more closely to the central surface peak than to the central trough.
Fig. 4

The initial surface shape with associated isobars and initial streamlines of a 2D dipole-field acceleration potential. The dipole is located in \((x,z) = (0,1)\) outside the fluid. Infinite fluid depth. Left: downward dipole flow with the maximum possible slope at a pointed apex crest. Dipole amplitude \(A = - 0.25\). Right: upward dipole flow inducing a surface with rounded trough. Dipole amplitude \(A = 0.25\)

Figure 5 represents the constant-depth counterparts to Fig. 4, where the acceleration field is given by the similar dipole field plus an image dipole satisfying the kinematic bottom condition at unit depth \(z=-1\)
$$\begin{aligned} \phi _1 = A \left( \frac{z - 1}{x^2 + (z - 1)^2} - \frac{z + 3}{x^2 + (z + 3)^2} \right) . \end{aligned}$$
(14)
The same amplitude \(\vert A \vert = 0.25 \) will produce a surface peak of height 0.5, similar to the case of infinite depth, and also a similar trough.
In Figs. 4 and 5, the streamlines and isobars are shown for two dipole flows with equal and opposite amplitude. In Fig. 6, two dipoles with opposite sign are superposed, revealing that the resulting initial shape with finite amplitude is not the sum of separate dipole-induced surface shapes. In Fig. 6, the upward dipole above the trough is chosen with almost three times the strength than the downward dipole above the peak, for generating a trough with approximately the same amplitude as the peak.
Fig. 5

The initial surface shape with associated isobars and initial streamlines of a 2D dipole-field acceleration potential. The dipole is located in \((x,z)=(0,1)\) outside the fluid. An impermeable bottom at \(z = -1\) is represented by an image dipole in \((x,z) = (0,-3)\) outside the fluid. Left: downward dipole flow with the maximum possible slope at a pointed apex crest. Dipole amplitudes are \(A = - 0.25\). The existence of the bottom does not change significantly the dipole amplitude that gives the highest crest with the peak, as it compensates exactly for the image dipole at \(z = -3\). Right: upward dipole flow inducing a surface with rounded trough. Dipole amplitudes are \(A = 0.25\)

Fig. 6

The initial surface shape with associated isobars and initial streamlines for a sum of two 2D dipole fields contributing to the acceleration potential. The depth is infinite, and the two dipoles have opposite signs, being located in the symmetric points \((x,z)=(-1,0.8)\) and \((x,z)=(1,0.8)\) outside the fluid. The peak has a local downward dipole flow of strength 0.363, while the trough has a local upward dipole flow of strength 1. As there is no superposition, the surface shape does not follow from combining the individual shapes induced by the single dipole flows of Fig. 4

3.2.1 Peaked Run-up at a Sloping Beach

The finite-amplitude Cauchy–Poisson problem may be associated with the run-up of an incoming solitary wave on a sloping beach [12]. We want to estimate the highest run-up that may result from a non-breaking solitary wave on a sloping beach. In oscillatory deep-water waves the energy is typically split equally between potential energy and kinetic energy. The concept of maximum run-up is hard to define precisely, but one thing is certain: zero kinetic energy represents the highest possible potential energy available for inviscid run-up. Maximal run-up may be reformulated backward in time as a finite amplitude CP problem. The above analysis showed how a downward dipole can pile up fluid to be released from rest. The basic acceleration field for extremal run-up is therefore a vertical dipole plus its image term for obeying the kinematic condition at the slope. Here we will only consider the slope angle \(\pi /4\), with infinite depth. A maximal run-up will then appear as a peaked surface to be released from rest. The dipole-type potential that satisfies these requirements has the form
$$\begin{aligned} \phi _1 = A \left( \frac{z - Z_1}{(x - X_1)^2 + (z - Z_1)^2} - \frac{x - X_2}{(x-X_2)^2 + (z - Z_2)^2} \right) . \end{aligned}$$
(15)
This is the sum of a downward vertical dipole located in the point \((X_1, Z_1)\) and an image dipole (horizontal) of the same magnitude, located in the point \((X_2, Z_2)\). The kinematic condition for the slope angle \(\pi /4\) implies \(Z_1 - Z_2 = X_1 - X_2\).
The induced surface shape has only one length scale, which we choose as the horizontal distance from the uniformly sloping beach to the surface peak. Figure 7 displays the computed streamlines and isobars for the dipole acceleration field (15) with the parameter choices \(A = 1\), \(X_1 = - X_2 = 1\), \(Z_1 = 2.4\), \(Z_2 = 0.4\). The approximate coordinates of the surface peak are (0.95, 1.45), roughly one length unit displaced horizontally from the slope. This released free surface meets the slope at the point (0.25, 1.15). The reference level of undisturbed surface in the far field is \(z=0\), which is marked with a blue line in Fig. 7.
Fig. 7

The peaked stagnant run-up shape with its initial streamlines and associated isobars, produced by a 2D dipole-field acceleration potential at a sloping beach with angle \(\pi /4\). The length scale is the horizontal displacement of the peak away from the slope. There is a vertical dipole located in the point \((x,z)=(1, 2.396)\) outside the fluid. We take care of the kinematic condition at the uniform slope by means of a horizontal image dipole outside the fluid, symmetrically located with respect to the slope. The depth is infinite, with the undisturbed free surface given as \(z=0\), asymptotically as \(\hbox {x }\rightarrow \infty \). The horizontal blue line to the left shows the reference level \((z=0)\) of the undisturbed surface, which is the far-field level of the free surface (Color figure online)

The stagnant mass of liquid released from rest, will represent a time reversal of the wave that caused the run-up, which may or may not represent a realistic wave motion. There are no length scales in this problem with a given angle, other than the length scale set by the peaked surface itself. This peaked run-up shape has a known Eulerian acceleration field, even though its displacement away from the undisturbed surface level \(z=0\) is great.

One basic objection may be raised to this dipole solution: any smooth initial surface shape can be released from rest, so why should we give any priority to this pure dipole field of acceleration? We should note that if the acceleration field is given as a multipole expansion, the dipole is the only option. No other multipole is able to lift the surface for generating a initial surface peak. The pressure distribution from a pure dipole field filters out flows involving almost free fall, which would have almost vertical streamlines around the highest initial elevation. A flow domain in almost free fall would lead to wave breaking, so it could not serve as a backward-time model for high run-up with gentle retardation.

3.2.2 Initial Surface Peak for Free Sloshing

The finite-amplitude Cauchy–Poisson problem in an open container belongs to the field of sloshing [6, 7]. We will now present such an example related to the run-up that we have just discussed. A basic question concerning free-surface oscillations in open containers is how high a free-surface elevation due to sloshing can become without surface breaking. We will now construct a sum of dipole potentials that generate an initial acceleration potential for a high initial surface peak in a finite 2D container. We start from the sum of two dipoles (15) being rewritten as:
$$\begin{aligned} \phi _1(x, z, X, Z) = \frac{z - Z}{(x - X)^2 + (z - Z)^2} - \frac{x + Z}{(x + Z)^2 + (z + X)^2}, \end{aligned}$$
(16)
where we have now put \(X_1= - Z_2=X\) and \(- X_2=Z_1=Z\), taking unit amplitude (\(A=1\)). This is the sum of a unit downward vertical dipole located in the point (XZ) and its image dipole (horizontal), located in the point \((-Z, -X)\). The kinematic condition for the slope of angle \(\pi /4\) through the origin is satisfied, while nothing is said about the free surface. This potential produces a local heap below the dipole location (XZ).
We aim to construct a dipole potential valid for a 2D container with length scale L and wedge shape with right-angle bottom at the origin, with two slope angles \(\pm \,\pi /4\). The kinematic condition at the left-hand slope (\(z=-x\)) is already satisfied by the potential (16), but we need to add two image dipoles in order to satisfy the kinematic condition at the right-hand slope (\(z=x\))
$$\begin{aligned} \phi _\mathrm{container}(x, z, L, \epsilon ) = A (\phi _1(x+L, z-L, \epsilon , \epsilon ) - \phi _1(x-L, z+L, -\epsilon , -\epsilon )), \end{aligned}$$
(17)
where we have reintroduced a flow amplitude A and placed in total four dipoles in the points \((-L+\epsilon ,L+\epsilon )\), \((-L-\epsilon ,L-\epsilon )\), \((L+\epsilon ,-L+\epsilon )\), \((L-\epsilon ,-L-\epsilon )\).
We show one calculated example in Fig. 8, choosing \(L=1\) and \(\epsilon = 0.5\), giving the four dipole locations \((- 0.5, 1.5)\), \((- 1.5, 0.5)\), \((1.5, - 0.5)\), \((0.5, - 1.5)\). The peaked surface results from choosing the amplitude \(A = 0.2887\). The initial streamlines and associated isobars are displayed. Any isobar may be reinterpreted as a possible initial surface shape, but we choose the steepest possible (peaked) surface shape as the free surface in the figure. The dipole locations give peaked initial surface shape with approximately the same shape at the left-hand slope as the run-up surface shape displayed in Fig. 7. There are two length scales for the peaked free surface in Fig. 8, while Fig. 7 had only one length scale. The new length scale in Fig. 8 is the length scale of the container.
Fig. 8

A peaked initial surface shape released from rest in a 2D wedge container composed of two sloping angles of \(\pm \,\uppi /4\). The flow is generated by a single vertical dipole potential plus its three image potentials to satisfy the two kinematic conditions at the slopes. The initial streamlines with their associated isobars are displayed. The initial surface shape has two length scales: the horizontal displacement of the peak from the slope, and the horizontal distance between the initial waterlines at each slope. We have not chosen a precise length scale in this figure, but we have put the peak in a sufficiently asymmetric position to be comparable with Fig. 7, and it shows reasonable agreement

Taylor expansion of the acceleration potential (17) around the right-angle bottom tip (0, 0) gives the local potential of the form
$$\begin{aligned} \phi _\mathrm{bottom}(x, z) = B x z,~~~~B = 2 A \frac{(L - \epsilon )(L+4 \epsilon +\epsilon ^2)}{(L+\epsilon ^2)^3}, \end{aligned}$$
(18)
representing the imaginary part of the complex potential \((x + i z)^2\), where i is the imaginary unit. The error term in complex variables is of order \((x+i z)^3\). The free-surface condition (10) gives us the isobars near the bottom tip
$$\begin{aligned} z (1 + B x) = C, \end{aligned}$$
(19)
after inserting the leading-order Taylor expansion (18). \(A = 0.2887\) for the computed example in Fig. 8, which gives \(B = 0.4804\) for the local potential around the bottom tip. Different values of C in formula (19) give different isobars. The local approximation (19) is in excellent agreement with the three lowest isobars in Fig. 8.

When the initial acceleration flow is of the downward-dipole type, there is only one possible peaked run-up shape of fluid at rest for a given slope angle, and the shape valid for the slope angle \(\pi /4\) is shown in Fig. 7 above. For the given wedge container, there is a broader family of initial peaked surface shapes that can be released from rest. The single parameter that classifies the members of this family can be chosen as the relative displacement of the surface peak from the left-hand waterline of the free surface, compared by the total horizontal distance between the two waterlines where free surface meets the slope. This ratio between the peak location and the total horizontal distance with wetted initial boundary (both measured from the left-hand waterline) is about 0.25 in Fig. 8. An explicit parametrization of the family of surface shapes can be given by the ratio \(\epsilon /L\), which was chosen as 0.5 in Fig. 8. The value of A that gives the peaked initial surface must be found numerically in each particular case. The peak is located at the left-hand side of the z axis (with x negative) for \(\epsilon < L\), and it is has a positive value of x when \(\epsilon \) is chosen greater than L. A symmetric initial surface shape requires \(\epsilon = L\).

3.3 On the Distribution of the Vertical Acceleration

Interesting comparisons between cases of liquid masses released under gravity are provided in Fig. 9, where we plot the distribution of the vertical acceleration \(\partial _z \phi _1(0,z)\) along the z axis for symmetric 2D flow calculated above. The left-hand portion of Fig. 9 shows the two Fourier mode solutions with and without bottom, in comparison with the linear acceleration profile that a wedge dam-break flow possesses [17]. The unit elevation height is achieved for the infinite-depth potential, while the finite-depth version with the same wavelength compromises slightly on this unit height of the sharp peak. Apart from that, the surface shape and the near-field near the surface peak are quite similar with and without the bottom.
Fig. 9

Distributions of the downward vertical acceleration \(\partial _{z}\upphi _{1}\)(0,z) along the central vertical line beneath the sharp surface peak associated with each of the different symmetric acceleration potentials. Comparisons between finite and infinite depth are shown for the cases of a Fourier component and a single dipole. The peak falls initially with the dimensionless gravitational acceleration \(-1\). Left: Fourier component distribution with infinite depth and with finite depth, with respective flow amplitudes \(A = -0.2661\) and \(A = - 0.3678\). For comparison a dashed curve is added, with the linear acceleration profile of a triangle dam-break flow [17], with the location of the surface peak adjusted to fit with the peaks of the Fourier component potentials. Right: dipole distributions with infinite depth and with finite depth, both having the same flow amplitude \(A = - 0.25\). It is impossible to see any difference between a dipole field with finite depth and infinite depth, near the surface

The right-hand portion of Fig. 9 represents two cases of dipole potentials for the initial acceleration field: the single dipole with an isolated peak on infinite depth, and the similar double-dipole solution for a unit depth. The amplitude \(\vert A \vert = 0.25\) of the dipole responsible for the downward flow gives a sharp peak with the elevation 0.5, in both cases. The image dipole reduces the flow more and more the closer we get to the bottom. The dipole acceleration flows with and without the bottom are practically identical near the surface peak. The curvature of the distribution of vertical acceleration is greater for the dipole distribution of a single peak than it is for the Fourier-component case that is periodic in the horizontal direction. We recall that the simple dam-break flow for an initial wedge has zero curvature, with linear distribution of vertical acceleration. The curved acceleration field has to do with the amount of fluid needed to be pushed aside for the particles to accelerate downwards, under the weight of the fluid above.

4 Pressure Impulse on a Horizontal Surface

Pressure impulses that force a semi-infinite fluid mass with an initially horizontal surface into motion constitute the second basic subproblem. The subsequent free-surface evolution after the application of a pressure impulse will follow nonlinear theory, provided the pressure impulse is strong enough in comparison with gravity, which means that the dimensionless version of the pressure impulse amplitude \(P_0\)
$$\begin{aligned} \frac{P_0}{\rho g^{1/2} L^{3/2}}, \end{aligned}$$
(20)
must be of order 1 or greater, scaled in gravitational units. Here L represents the length scale of the pressure impulse distribution along the surface. If \(P_0 \ll \rho g^{1/2} L^{3/2}\), wave dispersion will make the subsequent evolution effectively governed by linear theory. The nonlinear free-surface evolution when \(P_0 > \rho g^{1/2} L^{3/2}\) will be the topic of follow-up papers. In the present paper we will study the initial flows that satisfy linear theory, forced by a pressure impulse on an initially horizontal surface.
Restricting ourselves to infinite depth, there is no length scale other than the length scale of the pressure impulse distribution along the surface. Here we go back to variables with dimension, since impulsive initial flows with horizontal surface are independent of gravity, and gravitational dimensionless units have no relevance. The surface pressure impulse distribution P(xy) sets the scales for the impulsive flow, when the surface is initially flat. The initial free-surface flow immediately after a pressure impulse has been applied to a horizontal surface (\(\eta _0 = 0\)) is given by
$$\begin{aligned} \eta _1 = \frac{\partial \phi _0}{\partial z},~~z=0, \end{aligned}$$
(21)
while the surface pressure impulse can be given as
$$\begin{aligned} P(x) = - \rho \phi _0 (x, 0). \end{aligned}$$
(22)
The examples that we will show now, are preliminary since they are restricted to linear theory. We concentrate on multipole fields, for two reasons. The combined case of nonlinear interaction between pressure impulse and initial elevation that will be computed below, is based on dipole fields. We are preparing follow-up papers on the fully nonlinear free-surface flow initiated by multipole pressure impulses.

We will not treat spatially periodic pressure impulses here, noting that the case of a sinusoidal pressure impulse has already been investigated by Longuet-Higgins and Dommermuth [10]. They simulated the fully nonlinear evolution of the surface from a state of zero potential energy, and found final states of high potential energy that could lead to surface breaking of standing waves. These limit shapes are remarkably close to our peaked surface shapes released from rest, as mentioned above.

4.1 The Locally Uniform Pressure Impulse in 2D

We consider first the 2D stepwise uniform distribution, where \(p_{-1}(x,0) = P(x) = P_0\) for \(\vert x \vert < L/2\). The remaining surface is free, with \(P(x) = 0\) for \(\vert x \vert > L/2\). This piston-type pressure impulse induces the flow field
$$\begin{aligned} p_{-1}(x, z) = \frac{\vert z \vert }{\pi } \int _{-\infty }^{\infty } \frac{P(s) \mathrm{d}s}{(x-s)^2 + z^2} = \frac{P_0}{\pi } \left( \arctan \frac{x - L/2}{z} - \arctan \frac{x + L/2}{z} \right) ,\nonumber \\ \end{aligned}$$
(23)
given by Poisson’s 2D integral formula for a half-plane \(-\infty< x< \infty ,~~z<0\). The vertical surface velocity is
$$\begin{aligned} w_0(x,0) = -\frac{1}{\rho } \left. \frac{\partial p_{-1}}{\partial z}\right| _{z=0} = \frac{L}{\rho \pi } \frac{1}{(x + L/2) (x - L/2)} = - \frac{W_0}{\left( 1 + \frac{2 x}{L}\right) \left( 1 - \frac{2 x}{L}\right) }, \end{aligned}$$
(24)
where \(W_0 = \vert w_0(0,0)\vert \) is the reference downward velocity in the central point of the 2D uniform pressure impulse. Figure 10 shows the locally uniform pressure impulse and its induced free-surface flow. The velocity amplitude \(W_0\) for the locally uniform flux is given by
$$\begin{aligned} \frac{\rho L W_0}{P_0} = \frac{4}{\pi } = 1.27324. \end{aligned}$$
(25)
We note the unpleasant fact that all the three separate integrals of upwelling/downwelling fluxes at the surface
$$\begin{aligned} \int _{-\infty }^{-L/2} w_0(x, 0) \mathrm{d}x,~~\int _{-L/2}^{L/2} w_0(x, 0) \mathrm{d}x,~~\int _{L/2}^{\infty } w_0(x, 0) \mathrm{d}x, \end{aligned}$$
(26)
diverge.
Fig. 10

A locally uniform surface pressure impulse of unit size over a unit length (marked by arrows), and its induced zeroth-order vertical velocity \(w_{0}(x,0)\) at the initially horizontal surface \(z=0\). The surface velocity is singular at the to end-points \(x/L = \pm \,1/2\) where the pressure impulse terminates abruptly

The initial surface flow is singular in the end-points \((x,z)=(\pm \, L/2,0)\) of the uniform pressure impulse. This singularity means that the zeroth-order flow is an outer flow in a matched-asymptotics sense. These singularities limit the small-time Taylor expansion to one term only. Due to mass conservation, downwelling from the pressure impulse is surrounded by two upwelling domains with free surface. This locally uniform pressure impulse separates the two upwelling domains strictly from the forced downwelling domain. With a continuously distributed pressure impulse, the location of the border between downwelling and upwelling flows is not obvious. The formula (24) reveals a surrounding free-surface flow from this concentrated pressure impulse, with a dipole-type spatial decay, as \(\vert x \vert ^{-2}\) in the far field.

4.2 2D Symmetric Multipole Pressure Impulses

We consider the following class of symmetric multipole pressure impulses:
$$\begin{aligned} P(x) = p_{-1}(x,0) = \frac{P_0}{(1 + (x/L)^2)^n} = P_0 f_n(x/L, 0), \end{aligned}$$
(27)
where n is a positive integer. In the “Appendix” this dimensionless class of harmonic functions \(f_n(x,0) = (1 + x^2)^{-n}\) is elaborated, deriving their normal derivatives at \(z=0\).

4.2.1 A Simple Multipole Function in 2D

We may consider the simple multipole functions \(f_n(x,0) = (1 + (x/L)^2)^{-n}\) and the relationship between the surface forcing pressure impulse \(P(x) = P_0 f_n(x/L, 0)\) and its zeroth-order surface velocity \(w_0(x,0)\). In the present paper we consider only \(n=1\), representing a vertical-dipole forcing, giving
$$\begin{aligned} P(x) = p_{-1}(x,0) =\frac{P_0}{1 + (x/L)^2} = P_0 f_1(x/L,0), \end{aligned}$$
(28)
leading to
$$\begin{aligned} w_0(x,0) = - \left. \frac{1}{\rho } \frac{\partial p_{-1}}{\partial z} \right| _{z=0} = \frac{P_0}{\rho L} \left( f_1(x/L,0) - 2 f_2 (x/L,0)\right) , \end{aligned}$$
(29)
from which we extract the relationship between the central velocity \(W_0 = - w_0(0,0)\) and its forcing amplitude \(P_0\)
$$\begin{aligned} W_0 = \frac{P_0}{\rho L}. \end{aligned}$$
(30)
In Fig. 11 (left), this pressure impulse and its induced zeroth-order vertical velocity are displayed.
Fig. 11

Two surface pressure impulses P(x) of the multipole type, putting a semi-infinite fluid with initially horizontal surface into motion. The pressure impulses belong to the class of functions \(P(x)/P_{0 }= f_{n} (x,0) = (1+(x/L)^{2})^{-n}\) (where n is a positive integer). These pressure impulse distributions (marked by arrows) are displayed with their induced zeroth-order vertical velocity fields \(w_{0}(x,0)\), downwelling in the center. Left: a single dipole: \(P(x)/P_{0} = f_{1}\) Right: a sum of multipole functions: \(P(x)/P_{0} = (323 f_{7}-532f_{8}+224f_{9})/15, \) having the hand-picked properties \(P''(0) = P''''(0) = 0\). This gives a locally uniform pressure impulse around \(x=0\), for the purpose of approximating the uniform pressure impulse of Fig. 10

4.2.2 A Superposition of 2D Multipole Fuctions

We pick at a pressure-impulse distribution designed to become more uniform locally around \(x=0\). We consider a sum of three multipole-type functions
$$\begin{aligned} P(x) = P_0 (36 F_7(x) - 63 F_8(x) + 28 F_9(x)), \end{aligned}$$
(31)
which has the properties \(P''''(x) = P''(x)=0\) combined with a rapid spatial decay as \(\vert x \vert > 1\). We have calculated the associated formula for \(w_0(x,0)\) from the “Appendix”, but we omit it in the text. The surface velocity amplitude versus the pressure forcing is represented by the dimensionless parameter
$$\begin{aligned} \frac{\rho L W_0}{P_0} = \frac{\rho L \vert w_0 (0,0) \vert }{P_0} = \frac{9009}{8192} = 1.09973, \end{aligned}$$
(32)
to be compared with the value 1.27324 for a locally uniform pressure impulse (25).

This case is displayed in Fig. 11, together with the pure dipole-type pressure impulse. We compare with Fig. 10 to see how this continuous but almost uniform pressure impulse in the neighborhood around \(x=0\) approaches the previous case of locally uniform pressure impulse. The forced downward vertical velocity increases locally with \(\vert x \vert \) around \(x=0\), as a result of the mass balance constraint when the pressure impulse is almost constant: the fluid is put into horizontal motion also, in addition to its forced vertical surface velocity. There is nowhere for the fluid to escape from its surface forcing, and both the horizontal and vertical flow components increase with increasing \(\vert x \vert \) beneath the domain of the pressure impulse.

4.3 A Dipole Pressure Impulse Near a Sloping Beach

We have now considered pressure impulses with infinite depth, where it is obvious from momentum conservation that the entire force impulse exerted on the surface will give momentum to the fluid. A question that arises once we have a finite depth, is how great fraction of the imposed pressure impulse that will actually set the fluid into local vertical motion. Local momentum balance requires that the portion of the pressure impulse that does not induce vertical fluid momentum must be received by the bottom.

We will make this kind of analysis for a sloping beach of angle \(\pi /4\), where we consider a vertical dipole potential plus a horizontal image potential
$$\begin{aligned} \phi _0(x, z, X, Z) = A \left( \frac{z - Z}{(x - X)^2 + (z - Z)^2} - \frac{x + Z}{(x + Z)^2 + (z + X)^2}\right) , \end{aligned}$$
(33)
recalling that \(p_{-1} = - \phi _0\). The vertical dipole is located in the point (XZ), while its horizontal image dipole has the coordinates \((-Z, -X)\). Thereby the kinematic condition at a slope of angle \(\pi /4\) is satisfied, with symmetry of the dipoles around the slope \(z+x=0\). We choose \((X_1, Z_1) = (1,1)\), defining the distance from the free surface to the dipole as length scale. The induced dimensionless surface velocity is
$$\begin{aligned} w_0(x,0) = A \left( \frac{2}{(1+ (x-X)^2)^2} - \frac{2(x+X)}{(1+(x+X)^2)^2}+ \frac{1}{1+(x-X)^2}\right) . \end{aligned}$$
(34)
The pressure impulse along the bottom \(z=-x\) can be expressed as a function of x and X
$$\begin{aligned} p_{-1}(x, -x) = A \left( \frac{x + X}{(x + X)^2 + (x - 1)^2} - \frac{x + 1}{(x - X)^2 + (x + 1)^2} \right) , \end{aligned}$$
(35)
and it will be compared with the pressure impulse imposed on the initial surface \(z=0\), which is
$$\begin{aligned} p_{-1}(x, 0) = A \left( \frac{x + X}{(x + X)^2 + 1} - \frac{1}{(x - X)^2 + 1} \right) . \end{aligned}$$
(36)
Figure 12 shows the pressure impulse and the induced surface velocity (left), and also the initial streamlines and isobars (right). In Fig. 13, we have computed the ratio between the pressure impulse received at the sloping bottom and the exerted surface pressure impulse. This ratio is obviously equal to one where the slope starts at \(x=0\), and it also goes to one in the decaying far field (\(x\rightarrow \infty \)).
Fig. 12

A surface pressure impulse P(x) of the vertical-dipole type is present near a sloping beach \((z+x=0)\) with slope angle \(\pi /4\). Left: the surface pressure impulse P(x) due to a vertical dipole located at (1,1) is marked with arrows. Its induced zeroth-order vertical velocity \(w_{0}(x,0)\) is plotted. Right: initial streamlines and pressure-impulse isobars for this vertical dipole located at (1,1).

Fig. 13

A surface pressure impulse P(x) of the vertical-dipole type is present near a sloping beach \((z+x=0)\) with slope angle \(\pi /4\). This plot shows the fraction of the surface pressure impulse that is received and absorbed by the slope. Four dipole locations \((X,Z) = (0.5,1), (1,1), (2,1), (4,1)\) are displayed. The values for small \(x>0\) increase with increasing X. All curves tend to the asymptotic limit 1 as \(\hbox {x}\rightarrow \infty \), coinciding with the trivial value at \(x=0\)

5 The General Class of Nonlinear CP Problems

The full nonlinear CP problem considers an instantaneous pressure impulse on a surface that is already deformed. We will discuss this general case with the dimensionless description that has been introduced in Sect. 2.

We show one calculated example where the pressure impulse field is chosen, by specifying the zeroth-order potential \(\phi _0(x, y, z)\). In addition and independently, we specify the initial elevation \(\eta _0(x, y)\). By inserting both these causing agents into the nonlinear kinematic condition (3), we have
$$\begin{aligned} \eta _1 = - \nabla \phi _0 \cdot \nabla \eta _0 + \frac{\partial \phi _0}{\partial z},~~z=\eta _0(x,y), \end{aligned}$$
(37)
and the resulting surface velocity (first-order elevation) \(\eta _1\) can be calculated. Moreover, the first-order potential is known from inserting these functions into Eq. (4)
$$\begin{aligned} \phi _1 = - \frac{1}{2} \vert \nabla \phi _0 \vert ^2 - \eta _0,~~z=\eta _0(x,y), \end{aligned}$$
(38)
but this Dirichlet condition at a curved boundary can only approximately be extended analytically to the fluid domain.

5.1 A Dipole-Type 2D Problem

We consider the full nonlinear Cauchy–Poisson problem for one specified example of 2D initial flows. We will compare the fully nonlinear initial velocity with increasing orders of an asymptotic expansion.

We consider a flow that is symmetric around the z axis, with infinite fluid depth. A dipole potential offers a simple 2D flow
$$\begin{aligned} \phi _0(x, z) = - A \frac{1 - z}{x^2 + (z-1)^2} = - A f_1(x,z), \end{aligned}$$
(39)
taking the notation of the “Appendix”. We choose an initial elevation given as
$$\begin{aligned} \eta _0 = \frac{\epsilon }{x^2 + 1}, \end{aligned}$$
(40)
which describes a central localized mound with dimensionless amplitude \(\epsilon \). The restriction \(\epsilon <1\) is needed because the chosen potential (39) has its dipole singularity in the point \(z=1\), which must be located outside the initial fluid domain. The insertion of the deformed surface \(z = \eta _0(x)\) changes the dimensionless surface pressure impulse P(x) from the simple form \(A/(1 + x^2)\) (at \(z=0\)) to the more complicated expression
$$\begin{aligned} P(x) = - \phi _0(x, \eta _0(x)) = A \frac{1 - \epsilon (x^2 +1)^{-2}}{x^2 + (1 - \epsilon (x^2 + 1)^{-2})}, \end{aligned}$$
(41)
which expresses that the dipole distribution \(- \phi _0(x,z)\) for the pressure impulse is evaluated at a displaced location and not at the undisturbed level \(z=0\). The entire pressure-impulse field is given while its vertical coordinate takes the surface value \(z=\eta _0(x)\). A computationally less convenient alternative would have been to take the surface pressure field P(x) as a fixed function of x, independent of the initial elevation amplitude. The way we have formulated this fully nonlinear problem, the chosen initial surface shape does not coincide with the pressure impulse distribution, other than in the linearized limit \(\vert \epsilon \vert \ll 1\). The initial surface shape will only coincide with the pressure impulse distribution as far as linear theory is valid, to the order \(\epsilon ^1\).
The induced exact initial surface velocity (first-order elevation in time) is found by inserting these chosen fields into the kinematic condition (37)
$$\begin{aligned} \eta _1 = A (1 + x^2) \frac{(x^2 + 1)^3 (x^2 - 1) + \epsilon (2 + 8 x^2 + 6 x^4) - \epsilon ^2 (1 + 5 x^2)}{((1 + x^2)^3 - 2 \epsilon (1 + x^2) + \epsilon ^2)^2}, \end{aligned}$$
(42)
and it interesting to compare this exact initial velocity with its second-order approximation in terms of the parameter \(\epsilon (< 1)\)
$$\begin{aligned} \eta _{1,\mathrm{approx}} = A (\eta _{10}(x) + \epsilon \eta _{11}(x) + \epsilon ^2 \eta _{12}(x)). \end{aligned}$$
(43)
The formulas for the terms to each order in this approximation are
$$\begin{aligned} \eta _{10}(x) = \frac{x^2 - 1}{(1 + x^2)^2},~~~~~ \eta _{11}(x) = 2 \frac{5 x^2 - 1}{(1 + x^2)^4},~~~~~ \eta _{12}(x) = \frac{- 7 x^4 + 30 x^2 - 3}{(1 + x^2)^6}. \end{aligned}$$
(44)
In Fig. 14 (left), the exact initial surface velocity \(\eta _1(x)\) is displayed for unit amplitude of the pressure impulse \((A=1)\), together with its three successive approximations \(\eta _{10}\), \(\eta _{10} + \epsilon \eta _{11}(x)\), \(\eta _{10} + \epsilon \eta _{11}(x) + \epsilon ^2 \eta _{12}(x) \), with the choices \(\epsilon = \pm \, 0.3\). When \(\epsilon =0.3\) (left) the central amplitude \(\eta _1(x)\) of the induced downward motion is twice as large as that predicted by linear theory, even for this moderate amplitude of the initial elevation. The case \(\epsilon =-0.3\) (right) gives a smaller and wider trough, because the fluid is pushed in the outward direction. Linear theory gives opposite sign, but no other difference when \(\epsilon \) is replaced by \(-\epsilon \).
Fig. 14

The initial surface velocity \(\eta _{1}(x)\) is here displayed for a full nonlinear 2D problem, specifying both an initial elevation of the dipole type and a simultaneous dipole-type pressure impulse distribution. The non-dimensional pressure impulse that is chosen, induces the dipole flow potentia—\(\varphi _{0} = A (1 - z)/(x^{2} + (z-1)^{2}).\) The initial elevation is chosen as \(\eta _{0} (x) = \varepsilon /(1+x^{2})\), represented by the dotted curves. The solid lines give the exact induced initial surface velocity \(\eta _{1}(x),\) with dimensionless amplitudes \(\varepsilon =\pm \,0.3\). The sets of three dashed lines represent successive asymptotic approximations with increasing accuracy, to the orders \(\hbox {O }(\varepsilon ^{0})\), \(\hbox {O}(\varepsilon ^{1})\), \(\hbox {O }(\varepsilon ^{2})\) of the dimensionless initial elevation amplitude \(\varepsilon \). The zeroth-order distributions of surface velocity represent linear theory. Left: positive initial elevation (\(\varepsilon =0.3\)). Dotted curve shows the initial elevation \(\eta _{0}(x).\) Solid curve shows the exact induced first-order surface velocity \(\eta _{1}(x).\) Linear theory sets the unit velocity amplitude at \(x=0\). Right: negative initial elevation (\(\varepsilon = -0.3\)). Dotted curve shows the initial elevation \(\eta _{0}(x).\) Solid curve shows the exact induced first-order surface velocity \(\eta _{1}(x).\) Linear theory sets the unit velocity amplitude at \(x=0\)

The pressure impulse P(x) has the approximate shape given by
$$\begin{aligned} P(x) = A\left( \frac{1}{1 + x^2} + \epsilon \frac{1 - x^2}{(x^2 + 1)^3} + \epsilon ^2 \frac{1 - 3x^2}{(x^2 + 1)^5} + O(\epsilon ^2) \right) , \end{aligned}$$
(45)
which is an amplitude expansion derived from Eq. (41).
The early acceleration field (or first-order flow field) is given by the dynamic condition (38) as:
$$\begin{aligned} \phi _1(x,z) = - z - \frac{A^2}{2 (x^2 + (z-1)^2)^2},~~~~z = - \eta _0(x) = \epsilon /(1+x^2), \end{aligned}$$
(46)
which we may expand to second power of \(\epsilon \),
$$\begin{aligned} \phi _1(x,0) = - \frac{A^2}{2 (x^2 + 1)^2} - \frac{\epsilon }{1+x^2} - \epsilon A^2 \frac{2}{(1 + x^2)^4} + \epsilon ^2 A^2 \frac{x^2 - 5}{(1 + x^2)^6} + O(\epsilon ^3). \end{aligned}$$
(47)
We will not take this analysis further. We note that nonlinearity is expressed by the squared amplitude \(A^2\) that corresponds to the dimensionless parameter \(P_0^2/(\rho ^2 g L^3)\), introduced by the expression (20) above.

6 Summary and Conclusions

The present paper attempts a unified causal approach to the classical Cauchy–Poisson problem of initiation of water waves, including the full nonlinear effects of an initially deformed surface. The classical work in this field is restricted to linear theory. We concentrate on the initial stage of the flow, since nonlinearity is most important immediately after the initiation. Before exemplifying the full nonlinear problem, we have analyzed the two basic subproblems: (i) an initial finite surface deformation, initially released from rest to flow freely under gravity; (ii) an initially horizontal surface where an impulsive pressure is applied to force the fluid into motion.

A previous paper that gives links to the present work was written by Longuet-Higgins and Dommermuth [10]. They considered the initial value problem of a spatially periodic wave forced impulsively into motion from an initially horizontal surface (our second subproblem), and arrived at a state of the highest non-breaking wave (approaching our first subproblem).

The first class of flows starting from rest with finite elevation give exact local acceleration fields that satisfy Laplace’s equation. These initial acceleration flows are also valid solutions for Newtonian liquids. Both infinite-depth problems and problems with a horizontal bottom are investigated. The influence of the bottom is remarkably small on the surface flow field with maximum amplitude. The limit shapes for the free surface are almost identical with finite and infinite depth. Initial isobars for given acceleration flows can be reinterpreted as initial free surfaces. Initial dam-breaking can therefore be considered as a class of nonlinear Cauchy–Poisson flows released from rest.

Our analysis confirms the well-known tendency that nonlinear water waves have up-down asymmetries in the vertical direction: the wave crests tend to be sharper than the more rounded wave trough. This asymmetry is well known from Stokes waves, and it is most easily shown by the Lagrangian description of motion. The up-down asymmetry of deep-water waves is a nonlinear phenomenon, since each Fourier mode of a fully linearized wave has perfect up-down symmetry. This up-down asymmetry leads to peaking wave crests. This is the key element of the highest non-breaking waves, both for steady waves [4] and standing waves (Longuet-Higgins and Dommermuth [10]). In the present paper we have looked at three types of highest initial wave: (i) free waves released from rest in an infinite horizontal domain, with or without a horizontal bottom; (ii) maximal initial peak along a uniform slope, representing maximal non-breaking run-up; (iii) maximal wave elevation for free non-breaking sloshing in an open container.

The present work is only concerned with the initial flows in the Cauchy–Poisson problem, and the subsequent nonlinear free-surface evolution will be treated in follow-up papers. Our study of pressure impulses on an initially horizontal surface is therefore limited to linear theory. Dimensional arguments indicate that the free-surface flow will break during the time \(0<t<\sqrt{L/g}\), if the dimensionless pressure impulse \(P_0/(\rho g^{1/2} L^{3/2})\) is considerably greater that one. Here L denotes the length scale of the pressure impulse with amplitude \(P_0\).

We have computed only one case where a pressure impulse works on an already deformed fluid surface. The nonlinear effects of pushing an initial crest with a pressure impulse are quite large, in comparison with applying the same pressure impulse on an initial trough.

The theory of breaking deep-water waves was pioneered by Longuet-Higgins and Cokelet [9]. Their theory is related to the general nonlinear CP problem, since the waves could not break without applying a surface forcing on a surface that already has a finite deformation. The asymmetry required for initiating the gravitational breaking process in a semi-infinite fluid domain needs an external pressure on the surface, involving both the mechanisms of our nonlinear Cauchy–Poisson problem.

Water wave theory is somewhat biased to the extent that mathematical existence of solutions is given higher priority than asking for initial causes and their evolving effects. The search for water-wave instabilities has taken focus away from nonlinearities in the initial conditions. Everybody accepts the importance of nonlinearities in stability theory, since linear theory does not allow the instability concept at all. The fact that nonlinearity dominates non-causal instability theory of arbitrary disturbances, makes it paradoxical that nonlinearities in a chosen set of initial conditions have never occupied their legitimate position in the theory of water waves. The present work takes a modest step in that direction.

Notes

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Authors and Affiliations

  1. 1.Faculty of Mathematical Sciences and TechnologyNorwegian University of Life SciencesÅsNorway

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