## Abstract

The problem of edge waves as an example within classical water-wave theory is described by presenting an overview of some of the theories that have been offered for this phenomenon. The appropriate governing equations and boundary conditions are formulated, and then the important discoveries of Stokes and Ursell, concerning the travelling edge wave, are presented. (We do not address the corresponding problem of standing waves.) Thus, the linear problem and its spectrum are constructed; in addition, we also present the linear long-wave approximation to the problem, as well as Whitham's weakly nonlinear extension to Stokes' original theory. All these discussions are based on the same formulation of the problem, allowing an immediate comparison of the results, whether this be in terms of different approximations or whether the theory be for an irrotational flow or not.

Gerstner's exact solution of the water-wave problem is then briefly described, together with a transformation that produces an exact solution of the full equations for the edge wave. The form of this solution is then used as the basis for a multiple-scale description of the edge wave over a slowly varying depth; this leads to a version of the shallow-water equations which has an exact solution that corresponds to the edge wave. Some examples of the theoretical predictions for the run-up pattern are presented. We conclude with three variants of nonlinear model equations that may prove useful in the study of edge waves and, particularly, the interaction of different modes.

## 1. Introduction

In 1846, Stokes published his ‘Report on recent researches in hydrodynamics’ (Stokes 1846) which, in six sections, covered (without giving much mathematical detail) a number of aspects of fluid mechanics. The area of interest to us—in the second section—discusses various problems in wave propagation and, in particular, water waves over constant depth and also in a canal with sides inclined at 45°. This latter work was based on Kelland (1844), but Stokes added the comment that a one-sided canal, with the water increasing in depth away from the inclined canal wall, could accommodate propagation *along* the wall and restricted to the neighbourhood *of* the wall. This is the first reference to an edge wave. (This is an example of a ‘trapped wave’ because the amplitude decays rapidly away from the wall.) The essential features of this solution are described in Lamb (1932), although it was Ursell's (1952) work which gave the first complete description of the linear problem (being a mixture of continuous and discrete spectra).

This early work has been followed by many significant extensions. Thus, nonlinear corrections to the linear solution of Stokes have been given by Whitham (1976), and the inclusion of uniform depth at infinity has been added to this by Minzoni (1976). Further, the excitation of standing edge waves by a suitable incident wave normal to the beach has been analysed by Minzoni & Whitham (1977). Other mechanisms for the generation of edge waves are described by Evans (1988). The case of a slowly varying depth has been considered by Miles (1989), although this was restricted to an analysis of the linear problem. The inclusion of longshore currents, again for the linear problem, has been described by Howd *et al*. (1992); this paper includes a lot of general background information and also presents some of the current thinking on the importance of edge waves.

A particularly intriguing result was discovered by Yih (1966) and also by Mollo-Christensen (1982) who explored the nature of the problem a little further, which relates the edge wave to Gerstner's trochoidal wave (Gerstner 1802; Lamb 1932; a rigorous discussion of this solution can be found in Constantin 2001*a*). Indeed, an elementary transformation of coordinates demonstrates that Gerstner's exact solution of the full equations of classical water-wave theory recovers the edge-wave mode obtained by Stokes (although this alternative approach describes a rotational flow field and Stokes' was irrotational). However, this treatment of the problem of edge waves provides the solution (e.g. for the free surface) in an implicit form. The formulation does not produce any directly useful or convenient results, but this situation was remedied by Constantin (2001*b*) who presented an explicit version of the solution; in addition, Constantin also proved that the flow was dynamically possible (because the flow map is a diffeomorphism). Examples of the surface profiles, and of the quite delightful run-up patterns, are readily obtained from this presentation of the problem. More recently, an asymptotic approach for slowly varying depth, incorporating scalings that are consistent with the Constantin–Gerstner solution, was developed by Johnson (2005). This gave rise to a form of the (nonlinear) shallow-water equations which has an exact solution corresponding to the edge wave, but which still retains variable depth. Various modes are readily accessible, as is a representation of the surface profile and an equation that generates a number of versions of the run-up pattern.

In this paper, we will give an overview of some of the methods, and some of the results, that apply to the edge-wave problem. To accomplish this, first we present the formulation of the classical water-wave problem, being careful to make clear the modelling assumptions that are necessary. Further, all the results that we describe will be written in terms of our (standard) formulation of the problem, thereby ensuring that a comparison of the equations, and of the solutions, is altogether straightforward.

Finally, and before we commence this task, it would be wise to indicate why an understanding of this phenomenon, and the mechanisms that can bring it about, is important. Although from Stokes' time to at least the later editions of Lamb's text on hydrodynamics it was generally thought that the edge waves were a mathematical curiosity, this is no longer the case. It has been suggested, and there is some evidence to support this, that many near-shore processes are controlled (or significantly affected) by the presence of edge waves. Thus, beach erosion, together with energy and momentum transfer more generally, is the prime target for the influence of the edge wave (and a standing wave, in particular, is likely to have an observable effect on beach erosion and, perhaps, on the formation of sand bars close inshore). These, and other more practical aspects, are described in Howd *et al*. (1992) and in the many papers cited therein. Two examples of the run-up pattern produced by an edge wave are shown as sketches (loosely based on photographs) in figures 1 and 2; the second sketch, in particular, depicts the dramatic effect of the edge wave—probably a standing wave of some strength—on a shingle beach. Some excellent pictures of the run-up pattern generated by the edge waves can be found in Guza & Inman (1975) and Komar (1998).

## 2. Governing equations

We consider an incompressible inviscid fluid which is bounded above by a free surface (*z*=*h*(*x*, *y*, *t*)) and below by a fixed impermeable bed (*z*=*b*(*x*)) that varies only in the *x*-direction. The waves that we describe are one realization of gravity waves, and so the effects of surface tension are ignored in this model. In its undisturbed state, the free surface is *z*=0, and this intersects *z*=*b*(*x*) along *x*=0 (which is therefore the shoreline and the run-up ‘pattern’ in the absence of waves). The fluid extends to infinity as *x*→+∞ (which is towards the open ocean), otherwise we have −∞<*y*<∞, where *y* is the longshore coordinate. This configuration is shown in figure 3. We introduce a typical (or mean) wavelength, *λ*, of the edge waves and use this as our length-scale throughout; thus, we take as the speed scale (where *g* is the constant acceleration due to gravity). These two scales then define an appropriate time-scale, . The pressure (*P*) is written as the hydrostatic pressure distribution with a correction due to the passage of the wave(2.1)where *ρ* is the constant density of the water; *P*=*P*_{a}=constant is the pressure at the free surface; and *p* is the non-dimensional pressure perturbation.

The Euler equation, the equation of mass conservation and the boundary conditions, written in the corresponding non-dimensional variables are, respectively,(2.2)In addition, we assume that suitable initial data exist which will generate the edge waves. A typical mechanism is via a wind stress—perhaps initiated by a storm that moves along the shoreline (and this can often occur where, just inshore of the beach, there is a cliff face, forcing the wind to blow parallel to the shoreline; this topography is evident in figure 2). Finally, we note that the use of Euler's equation allows for the possibility that the flow field associated with the edge wave is rotational; we will find that some solutions are irrotational, but others are not.

We shall describe, in various forms, the conventional travelling edge wave, even though it is probable that the observed patterns (and almost certainly any erosion) are generated by standing waves; for this important aspect of the problem, see Minzoni & Whitham (1977) and Evans (1988). Thus, we have waves propagating in the *y*-direction with some appropriate period and speed (and, as we shall see, a sloping beach is essential for their existence).

## 3. Stokes' solution

In Stokes' (1846) publication, a seminal observation about the one-sided wave guide is made (which was a special case of Kelland's (1844) linear canal problem), so first we require the linearized version of equations (2.2). These are most conveniently obtained by transforming according to(3.1)to give, at leading order,(3.2a)(3.2b)with(3.3a)(3.3b)and(3.4)Further, following Stokes, we assume here that the flow field is irrotational, which therefore implies that(3.5)the solution that we seek is recovered by writing(3.6)(where *A* is an arbitrary constant), which requires that(3.7)We may elect to use either the real or imaginary part of this solution, or to add the complex conjugate, in order to obtain a real solution. Then from equation (3.2*a*), we obtainand so, from equation (3.3*a*), the surface wave is given byThe boundary condition (3.3*b*) is satisfied ifand, finally, the last boundary condition (3.4) requireswhere *b*′(*x*)=−tan *α*=constant (and 0≤*α*<*π*/2 for a beach), i.e. the bed has a constant slope. Thus, we have a solution (with ) described by(3.8)(3.9)which exists as a wave only if *α*>0, i.e. a sloping beach is essential for the presence of an edge wave. We also confirm that this is a trapped wave by virtue of the exponential decay of the amplitude away from the shore (*x*→+∞). This is the solution obtained by Stokes (although he used *α*=π/4, the slope of the canal wall chosen by Kelland). It should be noted that this formulation has not required the additional assumption of shallow water (or, equivalently, long waves) for which the *z*-momentum equation is replaced by ∂*p*/∂*z*=0; this important special case is discussed in §5.

We have presented Stokes' solution which, we note, is an exact (single-mode) solution of the linearized classical water-wave problem. To take this analysis further and examine the spectral problem in more detail, we now follow the route laid down by Ursell (1952).

## 4. The spectrum

In order to investigate this aspect, the problem defined by equations (3.2)–(3.5) is now solved by writing(4.1)so that(4.2)and(4.3)with(4.4)(4.5a)(4.5b)We already know that a solution of this system (see equation (3.8)) is given bywith (and *A*_{0} is an arbitrary constant), and then equations (4.3) and (4.4) determine, respectively, the pressure (*p*) and the surface profile (*h*). To proceed, we choose to fix *k* (although we could sum/integrate over this parameter and so obtain a more general solution); then we seek a solution of equation (4.2), satisfying conditions (4.5*a*) and (4.5*b*), in the form(4.6)*A*_{mn} are constants and, with *A*_{m0}=0, the choice *n*=0 recovers Stokes' solution. For an edge wave on a conventional beach, we require 0<*α*<*π*/2 and, further, for the construct (4.6) to represent a realizable solution (i.e. the velocity remains finite), we must have(4.7)which determines the maximum *n* for a given *α*.

Expression (4.6) is an exact solution of equation (4.2) and it also satisfies, for arbitrary *A*_{mn}, condition (4.5*b*) (essentially because *E*_{+m}=*E*_{−m} on *z*=−*x*tan *α*). The other boundary condition (4.5*a*) requires(4.8)and then(4.9)

Thus, for any given *n* satisfying equation (4.7), we have a discrete mode and, indeed, a set of *N*+1 discrete modes, where *n*=0 is the Stokes mode (given in equation (3.8)) and *N* is the maximum *n* satisfying equation (4.7). As *α* decreases, more discrete modes become available; furthermore, as expounded in Ursell (1952), there is additionally a continuous spectrum in *k*≤*ω*^{2}<∞, and so the discrete frequencies found above sit outside (below) the continuous spectrum. One upshot of this important work is that waves generated by, for example, a moving pressure distribution on the surface of the water will require a representation based on this mixed—discrete and continuous—spectral problem, not a routine or straightforward calculation. The apparently simple, rather innocuous, solution found by Stokes has turned out to be the tip of a mathematical iceberg.

## 5. Linear long waves

A significant simplification, with important consequences for some of our later discussion, occurs when we impose the assumption of long waves on the linear theory. This also makes transparent, we suggest, some aspects of the spectral problem. Formally, the procedure requires that different length-scales are introduced *ab initio,* one for the horizontal motion and another for the vertical motion. Hence, if the length-scale chosen to define *x* and *y* is much greater than that chosen for *z*—the long-wave or shallow-water approximation—then the linear equations (3.2)–(3.4) reduce to(5.1)to leading order. Here, ⊥ denotes the vector in the plane at right angles to the direction of the *z*-coordinate. The relevant solution takes the form(5.2)where *E*=exp[i(*ky*−*ωt*)], givingwhich satisfies the boundary condition on *z*=0. The final boundary condition on *z*=*b*(*x*)=−*x* tan *α* then yields the equation for *A*(*x*),or(5.3)This can be recast into a standard form by writing *A*(*x*)=e^{−kx}*L*(2*kx*), so that *L*(*Y*) satisfies(5.4)where the prime now denotes the derivative with respect to ,(5.5)Equation (5.4) has as its solution the Laguerre polynomials, *L*_{n}(*Y*), whenever *γ*=*n* (*n*=0, 1, 2, …), and these are the only solutions which ensure that *A*(*Y*) is bounded as *Y*→+∞, i.e. as *x*→∞ (with *k*>0). For these solutions, we see from equation (5.5) that(5.6)which is the analogue, for long waves, of Ursell's condition (4.8); indeed, for *α*→0, they coincide precisely. This connection with the Laguerre equation (5.4) provides a simple structure for the eigenmodes in the case of long waves. However, this close correspondence hides some important differences.

In the case of long waves, the solution described by equation (5.2) represents a rotational flow field; the vorticity isthis, as expected, decays as *x*→+∞ (by virtue of the exponential decay of *A*(*x*)). We also see that the component of the vorticity in the vertical (*z*) direction is zero—the vorticity is restricted to the horizontal plane. It has already been noted that Stokes' solution, and Ursell's extension of it, is irrotational.

Finally, we observe that the eigenfunctions e^{−kx}*L*_{n}(2*kx*) (=*A*(*x*)) form a complete set for any 0<*α*<*π*/2, i.e. there is no continuous spectrum. Indeed, the condition here that corresponds, in Ursell's theory, to ∞>*ω*^{2}≥*k* becomes ∞>*ω*^{2}≥*k* tan *α* with *α*=π/2, which is impossible, thus confirming that no continuous spectrum exists. The long-wave version of the problem has an infinite number of eigenmodes, *n*=0, 1, 2, …, for every *α*∈(0, *π*/2), whereas Ursell's eigenmodes increase in number as *α* decreases.

## 6. The weakly nonlinear edge wave

In 1976, Whitham addressed the problem of finding nonlinear corrections to the Stokes solution; this was accomplished for both the long-wave (shallow-water) and the full equations, under the assumption of irrotationality. Here, we shall outline the solution discussed in Whitham (1976) for the most general case, i.e. the full water-wave equations. The appropriate solution of equations (2.2) is expressed as(6.1)where ** u**=∇

*ϕ*and

*θ*=

*ky*−

*ωt*, with

*ω*=

*ω*(

*k,δ*); correspondingly, we write(6.2)both valid as

*δ*→0 (cf. equation (3.1)). The calculation performed by Whitham was then carried as far as O(

*δ*

^{2}), so that the errors in computing both

*ϕ*and

*h*are O(

*δ*

^{3}). To this end, we find at this order of approximationwith

*F*

_{2}=

*F*

_{3}=0 and

*F*

_{1}a suitable solution of(6.3)Note that the surface boundary condition has been transferred from

*z*=

*h*to

*z*=0, and this is most readily accomplished by invoking Taylor expansions about

*z*=0 (which is a valid manoeuvre here since

*h*=O(

*δ*) and owing to the form of solution that we seek).

The asymptotic solution of equation (6.3), as *δ*→0, can be expressed aswith the choicewhere *γ* is a constant independent of *δ*; we see that this form recovers the Stokes result for *ϕ*, *ω* and *h*, when we set *δ*=0 (e.g. equations (3.8) and (3.9)). A solution for which is no more singular than exp[−*kx* cos *α*] as *x*→+∞ requires that *γ*=1/2 (by imposing an orthogonality condition), and thenbut the term in (*z*+*x* tan *α*) leads to another non-uniformity as |*z*+*x* tan *α*|→∞. However, this can be removed by writing the solution in the formwhich, at this order, produces an altogether satisfactory asymptotic solution for 0<*α*<*π*/2, although it is evident that we cannot allow *α*→*π*/2—but then this does not describe a conventional beach, so it is of little interest here.

## 7. Gerstner's wave and edge waves

In 1802, Gerstner produced the only known, exact non-trivial solution of the full classical water-wave equations (valid for water of infinite depth; see Gerstner 1802, Lamb 1932 and Constantin 2001*a*). We shall provide a brief description of this solution, and then show how it can be interpreted to produce a corresponding exact solution (of the full equations) for the edge waves. We consider one-dimensional waves that are propagating in the *x*-direction, with some appropriate behaviour in the *z*-direction; the governing equations are therefore the set (2.2), but with *y* and *v* absent—so ** u**=(

*u*,

*w*) with

**=(**

*x**x*,

*z*)—and the bottom condition replaced by

*w*→0 as

*z*→−∞.

Gerstner's solution is expressed in terms of particle paths, i.e. it is a Lagrangian formulation written as(7.1)where *a*, *b*, *k* and *c* are constants (and the pair (*a*, *b*) is used to identify the initial position of the particle—but note that this position is not simply (*a*, *b*)!). The acceleration of the particle can be computed asand then the non-dimensional pressure perturbation is found as(7.2)Thus, the pressure (equation (2.1)) along a particle path, which will include the surface, is constant ifis constant, which is possible only if *c*^{2}=1/*k*. Furthermore, it is easily confirmed that the equation of mass conservation is satisfied in the formexpressed in Jacobian form, with and . Thus, we do indeed have an exact solution, which describes the surface profile as a trochoid (or a cycloid if the free surface corresponds to the choice *b*=0). The flow field is not irrotational; the vorticity associated with the particle (*a*, *b*) is(7.3)We have described, in outline, Gerstner's solution, so how does this relate to edge waves?

It was discovered by Yih (1966) and Mollo-Christensen (1982) that a simple transformation of the Gerstner solution produces an exact solution to the edge-wave problem. However, a more complete and useful analysis of this aspect of the Gerstner solution was developed by Constantin (2001*b*), who not only gave a simple realization of the solution, but also demonstrated its mathematical correctness. It is this latter paper that we shall look into for guidance. The essential idea is to rotate the coordinate frame so that *z* points generally upwards but at right angles to the constant-slope beach/bottom; *x* is directed towards deep water, but parallel to the constant slope; *y* is then along the beach (being the conventional longshore coordinate, exactly as in figure 3). It should be noted that this choice of coordinates is not that used by Constantin (2001*b*); we have essentially interchanged his *x* and *y*, the reason being that we prefer to use the same general choice throughout all our different presentations. The slope of the bottom is taken as tan *α* (as before), so that the components of gravity in this new coordinate system are (*g* sin *α*, 0, −*g* cos *α*). The dimensional pressure (cf. equation (2.1)) is written aswhich then leads to our full set of equations (2.2). This set has the exact solution (cf. equation (7.1))it then follows that(7.4)andIt should be noted that, in this frame, the *z*-component of the path is simply *z*=constant on a path. The non-dimensional pressure perturbation (cf. equation (7.2)) is thenand the pressure is constant on particle paths only if (more than 0 has been chosen) which is precisely the expression used in Stokes' solution (cf. equations (3.8) and (3.9)). It is easily confirmed that the equation of mass conservation is also satisfied, so we have an exact solution of the classical water-wave equations. The free surface is then described, in parametric form, byfor *a*∈R, *b*≤*b*_{0} and *t*≥0; the run-up pattern corresponds to *z*=0, so we must have *b*=*b*_{0} and then this run-up pattern is described parametrically (parameter *a*) by

The vorticity follows from our previous calculation asand further details of this solution, together with examples of the run-up pattern, can be found in Constantin (2001*b*)—a cycloid for *b*_{0}=0 and a trochoid (sometimes called a *curtate cycloid*) for *b*_{0}<0. Some examples of the theoretical run-up patterns will be shown later.

## 8. Edge waves over slowly varying depth

All the theories that we have presented so far, for the propagation of edge waves, have required uniform slope, implying that the water depth increases linearly with distance from the shore; this is manifestly not realistic. The inclusion of variable depth, and most particularly finite depth at infinity, into Whitham's (1976) weakly nonlinear theory (§6) was developed by Minzoni (1976). This valuable contribution demonstrated that the eigenvalue problem was essentially unaltered from that associated with the constant slope everywhere. With this result in mind, we will now describe, in outline, an approach (Johnson 2005) that uses the method of multiple scales for slowly varying depth, based on the dependence on slope angle *α* (taken to be small) implied by the Gerstner–Constantin exact solution (§7).

First, we revert to our original coordinate frame (used for equations (2.2) and represented in figure 3) and then observe (from equation (7.4) with a rotation of coordinates) that(8.1)with both *p* and *h* proportional to sin *α*. Let us write |*b*′(0)|=*ϵ*, the slope of the beach, and then assume that the bottom profile is given by *z*=*b*(*x*)=−*B*(*X*) with *X*=*ϵx* (and *B*′(0)=1). We may now use *ϵ* as the basis for scaling of the variables that is consistent with equation (8.1), namelyIn addition, we need appropriate independent variables to represent the slow evolution of a propagating wave; in the light of the discoveries that we have already described, and to be consistent with the exact solution in §7, we introduce(8.2)where *k* (>0) is a given wavenumber, *ω* (equal to constant) is to be determined, as is *Θ*(*X*;*ϵ*). The equations (2.2), expressed in terms of *ξ*, *X*, *θ* and *z*, then become(8.3)whereSubscripts have been used, where convenient, to denote partial derivatives. It is immediately evident that, as *ϵ*→0, the equations remain nonlinear: the scaling (8.1) does not produce a linear problem in the limit (although linearization could be superimposed, if that was thought useful; cf. equation (3.1)).

The problem posed in equations (8.3) is solved by seeking an asymptotic solution in which all the functions (including *Θ*) are expanded with respect to the asymptotic sequence {*ϵ*^{n}}, *n*=0, 1, 2, …. The leading order (denoted by the subscript zero) is then given by the set(8.4)we observe that these are a version of the shallow-water equations, but with the determination of *w*_{0} relegated to the next order. We seek a solution in a form consistent with equation (7.4) and Stokes' original solution,This choice leads directly to an exact solution of the set (8.4), with(8.5)provided we make the choice *Θ*_{0}=−*k* (so that *θ*∼−*kX*/ϵ, i.e. e^{θ}→0 as *X*→+∞), and then both *A*_{0}(*X*) and *ω* are arbitrary (at this stage). It is not necessary also to expand *ω* in terms of *ϵ*, at least for the purposes of determining completely the leading-order solution. The solution (8.5) was reported (Johnson 2005) as a new exact solution of the shallow-water equations—and a solution that does not exhibit wave steepening. This solution is consistent with an initial surface profile which takes the formfor some *A*_{0}(*ϵx*). We note in passing that we clearly have a trapped wave, by virtue of the exponential decay as *θ*→−∞(i.e. *x*→+∞).

The construction of a suitable solution at the next order is not straightforward—the details can be found in Johnson (2005)—but the upshot is that these asymptotic expansions, to be uniformly valid as *θ*→−∞ and as |ξ|→∞, require that *A*_{0}(*X*) be a solution offor given *B*(*X*). Thus,(8.6)which is a generalization of a result obtained by Miles (1989) as a contribution to his uniformly valid expansion for the linear dominant mode for edge waves in the presence of a small uniform slope. The beach, at the shoreline, is described by *B*(*X*)∼*X* as *X*→0^{+}, and then equation (8.6) yields(8.7)where *K* is a constant which is fixed by the amplitude of the wave for some *X*>0. Now if *A*_{0}(*X*), and all its derivatives, exists as *X*→0^{+}, then we requireor *ω*^{2}=(1+2*n*)*k* which is equation (5.6) (when we note that tan *α*=*ϵ* and that the factor is already incorporated into the definition of *ξ*; see equation (8.2)). Finally, it transpires that another non-uniformity is evident as *B*→0, but this is removed if remains bounded as *X*→0, and so we must use *n*>1/2. This condition therefore permits only *n*=1, 2, …, i.e. the lowest mode (*n*=0) cannot be accessed—and this is precisely the mode that is associated with the Gerstner–Constantin exact solution! The case *n*=0 is discussed in Johnson (2005), where some of the properties of the solution are discussed but, sadly, an exact solution at leading order is not available.

As an example of what can be extracted from this asymptotic solution, we present an equation for the run-up pattern. This is given by the intersection of the surface wave and the bottom profile at the beach, i.e.with *B*(*X*)∼*X* as *X*→0 describing the beach profile; to the leading order, this produces the equation(8.8)with *A*_{0}(*X*) taken as *A*_{0}(*X*)=*KX*^{n} (see equation (8.7)). A suitably normalized version of equation (8.8) is(8.9)where *Z*=*kx*, *μ*=*Kϵ*^{n}/*k*^{n−1} (*n*=1, 2, …) and the root *Z*=0 has been eliminated. Equation (8.9) appears to capture all the observed properties of the run-up patterns that are generated by the edge waves. It can be shown that the solutions of equation (8.9) which are continuous, bounded and periodic exist for *μ*≥*μ*_{n}>0 and for appropriate *μ*_{n}. The solution for *μ*<*μ*_{n} comprises only closed curves, spaced periodically, some of which coalesce as *μ*→*μ*_{n} to form two curves (that have the form of cycloids) that meet at their cusps; for *μ*>*μ*_{n}, these curves separate to become a pair of curves that take the general form of trochoids. (For some *n*, other curves exist that remain as closed curves.) A routine numerical investigation shows that *μ*_{1}≈18.57, *μ*_{2}≈21.54, *μ*_{3}≈12.27, *μ*_{4}≈4.55 and so on; a sequence of three sets of solutions of equation (8.9) is shown in figure 4 (for *n*=2 with *μ*=15, *μ*=*μ*_{2}≈21.54 and *μ*=25). All these continuous profiles correspond to those given in Constantin (2001*b*), although our equation (8.9) generates a pair, one pointing towards the shore and another towards the open ocean—either is an acceptable solution for the run-up pattern. Different shapes of this pattern are observed, which appear to confirm that each variant can occur in nature (e.g. Guza & Inman 1975; Komar 1998). An example of a typical solution for the surface wave is depicted in figure 5.

## 9. Nonlinear model equations

Based on the multiple-scale approach described in §8, the leading order for *ϕ* (where ** u**=

*∇ϕ*) and

*h*satisfies the pair of equations(9.1)which corresponds to the solution described earlier if we select

*Θ*

_{0}=−

*k*(and we should note that the dependence on

*X*is purely parametric here). Equations (9.1) have an exact solutionfor arbitrary

*A*(

*X*). This formulation, however, simply uses any appropriate solution of Laplace's equation to generate directly a solution for the surface. An improvement occurs if we assume small-amplitude waves (by making use of an additional parameter,

*δ*, as in equation (3.1)), retain terms as far as O(

*ϵ*) and O(

*δ*), but neglect terms O(

*ϵδ*),(9.2)This pair also has an exact solution, for any given and suitable

*B*(

*X*), with

*Θ*=−

*k*,whereFinally, if we eliminate

*h*between the two equations in equation (9.2), and further simplify (by taking advantage of

*ϵ*→0), we could produce a normalized version of the equation written in the form(9.3)which possesses the exact solution

*Φ*=

*A*(

*X*)e

^{θ}sin

*ξ*for arbitrary

*A*(

*X*).

## 10. Summary

We have presented some of the classical ideas, together with some of the more modern ones, that are available for the analysis of the edge-wave phenomenon in water-wave theory. The presentation has brought together different aspects of the problem, but all under the umbrella of a consistent formulation, allowing direct comparison of the various results and solutions. In particular, we have moved from Stokes' original work, through Ursell's important observations and the extension of Gerstner's exact solution, eventually to some observations about an appropriate fully nonlinear, variable depth theory. We have concluded with three model equations that may prove useful in extending the simple single-mode solutions described here—or simply as partial differential equations that may prove worthy of further study.

## Footnotes

One contribution of 13 to a Theme Issue ‘Water waves’.

- © 2007 The Royal Society