# Initial Stage of the Finite-Amplitude Cauchy–Poisson Problem

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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 *x*, *y*, *z* are introduced, where the *z* axis is directed upwards in the gravity field and the horizontal *x*, *y* 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.

*P*(

*x*,

*y*) is forcing the surface into a finite-amplitude flow described by the velocity potential \(\Phi (x,y,\eta _0,0^+)\). The pressure impulse

*P*(

*x*,

*y*) has the dimension of pressure multiplied by time.

*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.

*P*(

*x*,

*y*) received by the surface during the infinitesimal time interval is generally defined by

## 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 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

*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.

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].

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

#### 3.2.1 Peaked Run-up at a Sloping Beach

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

*X*,

*Z*) 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 (

*X*,

*Z*).

*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 (\(z=x\))

*A*and placed in total four dipoles in the points \((-L+\epsilon ,L+\epsilon )\), \((-L-\epsilon ,L-\epsilon )\), \((L+\epsilon ,-L+\epsilon )\), \((L-\epsilon ,-L-\epsilon )\).

*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

*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

*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.

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

*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.

*P*(

*x*,

*y*) 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

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

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

*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

#### 4.2.2 A Superposition of 2D Multipole Fuctions

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.

*X*,

*Z*), 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

*x*and

*X*

## 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.

### 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.

*z*axis, with infinite fluid depth. A dipole potential offers a simple 2D flow

*P*(

*x*) from the simple form \(A/(1 + x^2)\) (at \(z=0\)) to the more complicated expression

*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 \(\epsilon ^1\).

*P*(

*x*) has the approximate shape given by

## 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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