The Boundary Value Problem for the Nonlinear Shallow Water Equations

The Boundary Value Problem for the Nonlinear ShallowWater Equations

By Matteo Antuono and Maurizio Brocchini

We propose an approximate analytical solution of the boundary value problem(BVP) for the nonlinear shallow waters equations. Our work, based on the Carrierand Greenspan [1] hodograph transformation, focuses on the propagation ofnonlinear nonbreaking waves over a uniformly plane beach. Available resultsare briefly discussed with specific emphasis on the comparison between theInitial Value Problem and the BVP; the latter more completely representingthe physical phenomenon of wave propagation on a beach. The solution of theBVP is achieved through a perturbation approach solely using the assumptionof small waves incoming at the seaward boundary of the domain. The mostsignificant results, i.e., the shoreline position estimation, the actual wave heightand velocity at the seaward boundary, the reflected wave height and velocity atthe seaward boundary are given for three specific input waves and comparedwith available solutions.

1. Background

Since the pioneering work of Carrier and Greenspan [1] analytical solutionsof the “beach problem,” i.e., NSWE forced by a constant term representinga uniformly sloping beach, have proven both very useful/attractive and,

Address for correspondence: M. Antuono, D.I.Am., Universita di Genova, via Montallegro 1, 16145Genova, Italy; e-mail: [email protected]

STUDIES IN APPLIED MATHEMATICS 119:73–93 73C© 2007 by the Massachusetts Institute of TechnologyPublished by Blackwell Publishing, 350 Main Street, Malden, MA 02148, USA, and 9600 GarsingtonRoad, Oxford, OX4 2DQ, UK.

74 M. Antuono and M. Brocchini

Figure 1. Sketch of geometry and flow variables for the beach problem.

unfortunately, very elusive. We here provide a further contribution to thisfascinating research by inspecting in detail the fundamental features of theboundary value problem (BVP).

For wave fronts approaching a uniformly sloping, frictionless beach with asmall incident angle [2, 3], the onshore problem of weakly two-dimensionalnonlinear shallow water Equation (NSWE) (in dimensionless form) is{

ηt + [u(η + h)]x = 0,

ut + uux + ηx = 0,(1)

in which u is the onshore velocity, η is the free surface elevation, h = −x isthe still water depth, d = η + h is the total water depth and subindices givepartial differentiation. The axes origin is posed at the undisturbed shoreline;the x-coordinate gives the onshore direction and points in the landwarddirection, the y-coordinate represents the longshore direction, (x , y, z) forminga right-handed Cartesian reference frame (see Figure 1). The scale factors forvertical lengths, horizontal lengths, and times are respectively the still waterdepth at the seaward boundary h0 = h(x = −1), h0/s and t0 = s−1√h0/g, sbeing the beach slope. More details can be found in Brocchini and Peregrine [3].

The characteristic form of the hyperbolic set (1) is

d

dt(2c + u + t) ≡ dα

dt= 0 along curves such that

dx

dt= u + c,

d

dt(2c − u − t) ≡ dβ

dt= 0 along curves such that

dx

dt= u − c,

(2)

in which c = √d = √

η − x . The curves in (2) giving the directions ofintegration in the (x , t) plane are said “characteristic curves” while α and β arethe so-called Riemann invariants since they do not change their value movingalong the characteristic curves. Assuming the flow to be subcritical (i.e.,u + c > 0 and u − c < 0), the former type of characteristic curves propagatesignals toward the shore while the latter ones toward the deep waters. Since we

BVP for the NSWE 75

are interested in studying the flow motion only in the sloping region (seeFigure 1), we call the first ones “incoming” and the latter ones “outgoing.”Note that by defining the shoreline as the curve along which d = 0, that isc = 0, all characteristic curves coincide along this special curve except whenbores meet the shoreline, a case which is not considered here.

The major problem being that of solving the NSWE (2), various approacheshave been tried to handle nonlinearities: by transferring them from the equationsto a suitable coordinate transformation [1], by considering linearized BVPs [4]and linearized Initial Value Problems (IVPs) [5].

Basis of all the approaches is the analytical solution for the onshorepropagation of non-breaking waves [see the first two equations in (2)] givenby Carrier and Greenspan [1], hereafter CG, by means of the followinghodographic transformation:

{λ = α − β = 2(u + t),

σ = α + β = 4c,⇒

t = 1

2λ − u,

x = η − 1

16σ 2,

(3)

and use of the pseudo-potential

u = φσ

σ. (4)

[Note that (i) σ = 0 corresponds to c = 0 and, thus, represents the shoreline,(ii) the (x , t) → (σ , λ) transformation is single-valued only when the JacobianJ = c(t2

σ − t2λ ) does not vanish.]

The CG approach makes the first-order nonlinear equations of the onshoreproblem collapse in the following second-order linear equation:

σφλλ = (σφσ )σ . (5)

Solutions can be found for given potentials which satisfy (5) and specificinitial/boundary conditions. They all prescribe:

η = 1

4φλ − 1

2u2. (6)

The most notable solutions, used both to investigate the physics and tovalidate numerical results, are the standing and transient wave CG solutions,the solitary wave solution of Synolakis [4] and the leading-depression N-wavesolution of Carrier et al. [6]. Apart from those who solve an IVP (which canbe solved exactly [6]), up to now the BVP solutions are obtained with the helpof the linear shallow water equations [4, 7].

Our aim is to obtain a BVP solution without using linear theory.Hereafter the BVP solution of the NSWE denotes a solution in the domain(x, t) ∈ (−1, ∞) × R obtained by assigning data at the seaward boundary ofthe domain, i.e., at x = −1, ∀t ∈ R.


Sections 2 and 3 discuss the main conceptual problems related with assigningdata at the seaward boundary of the chosen domain. Differences between theBVP and the more standard IVP are highlighted in the fourth section, while thefundamental data assignment in the transformed (σ , λ)-plane is discussed inSection 5. The general solution procedure and examples of application to themost-known available solutions are proposed in final section which is closedby a short summary of the work.

2. The assignment problem

Notwithstanding its simplicity, the resolution of the BVP for Equation (5) isnot straightforward. The first problem concerns the transfer of the boundarydata from the (x , t)-plane to the (σ , λ)-plane. Indeed Equations (3), linkingthe mentioned pairs of variables, are nonlinear. Moreover they depend on thesolution itself; this means that if we want to carry data in the (σ , λ)-plane andthen solve the BVP, we already have to know the solution. This seems aninsuperable problem. However, to go round it, Synolakis [4] has assumed η

and u to be small and then has linearized (3) to get t � λ/2 and x � −σ 2/16.Such approximate relationships provide a good starting point for the

resolution of the BVP. In fact, at the seaward boundary of the domain, theassumption of small η and u is not restrictive (see Section 4 for a moredetailed analysis) and allows to carry the boundary data in the (σ , λ)-plane. Itis clear that inaccuracies are made when using such approximations; howeverwe provide a method to control and eventually avoid them in the next section.

Let us assume to have successfully carried the boundary data in the(σ , λ)-plane. The seaward boundary of the domain is x = −1, whichcorresponds to σ = 4 by means of x � −σ 2/16. Until now the really unsolvedproblem is the assignment of such boundary data. We show a simple exampleto understand it.

The only bounded solution of (5) is given by

φ(σ, λ) =∫

R

A(ξ )J0(ξσ )eiξλ dξ, (7)

in which J m is the first kind Bessel function of the mth order. Note that onlyone datum can be assigned since the only function which must be definedis A(ξ ). Now assume to know the boundary datum f (t) = u(−1, t). Sincet � λ/2, we obtain f (λ/2) = u(−1, λ/2) in the (σ , λ)-plane. Using (4) andimposing the boundary datum, we have

F[ f (ξ )] = −π

2ξ A(ξ )J1(4ξ ), (8)

in which F denotes the Fourier transform. We need an explicit expression forthe function A(ξ ) to solve the problem. We, thus, would like to write:

BVP for the NSWE 77

A(ξ ) = − 2

πξ

F[ f (ξ )]

J1(4ξ ). (9)

Unfortunately, such expression has no meaning: the function A(ξ ) defined by(9) is unbounded and not integrable since J 1(4ξ ) has a numerable set of zeros.Similar problems also arise when trying to assign f (t) = η(−1, t).

Such difficulty in the assignment of the boundary data for the BVP hasforced many researchers to use the linear shallow water equation or to combinelinear and nonlinear theory together [4, 7].

The proper way to solve the assignment problem without using the lineartheory at all is not to wonder how to assign the boundary datum but to wonderwhich boundary datum has to be assigned. We show in the following sectionsthat the proper assignment is given not on η nor on u but on the incomingRiemann variable α.

3. The boundary datum α

Before introducing the solution procedure, based on the CG hodographictransformation, for the NSWE-BVP we inspect in detail the features of theproblem, for the geometric configuration of Figure 1, by looking at thecharacteristics paths.

We want to solve the BVP, that is, we want to find a solution of the NSWE inthe domain (x, t) ∈ (−1, ∞) × R by assigning data at the seaward boundaryof the domain, i.e., at x = −1, ∀t ∈ R. We prove that, once the solution isimposed to be bounded, we can solve the BVP assigning only one datum at theseaward boundary: α(−1, t) ∀t ∈ R.

The incoming characteristics travel from the seaward boundary up to theshoreline carrying the datum α; at the shoreline it is c = (α + β)/4 = 0 andthe incoming characteristic curves coincide with the outgoing ones. Thus, atthe shoreline it is β = −α; this implies that each incoming characteristic,carrying the specific value α0 and the related outgoing characteristic, carryingthe specific value β0 = −α0, can be regarded as branches of a unique curvewhich goes from the seaward boundary till the shoreline and then goes backoutside the domain (see Figure 2). This suggests use of one single datum at theseaward boundary, i.e., α, regarding β as an unknown function of such datum.Given α(−1, t), we can theoretically know α(x , t), β(x , t) and then η(x , t) andu(x , t) by use of (2); the BVP is then solved. Since α is associated with theincoming waves, this also means that the BVP is completely determined bythem. On the contrary, β is related to the reflection of the incoming waves.From a physical point of view, this means that α is carrying informationcoming from the offshore region with uniform depth into the sloping region,while β is carrying information on the presence of the shoreline out of the


Figure 2. Example of characteristics (thin lines) collapse at the shoreline (thick line). Periodicwave described in Section 6.3.3 with ε = 0.05 and ω = 1.

sloping region. When we try to assign η or u we cannot distinguish between theincoming or the reflected component of the datum; this causes the assignmentproblem which has been treated in Section 2. We can see it by means of thefollowing example. Consider a simple solution of (5):

φ(k)(σ, λ) = AJ0(kσ )eikλ with k ∈ R. (10)

Using (4), we obtain:

u(k)(σ, λ) = −k AJ1(kσ )

σeikλ with k ∈ R. (11)

This is a standing wave solution. However, using the third kind Bessel functionsH (1)

1 , H (2)1 , the Bessel function J1 can be split in two components [8] to give:

u(k)(σ, λ) = −k AH (1)

1 (kσ )

2σeikλ − k A

H (2)1 (kσ )

2σeikλ. (12)

The previous two signals are both solutions of (5) and can be theoreticallyassociated one to an incoming wave and the other to a reflected one. If, using(11), we assign the following boundary datum u(4, λ) = U 0eikλ, we obtain:

−kA

4J1(4k) = U0. (13)

Unfortunately, some special values k = k0 can exist such that J 1(4 k0) = 0 (notethat such value is not unique). In this case the equality in (13) is never fulfilled

BVP for the NSWE 79

except for U 0 = 0. From a physical point of view this means that we are tryingto assign a datum at a node of the standing wave (11), that is, at a point whereu ≡ 0. On the contrary, the two signals in (12) are never simultaneously equalto zero [8]. In other words even if u(4, λ) ≡ 0, we can always find an incomingwave and a reflected one. In this special case they are out of phase just at σ = 4.

Previous results imply that u is not the right choice for the assignment.In fact, from a physical point of view we cannot adequately characterizethe incoming and reflected waves while from a mathematical point of viewwe loose the uniqueness of the BVP solution. Indeed the boundary datumu(4, λ) ≡ 0 gives many possible solutions:

u(σ, λ) =∞∑

n=0

BnJ1(knσ )

σeiknλ ∀kn | J1(4kn) = 0, (14)

one for each choice of the coefficients {Bn}. A similar argument holds for η;then the assignments on η and on u are both ill posed.

4. The BVP versus the IVP

As already mentioned, solutions of the IVP based on Equation (5) are available.For example, Carrier et al. [6] solve the problem by assigning two independentdata on φ(σ , 0) and φλ(σ , 0) for σ > 0. This represents the fundamentaldifference between the IVP and the BVP: both in the (x , t) and (σ , λ)-plane,two data are needed for the IVP while, once the solution is imposed to bebounded, only one is required (α at the seaward boundary) for the BVP one.

In addition there are numerous mathematical problems concerning theassignment of the data both in the (x , t) and (σ , λ)-plane. For example, whensolving an IVP, data must be assigned on x ∈ R or, in our case, at least on x < 0.The problem is that the sloping region is limited at x = −1 and for x < −1 wehave a region with constant depth in which the CG transformation does nothold. We are then forced to choose data which have a limited support or whichare very small outside of a limited space interval; that is, we are forced tochoose pulse-like waves.

Solutions of the IVP for the beach problem configuration are only possibleif simplifications, such as the x � −σ 2/16 linearization proposed in [4], areused to carry data from the (x , t)-plane to the (σ , λ)-plane. It is simple tosee that, using such approximation, data can only be assigned for x < 0(i.e., we are forced to start with conditions of no motion). Moreover thementioned simplification is based on the assumption that −σ 2/16 � η, whichis only true near the seaward boundary. Therefore a simplified transformationis used which is not uniform and characterized by a decreasing accuracy whilemoving shoreward.


The BVP has also to be preferred to the IVP since it can be easily matchedto laboratory and field experiments where data are available at the seawardboundary at all times. On the contrary, the IVP would require knowledge ofthe initial flow field all over the spatial domain.

These considerations support our choice to study the NSWE-BVP. In thefollowing we use a perturbation approach which helps clarify that, at each orderof approximation, the method allows for an appropriate data transfer from the(x , t)-plane to the (σ , λ)-plane one and allows for the solution of the problem.

5. Assigning the data in the (σ,λ) plane

The main problem in handling the hodographic transformation is represented,as rightly pointed out in [5], by the difficulty of carrying the boundary datafrom the (x , t)-plane to the (σ , λ)-plane. Unfortunately when solving theNSWE-BVP one further difficulty arises.

Generally boundary data (that is, data evaluated at x = −1) are known andgiven in terms of the primitive variables η and u rather than in terms of thederived variable α. Once η and u are known, the problem arises of how totreat two data while, on the contrary, the BVP can satisfy only one datum.Moreover, we know that the assignments on η and u are both ill posed sincethey cannot properly take into account the incoming and the reflected wavecomponents. Therefore, we have to find a new variable that can adequatelyrepresent the BVP and, at the same time, has a well-posed assignment. Asdiscussed in Section 4, the appropriate datum to assign seems to be α. In thefollowing, we show that this is indeed the right choice.

First, we evaluate α using η and u. In this way we can isolate the incomingsignal hidden in the primitive variables η and u. Now a further problem arises.Let us assume to solve the BVP using the datum α evaluated through η and u.Once the solution is known, we can also obtain β at the seaward boundary as aresult of the solution itself . Unfortunately, such a value can be different fromthe value of β obtained directly using η and u. By (2) such statement meansthat the computed η and u at the seaward boundary can be different from thestarting data η and u. This can seem strange but it is not. From a mathematicalpoint of view, this is a consequence of using only one datum (α) instead of twodata (starting η and u).

The right way to make the BVP coherent is to introduce some modifieddata η and u such that the incoming component of the starting data η and uis preserved and such that they agree with the BVP solution at the seawardboundary.

To do this, we first assume the starting boundary data to be small and,consequently, define η = εη0 and u = εu0 with η0, u0 known functions of tand ε � 1 a perturbation parameter. Typically ε = H/h0 in which H is

BVP for the NSWE 81

the wave height and h0 is the still water depth both evaluated at the seawardboundary of the domain, that is, at x = −1. We then compute α from the dataand expand it in series of ε:

α = 2√

1 + εη0 + εu0 + t = (2 + t) + ε(u0 + η0) − ε2 η20

4+ O(ε3)

= (2 + t) + εα1 + ε2α2 + O(ε3). (15)

In the same way, we can obtain β and then β1, β2 but, generally, such valuesare not compatible with the solution of the problem.

To solve the problem, we proceed as follows. We modify the data to preservethe value of α and, at the same time, to obtain a suitable value of β, that is, avalue which coincides with that of β obtained through the BVP solution. Weindicate the modified data with η and u and assume the following expansion inseries of ε:

η(t) = εη0(t) + ε2η1(t) + O(ε3), u(t) = εu0(t) + ε2u1(t) + O(ε3). (16)

Substituting (16) into the expression of α and β, we obtain:

α = (2 + t) + ε (η0 + u0) + ε2

(η1 + u1 − η2

0

4

)+ O(ε3), (17a)

β = (2 − t) + ε (η0 − u0) + ε2

(η1 − u1 − η2

0

4

)+ O(ε3). (17b)

Since we want to preserve the incoming datum, we have to pose αn = αn ∀n ∈ N,hence obtaining the following relations:

O(ε) (η0 + u0) = α1, (18a)

O(ε2) (η1 + u1) = α2 + η20

4. (18b)

Both η and u are unknown which are to be found by solving the BVP. Oncethe BVP is solved, they will be regarded as the physical data of the problem.

We now are ready to tackle the problem of transferring the data from the(x , t)-plane to the (σ , λ)-plane. We evaluate the first equation of (3) at theseaward boundary obtaining t = λ/2 − u with u given by (16). Then we lookfor a solution of the form t = t0 + εt1 + ε2t2 + O(ε3). We obtain

t0 + εt1 + ε2t2 = λ

2− εu0

(t0 + εt1 + O(ε2)

) − ε2u1(t0 + O(ε)) + O(ε3).

Using a Taylor expansion and matching the orders of magnitude, we find:

t(λ) = λ

2− εu0

∣∣λ/2

+ ε2(u0˙u0 − u1)

∣∣λ/2

+ O(ε3), (19)


and then:

u(λ) = εu0

∣∣λ/2

− ε2(u0˙u0 − u1)

∣∣λ/2

+ O(ε3), (20a)

η(λ) = εη0

∣∣λ/2

− ε2(u0˙η0 − η1)

∣∣λ/2

+ O(ε3), (20b)

the dot indicating the t-derivative. In the same way we obtain:

α(λ) =(

2 + λ

2

)+ εη0|λ/2 + O(ε2), β(λ) =

(2 − λ

2

)+ εη0|λ/2 + O(ε2).

(21)

For the sake of brevity, we define:

ηn|λ/2 ≡ ηn(λ), un|λ/2 ≡ un(λ), n = 0, 1. (22)

Note that in (19) and (20 a,b) some t-derivatives of η0(t) and u0(t) are evaluatedat λ/2. We want to express such t-derivatives as λ-derivatives of the η0(λ) andu0(λ) defined in (22). Then, since[

d f (t)

dt

]∣∣∣∣t=λ/2

= 2d

dλf

(λ

2

), (23)

we can rewrite (19) and (20 a,b) as follows:

t(λ) = λ

2− εu0 + ε2(2u0

˙u0 − u1) + O(ε3), (24)

and then

u(λ) = εu0 − ε2(2u0˙u0 − u1) + O(ε3), (25a)

η(λ) = εη0 − ε2(2u0˙η0 − η1) + O(ε3), (25b)

the dot indicating the λ-derivative and ηn, un given by (22). It is also easy toprove that relations (18a, b) can be transported in the (σ , λ)-plane by simplyevaluating them at t = λ/2. For the sake of brevity in the following weuse the same notation to indicate the same function evaluated at either t orλ/2. Moreover, using α and β, we can obtain an expression for the seawardboundary of our domain:

σsea ≡ α + β = 4 + 2εη0 + O(ε2). (26)

It is simple to see that σ sea is not constant but is a function of λ, i.e., itrepresents a curve in the (σ , λ)-plane which depends on the solution of theBVP itself. Since the shoreline corresponds to σ = 0, the starting domain(x, t) ∈ (−1, +∞) × R becomes (σ, λ) ∈ (0, σsea) × R and the data have to be

BVP for the NSWE 83

assigned along σ = σ sea. Since we have used a perturbation expansion for theboundary data, we also have to assume:

φ = εφ(0) + ε2φ(1) + O(ε3), (27)

to associate the boundary data to the right order solution. Inspection of (18a, b),suggests the correct way to solve the problem. Using (4) and (6), we consider:

η + u = φλ

4

∣∣∣∣σsea

+ φσ

σ

∣∣∣∣σsea

− u2

2. (28)

Using (26) and expanding (28) in series of ε, we obtain:

O(ε)(φ

(0)λ + φ(0)

σ

)∣∣σ=4

= 4α1, (29a)

O(ε2)(φ

(1)λ + φ(1)

σ

)∣∣σ=4

= 4 (α2 + N ) , (29b)

in which

N = η20

4− 2u0α1 + u2

0

2− η0

2

[φ

(0)σλ + φ(0)

σσ − φ(0)σ

4

] ∣∣∣∣∣σ=4

. (30)

Note that with the perturbation approach we can make a suitable dataassignment along σ = 4 at each and all order of approximation required.

6. Solutions

Now we can solve the problem. The domain is (σ, λ) ∈ (0, 4) × R and the dataare assigned at σ = 4. At each order, the bounded solution of (5) is

φ(n)(σ, λ) =∫

R

An(ξ )J0(ξσ )eiξλ dξ, (31)

in which J m is the first kind Bessel function of the mth order. We, thus, obtain(φ

(n)λ + φ(n)

σ

)∣∣σ=4

=∫

R

iξ An(ξ ) [J0(4ξ ) + i J1(4ξ )] eiξλ dξ. (32)

The first- and second-order solutions are, respectively:

A0(ξ ) = − 2i

πξ

F(α1)

[J0(4ξ ) + i J1(4ξ )], A1(ξ ) = − 2i

πξ

F(α2) + F(N )

[J0(4ξ ) + i J1(4ξ )], (33)

in which F represents the Fourier transformation and N is known once we solvethe first-order problem. The first-order solution also allows for computation ofu0 and η0. We find:

u0(σ, λ) = φ(0)σ

σ, u1(σ, λ) = φ

(1)σ

σ, (34a)


η0(σ, λ) = φ(0)λ

4, η1(σ, λ) = φ

(1)λ

4− u2

0

2. (34b)

Then using (26) and the previous expressions, we, finally, obtain an explicitsolution for the “modified” data η0(λ) and u0(λ) defined in (22): we haveη0(λ) ≡ η0(4, λ) and u0(λ) ≡ u0(4, λ). At this step, we can also verify whetherthey are O(1) (i.e., if our perturbation method is consistent with the initialassumption).

We can also simplify the expression of N . In fact, we have[φ

(0)σλ + φ(0)

σσ − φ(0)σ

4

] ∣∣∣∣∣σ=4

=∫

R

ξ 2 A0(ξ )

[J1(4ξ )

2ξ− J0(4ξ ) − i J1(4ξ )

]eiξλ dξ.

Then after lengthy computations, we get[φ

(0)σλ + φ(0)

σσ − φ(0)σ

4

] ∣∣∣∣∣σ=4

= −2u0 + 4α1,

and, therefore

N (λ) = α21

2− 2α1α1 − η2

0

4. (35)

Now we can give an estimate of the shoreline position. We recall that theshoreline is represented by σ = 0 and note that along this curve it is x = η.Then we can write:

xs(λ) = εx (1)s (λ) + ε2x (2)

s (λ) + O(ε3) = εη0(0, λ) + ε2η1(0, λ) + O(ε3), (36)

in which

η0(0, λ) = 1

2π

∫R

F(α1)

[J0(4ξ ) + i J1(4ξ )]eiξλ dξ,

η1(0, λ) = 1

2π

∫R

F(α2) + F(N )

[J0(4ξ ) + i J1(4ξ )]eiξλ dξ

− 1

2

[1

π

∫R

iξF(α1)

[J0(4ξ ) + i J1(4ξ )]eiξλ dξ

]2

.

Some observations are now needed:

1. the first is concerned with Equation (19). If the derivative of u0 is large (forexample, ˙u0 = O(ε−1)), the perturbation expansion is not uniform and failsin the evaluation of the order of magnitude. Actually this is not a problemsince if the derivative of u0 is large, breaking occurs and the CG hodographtransformation cannot be used;

2. many authors (e.g. [4]) have linearized the expressions in (3) assumingt � λ/2 (for the BVP) and x � −σ 2/16 (for the IVP) to transfer datafrom the (x , t)-plane to the (σ , λ)-plane. Both the previous approximations

BVP for the NSWE 85

lead to neglect terms which are O(ε) inside the transformation itself; it issimple to show that this brings in second-order errors. On the contrary, theproposed method is correct up to the desired order of computation.

6.1. Further results

Now we can prove xs = us to hold up to the second order (us is the onshorevelocity at the shoreline). Since we have u(0, λ) ≡ us(λ), we put:

us(λ) = εus,0(λ) + ε2us,1(λ) + O(ε3) with un(0, λ) = us,n(λ). (37)

Then, evaluating the first equation of (3) at the shoreline, we obtain

t(λ) = λ

2− εus,0(λ) + O(ε2). (38)

Now we have to compute the t-derivative of the shoreline position. Using (38)we can write the following differential operator:

d

dt= t−1 d

dλ= 2

(1 + 2ε ˙us,0 + O(ε2)

) d

dλ. (39)

We also observe the following equality to hold:(φ

(n)λλ

4

) ∣∣∣∣∣σ=0

=(

un

2

) ∣∣∣∣σ=0

≡(

us,n

2

). (40)

Then applying (39) to (36) and using (40), the thesis is proved immediately.

6.2. The inverse transformation

Once solved the problem, we want to go back to the (x , t)-plane. We call“inverse” the transformation which enables us to carry the solution to the(x , t)-plane. Such a transformation is very simple at the seaward boundary ofthe domain and at the shoreline. We start from the usual equation:

t = λ

2− εu0(σ, λ) + O(ε2). (41)

If we want to achieve the transformation at the seaward boundary, we have toevaluate (41) along σ = σ sea obtaining:

t = λ

2− εu0(4, λ) + O(ε2). (42)

Then we search for a solution of the form λ = λ0 + ελ1 + O(ε2) and, finally,we obtain:

λ(t) = 2t + 2εu0(4, λ)|λ=2t + O(ε2). (43)

In the same way we can evaluate the transformation at the shoreline. We have

λ(t) = 2t + 2εu0(0, λ)|λ=2t + O(ε2). (44)


A simple example illustrates use of the transformation: we want to evaluatexs(t). We know xs(λ) from (36); then we can use (44) and expand in ε series.We get

xs(t) ≡ xs(λ)|λ(t) = εx (1)s (λ)|2t + ε2

[x (2)

s (λ) + 2u0(0, λ)x (1)s (λ)

]∣∣2t

+ O(ε3).(45)

From Section 6.1 we have 2x (1)s (λ) = u0(0, λ) = us,0(λ). Then Equation (45),

finally, becomes

xs(t) = εx (1)s (2t) + ε2

[x (2)

s (2t) + u2s,0(2t)

] + O(ε3), (46)

the dot indicating the t-derivative.Note that all the inverse transformations can be used only when waves are

far from breaking. Since we have assumed the boundary data to be small, wecan reasonably take the waves to be non-breaking far from the shoreline. Onthe contrary, we can have waves close to breaking at the shoreline, since theysteepen as they travel shoreward. Then, let us consider the first-order shorelinetransformation:

t = λ

2− εus,0(λ). (47)

The inversion is possible only when the λ-derivative of (47) is different fromzero, that is, when

dt

dλ= 1

2− ε

dus,0(λ)

dλ> 0. (48)

It is simple to prove that this expression represents the first-order breakingcondition (J = 0) for waves at the shoreline. When waves are close to breaking(48) is almost zero and this implies that dus,0/dλ = O(ε−1). This means thatwe cannot use the perturbation approach to invert relation (47) since it canintroduce large errors in the computation. In this case we can use numericalmethods such as, for example, the Newton method.

6.3. Examples

We briefly describe some results obtained through numerical computations ofsecond-order solutions for specific inputs of interest, such inputs are, typically,used in various similar analyses [4–6].

We are interested in the case of waves propagating from the constant-depthsshelf to the sloping region. In our specific case we consider small waves ofpermanent shape propagating shoreward. Outside the sloping region such wavesare associated with functions depending on the composite variable (x − t)and, at the first order of approximation, it is η = u (e.g. [9]). Then at theseaward boundary of the sloping region, we can assume η0 = u0. This relation

BVP for the NSWE 87

is very useful since it allows to compute α using just η0 or just u0 (note thatthis is the unique case in which we can do it).

In the following, we denote η0 = ηI0 and u0 = uI

0 to underline that η0 and u0

are associated with the incident waves. As incident waves enter the slopingregion, they are reflected. Then the final solutions η and u are, generally,given by a nonlinear superposition of incident and reflected waves. Suchsuperposition is simple to see at the seaward boundary where the incomingdata are just known by the assignment. Then at the seaward boundary wedefine components associated with the reflected waves by ηR = η − ηI andu R = u − uI at each ε order.

6.3.1. The gaussian wave. This is the most frequently used shape to evaluatethe propagation and run-up of tsunami wave. Recent applications cover the caseof landslide-generated tsunami [10]. We choose the following gaussian shape:ηI

0 = exp (−k2t2), with k = 0.6. The parameter k controls the steepness of thedatum (Figure 3c); if the datum is too steep the wave breaks and the previousanalysis fails. Figures 3e and f show η and u together with the incident wavedata εηI

0 and εuI0 at the seaward boundary. It is clear the superposition of

reflected waves (Figures 3g and h) with the incident ones to give the actual dataη and u. It is also evident the excellent match between εηI

0, εuI0, and η, u as

the gaussian wave starts entering the sloping region. This confirms the valueof the initial assumption ηI

0 = uI0 at the seaward boundary. Figure 3a shows the

shoreline position; its shape is coherent with the main results available inthe literature [5, 6]. The maximum run-up is approximately 1.5 times theamplitude of α at the seaward boundary. Also note that the reflected waves reachthe seaward boundary about four time units after the incident waves have leftthe seaward boundary. This is also true for the N-wave and the periodic wave.

6.3.2. The N-wave. The leading depression N-wave shape is, typically, usedto represent a seismic fault dislocation by subduction earthquake [11, 12].We choose the following N-wave shape: ηI

0 = exp(−k2(t − 2.5)2) − 0.5×exp(−k2t2), with k = 0.4, i.e., moderately flatter than the chosen gaussianwave. Results, reported in Figure 4 with the same style used for Figure 3,show qualitatively similar behaviors which are typical of pulse-type signals.

6.3.3. Periodic waves. The proposed model can properly describe the behaviorof solitary waves. The regularity assumptions force a choice of smooth data inL1 even if, using a distributional approach, one can assign periodic data, thatis, data which are in L∞. Obviously when we solve the BVP, we have to use aFourier series instead of a Fourier transform. We choose the following simplewave shape: ηI

0 = cos (ωt), with ω = 1. The latter constraint guaranteesthat the wave does not break. Results pertaining to this case are reported,with the same graphical arrangements of Figures 3 and 4, in Figure 5. Themost important variable, i.e., the shoreline position, of panel a) reveals that the


Figure 3. Gaussian wave solution up to the second order for ε = 0.05 and k = 0.6: (a) theshoreline position xs, (b) the shoreline velocity us . Signals at the seaward boundary of thedomain are: (c) α − (2 + t), (d) β − (2 − t), (e) η (solid line) and εη I

0 (dashed line), (f) u(solid line) and εuI

0 (dashed line), (g) the reflected elevation ηR , (h) the reflected velocityuR.

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Figure 4. N-wave solution up to the second order for ε = 0.05 and k = 0.4: (a) the shorelineposition xs, (b) the shoreline velocity us . Signals at the seaward boundary of the domain are:(c) α − (2 + t), (d) β − (2 − t), (e) η (solid line) and εη I

0 (dashed line), (f) u (solid line)and εuI

0 (dashed line), (g) the reflected elevation ηR , (h) the reflected velocity uR.


Figure 5. Periodic wave solution up to the second order for ε = 0.05 and ω = 1: (a) theshoreline position xs, (b) the shoreline velocity us . Signals at the seaward boundary of thedomain are: (c) α − (2 + t), (d) β − (2 − t), (e) η (solid line) and εη I

0 (dashed line), (f) u(solid line) and εuI

0 (dashed line), (g) the reflected elevation ηR , (h) the reflected velocityuR.

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Figure 6. Comparison between first-order (dashed lines) and second-order (solid lines)solution for the gaussian wave: (a) shoreline position, (b) η at the seaward boundary.

Table 1Comparison between the Solitary Wave Maximum Run-up as Obtained

Through Laboratory Experiments [4, 7] and Through Our Model. The Symbol1/s Represents the Inverse of the Beach Slope

Laboratory Present model1/s ε experiments results

3.732 0.050 0.173 0.1462.747 0.050 0.115 0.1302.747 0.098 0.252 0.3032.080 0.026 0.050 0.0572.080 0.063 0.148 0.1582.080 0.071 0.172 0.1822.080 0.089 0.221 0.2402.080 0.108 0.273 0.3032.080 0.113 0.304 0.3261.000 0.060 0.115 0.1291.000 0.100 0.212 0.2281.000 0.200 0.454 0.528

typical behavior of periodic wave washes (i.e., peaked rundowns and morerounded run-ups), is well reproduced by the proposed solution.

To give an immediate and clear estimate of the changes brought by theinclusion of the higher-order contribution, in Figure 6 we show a comparisonbetween the first- and second-order shoreline position and η at the seawardboundary for the gaussian wave. The differences are in the order of few percentand, essentially, lead to corrections in the wave profile.

Finally, in Table 1 we propose a quantitative comparison between the solitarywave maximum run-up obtained through our model and the results (laboratory


experiments) reported in Synolakis [4] and Li [7]. With the exception ofthe mildest-sloping beach, the model overpredicts the experimental data ofabout 10–20%, such overprediction increasing with the nonlinearity ε, hencesuggesting that better quantitative results can be achieved by increasing theaccuracy of the solution.

7. Conclusions

Using the Carrier and Greenspan hodograph transformation, an approximateanalytical solution of the BVP for the NSWE has been obtained. Such asolution is based on a perturbation approach which allows to properly accountfor the wave nonlinearities. The solely hypothesis of small waves at the seawardboundary is far to be restrictive since the Carrier and Greenspan hodographtransformation is limited to the case of non-breaking waves. Since waves growand steepen as they propagate shoreward, non-breaking waves at the shorelinewould imply small waves at the seaward boundary.

All the graphical examples reported and the quantitative computation of themaximum run-up values shown in Table 1 clearly illustrate that the proposedmodel gives excellent results in comparison to classical run-up solutions[4–6], with the advantage of providing solution also for the interaction ofincoming-outgoing waves at the seaward boundary of the chosen domain, thelatter being only possible when setting up and solving the BVP.

Finally, the BVP solution can be easily matched to laboratory and fieldexperiments where data are available at the seaward boundary at all times.

Acknowledgments

Insightful and useful criticism by Prof. D. H. Peregrine is gratefullyacknowledged. The authors thank the E.U. for the partial financial supportreceived through the INTAS Project 06-1000013-9236.

References

1. G. F. CARRIER and H. P. GREENSPAN, Water waves of finite amplitude on a sloping beach,J. Fluid Mech. 4:97–109 (1958).

2. S. C. RYRIE, Longshore motion generated on beaches by obliquely incident bores, J.Fluid Mech. 129:193–212 (1983).

3. M. BROCCHINI and D. H. PEREGRINE, Integral flow properties of the swash zone andaveraging, J. Fluid Mech. 317:241–273 (1996).

4. C. E. SYNOLAKIS, The run-up of solitary waves, J. Fluid Mech. 185:523–545 (1987).

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5. U. KANOGLU, Nonlinear evolution and runup-rundown of long waves over a slopingbeach, J. Fluid Mech. 513:363–372 (2004).

6. G. F. CARRIER, T. T. WU, and H. H. YEH, Tsunami runup and drawdown on a planebeach, J. Fluid Mech. 475:449–461 (2003).

7. Y. LI and F. RAICHLEN, Solitary wave runup on plane slopes, J. Wtrwy., Port., Coast. andOc., Engrg. 127(1):33–44 (2001).

8. M. ABRAMOVWITZ and I. STEGUN, Handbook of Mathematical Functions, Dover publications,Inc., New York, 1964.

9. C. C. MEI, The Applied Dynamics of Ocean Surface Waves, John Wiley and Sons(Current edition: World Scientific Publishing Co. Pte. Ltd., Singapore, 1989), 1983.

10. C. L. MADER, Modeling the la Palma landslide tsunami, Science of Tsunami Hazards19:150–170 (2001).

11. R. KH. MAZOVA and E. N. PELINOVSKY, The increasing of tsunami runup height withnegative leading wave, Proc. IV Intl Symp. on Geophys. Hazards, Perugia, Italy, p. 118(1991).

12. S. TADEPALLI and C. E. SYNOLAKIS, The run-up of N-waves on sloping beaches, Proc. R.Soc. Lond. A. 445:99–112 (1994).

UNIVERSITA DI GENOVA

UNIVERSITA POLITECNICA DELLE MARCHE

(Received December 29, 2006)

The Boundary Value Problem for the Nonlinear Shallow Water Equations

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