Friedlander–Keller ray expansions and scalar wave reflection at canonically perturbed boundaries

This paper concerns the reflection of high-frequency, monochromatic linear waves of wavenumber k(≫ 1) from smooth boundaries which are O(k−1/2) perturbations away from either a specified near-planar boundary or else from a given smooth, two-dimensional curve of general O(1) curvature. For each class of perturbed boundary, we will consider separately plane and cylindrical wave incidence, with general amplitude profiles of each type of incident field. This interfacial perturbation scaling is canonical in the sense that a ray approach requires a modification to the standard WKBJ ‘ray ansatz’ which, in turn, leads to a leading-order amplitude (or ‘transport’) equation which includes an extra term absent in a standard application of the geometrical theory of diffraction. This extra term is unique to this scaling, and the afore-mentioned modification that is required is an application of a generalised type of ray expansion first posed by F. G. Friedlander and J. B. Keller (1955 Commun. Pure Appl. Math. 6, 387–394).


Introduction and motivation
The interaction between monochromatic, propagating, linear waves (such as acoustic, elastic or electromagnetic waves) with obstructing boundaries is key to many topical scattering problems arising in modern applications of wave physics. Indeed, entire theories such as the geometrical theory of diffraction, relevant in the high-frequency domain, have been developed and applied very successfully for exactly such reasons.
One aim of this paper is to analyse examples from a relatively new class of timeharmonic, high-frequency scattering problems, in which standard methodologies have to be modified to account for reflection from obstructing boundaries having smooth perturbations (on a specified length scale) away from either a perfectly flat boundary or else one with a general O(1) curvature profile. These modifications are to the WKBJ-type expansion applied in order to use ray methods, necessitating a modified 'ray' expansion of the type first considered by Friedlander and Keller [1].
This application of Friedlander and Keller's theory is a second aim of this paper, and might even be the first such application for any specific value of a key parameter α that they introduce, apart from two other applicable values (α = 0 and α = 1/3) which those authors identified as having already arisen in examples pre-existing their generalised work. Indeed, Keller and Lewis [2] make this point explicitly following equation (1.225) of their paper, and so our current work will add α = 1/2 to the list of these only other known cases of practical relevance (α = 0 amounts to the standard ray expansion and α = 1/3 is relevant to, for example, creeping and whispering gallery wave propagation). We mention the case α = 1 later, commenting that this case is not really special since it can be encapsulated within complex ray theory without any further modifications being necessary.
The first problem that we consider concerns reflection at near-planar interfaces, an example of which is the problem analysed by Engineer et al. [3]. There, scattering by exterior plane wave incidence upon a two-dimensional 'canonically slender' obstruction was analysed in the high-frequency (i.e. k → ∞) limit. In that context, 'canonically slender' referred to a body of finite length whose transverse profile away from its 'tips' was of a width commensurate with the requirements that the inner diffraction problem within a length scale O(k −1 ) of either tip region was the full Helmholtz equation (∇ 2 + 1)φ = 0 (in terms of scaled inner coordinates x = k −1x , y = k −1ŷ and with φ(x, y) =φ(x,ŷ)), rather than some 'parabolic wave equation' reduction of it, subject to data prescribed on a locally parabolic boundary, rather than an exactly flat half-plane, so that the full effects of the 'inner' boundary curvature were accounted for.
That argument supposes that the left-hand tip (for example) has a locally parabolic profile given by the equationŷ 2 = 2β 2x (1.1) in which |x|, |ŷ| and the constant β are all O (1) quantities. This inner form of the boundary must match smoothly with its equation y = F (x; k) valid elsewhere for |x| = O (1), and supposing F (x; k) = k γ f(x) for some γ to be determined, this leads to the 'asymptotic matching conditions'ŷ from which γ = −1/2 and the condition f(x) ∼ β √ 2x as x → 0, a condition which supplies the value of β since f(x) is assumed known, both follow.
From this, we therefore take y = k −1/2 f(x) to be the generic profile of the upper surface away from the tip region of what we now term a 'canonically near-planar' interface and we now consider reflection phenomena associated with both plane and cylindrical wave incidence upon it. In both cases, we consider incident waves with arbitrary amplitude profiles represented by amplitude pre-factors carried straightforwardly through the ray calculations to follow, though we do emphasise that our analysis pertains to perturbations from general, fully infinite interfaces and not just from a semi-infinite half-plane as arises in the limiting case studied in [3].
Hence, the first part of this paper concerns the reflection of plane and cylindrical waves incident upon a Cartesian boundary profile y = k −1/2 f(x), and we retain this notation here so that direct comparison can be made to the work of Engineer et al. [3].
Generalising this, we then pose similar problems when the reflecting interface is a perturbation of the same length scale but from an underlying boundary with general curvature (rather than simply y = 0). If x = x 0 (s) = ((x 0 (s), y 0 (s))) is the original, unperturbed boundary ∂D, parameterised by arc-length s, then we consider reflection from a generalised profile given by ∂D : x = x 0 (s) + k −1/2x 0 (s) and in which the hatted coordinate variables are prescribed but general; notice that s is no longer arc-length along the perturbed boundary ∂D, and we note the following relationship between s andŝ, arc-length along the perturbed boundary, correct to O(k −1 ): Restricting attention temporarily to plane wave incidence upon the first class (y = k −1/2 f(x)) of reflecting boundaries (an exactly similar argument applies to cylindrical waves, and then to generally curved boundaries for both types of incident wave), for which we will see that the incident field is of the general form A I P (x, y)e ik(x cos θ−y sin θ) in the high-frequency limit, then exponentials of the general form A I P (x, y)e ikx cos θ±ik 1/2 f(x) sin θ are clearly inevitable when satisfying the imposed Neumann boundary condition. This guides us naturally towards seeking reflected fields with corresponding exponential factors e iku(x,y)+ik 1/2 v(x,y) and straightaway we see the onset of a Friedlander-Keller-type ray expansion (here with α = 1/2), which we describe further in the next section.

The Friedlander-Keller ray expansion
The underlying field equation throughout our discussion is the two-dimensional Helmholtz equation in which k is the large wavenumber and φ(x, y) is the field variable, which might be the potential function for a linear acoustic or elastic wave, or a component of an electromagnetic disturbance, for example. The constant k is given by ω/c, where we assume but suppress a multiplicative harmonic time dependence e −iωt throughout, c is the wavespeed of the medium. We solve (2.1) in the domainD adjacent to some given boundary ∂D, an O(k −1/2 ) perturbation away from another prescribed curve ∂D, on which a Neumann boundary condition for the total field is imposed (although we comment on the adjustments needed to account for Dirichlet data at appropriate points in the paper). Presented here are situations in whichD is the two-dimensional region given by (i) y > k −1/2 f(x) with ∂D being the boundary profile y = k −1/2 f(x) and ∂D being the x-axis, y = 0, and (ii) n > 0 with ∂D : where (s, n) are the standard curvilinear coordinates associated with the unperturbed boundary ∂D : x = x 0 (s), y = y 0 (s), s being arc-length and n the normal distance along and from ∂D, respectively. The hatted perturbation coordinates are all taken as given, as is f(x).
In the presence of the unperturbed boundary ∂D, we would ordinarily seek solutions to (2.1) in the singularly perturbed, high-frequency limit k → ∞ in the WKBJ form substitution into (2.1) and systematic comparison of terms at the various orders in k that arise then quickly yield the eikonal equation for u and the recursive family of transport equations for the amplitude functions A n (n = 0, 1, 2, . . . ; A −1 (x, y) ≡ 0). Such an expansion (2.2) is the basis of Keller's geometrical theory of diffraction as described in [4], applied in (for example) [5][6][7] and reviewed in [2]. Notice that the eikonal equation (2.3) for the phase decouples entirely from the amplitude equations (2.4), and is typically solved first (subject to appropriate boundary data) using Charpit's method (see, for example, [8] or [9]) by first introducing new dependent variables p = ∂u ∂x , q = ∂u ∂y and showing that these quantities are conserved (i.e. p = p 0 (s), q = q 0 (s), where the subscript zero once more denotes evaluation on ∂D) along the rays Γ (s) given parametrically in terms of arc-length τ along them by The eikonal equation (2.3) now becomes whilst differentiation of the boundary data u = u 0 (s) on ∂D gives u 0 (s) = x 0 (s)p 0 (s) + y 0 (s)q 0 (s). (2.7) With u 0 (s) given, or calculable, equations (2.6) and (2.7), together with a radiation condition at infinity to fix the sign (or choice of branch) of q 0 (s) = (1 − p 2 0 (s)) 1/2 , now provide p 0 (s) and q 0 (s), and hence the ray equations Γ (s) : x = x 0 (s) + τp 0 (s), y = y 0 (s) + τq 0 (s) (2.8) emanating from the boundary curve τ = 0 : x = x 0 (s), y = y 0 (s) and along which du dτ = 1 also now follow, implying u(s, τ) = u 0 (s) + τ (2.9) along them. The leading-order transport equation, corresponding to n = 0 in (2.4), reduces to a homogeneous first-order ordinary differential equation along each ray, with general solution in which a(s) = q 0 (s)x 0 (s) − p 0 (s)y 0 (s) q 0 (s)p 0 (s) − p 0 (s)q 0 (s) , (2.11) and this fully determines the leading-order solution; in principle, higher order amplitude terms A 1 ,, A 2 ,. . . can then be calculated from (2.4) in a recursive fashion.
Modifications to this ansatz are required whenever, for example, the rays associated with the incoming field are tangent to an obstructing boundary or else form a caustic; in the former case, for example, the exponent and denominator occurring in (2.2) are replaced by iku(x, y) + ik 1/3 v(x, y) and a factor proportional to k n/3 , respectively (see [10][11][12] and [13] and references cited therein for reviews of this and other related work.) The one-third power law in the exponent and amplitude expansion arises in that analysis because of the variable coefficients of the Helmholtz equation when written in local curvilinear coordinates near the point of tangential ray incidence.
This motivated Friedlander and Keller [1] to consider a generalised asymptotic ansatz for solutions φ of the Helmholtz equation in the modified WKBJ format in which we have changed their original notation to one consistent with ours in the work presented here; in order to reproduce Friedlander and Keller's original ansatz, we simply replace φ, u, v and A n in (2.12) by u, φ, χ and v n , respectively. Also, their term proportional to k α in the exponent of their exponential pre-factor had a real and negative co-efficient, whereas we have replaced by one that is pure imaginary in order to make this ansatz directly relevant to the wave scattering applications we consider later in the paper.
In [1], the field equations satisfied by u, v and the A n 's are listed for all possible ranges of α and for appropriate corresponding values of λ n , together with a commentary on special cases having important applications elsewhere in wave physics. For example, they observe that α can be restricted to the range 0 < α 6 1/2, with the case α = 0 coinciding with standard ray theory for the Helmholtz equation, and those cases for α > 1/2 essentially result in v being constant so that this exponent can be absorbed into the multiplicative amplitude coefficients. It is important to remark that the constancy of v in those situations is a direct consequence of there only being one extra term (additional to the standard term u) within the exponent of the ansatz (2.12). Had we included more such terms, which would amount to a generalisation of (2.12), then we would not be able to draw this conclusion and we return to this point in the discussion at the end of the paper.
Another special case in this range deserving special attention is α = 1 but this coincides with a Luneberg-Kline expansion, also cited in [1]. In this case, the functions u and v can be combined to form a single exponent (whereby u is replaced by u + iv), and then a theory of complex rays applied [14]. Friedlander and Keller also cite the particular case of α = 1/3 which, as we have already noted, is relevant for creeping and whispering gallery mode propagation around convex bodies of O(1) curvature.
The results presented here complement this work of Friedlander and Keller in that it provides a concrete example of an application of their expansions for the case α = 1/2 (which they do not consider), and also we go on and solve the resulting equations for the two exponent functions and the leading-order amplitude; Friedlander and Keller list the equations they satisfy but never actually solve them, either in general terms or via specific application.

R. H. Tew
We have already presented arguments which motivate an ansatz for the reflected fields φ R for either of the two classes of problems we consider of the form (2.13) Substitution into the Helmholtz equation (2.1) shows that u still satisfies the eikonal equation (2.3), and subsequently that v is coupled to u via the equation ∇u · ∇v = 0, (2.14) in agreement with Friedlander and Keller [1]. The amplitude A(x, y; k), which still carries k as a parameter, now satisfies which, in turn, prompts a modified 'ray', or Friedlander-Keller type, expansion Substitution of (2.16) into (2.15) and extracting like powers of k, we obtain the recursive system of transport equations  (10), the term ∇ 2 χ should, in fact, be (∇χ) 2 . This is an isolated error and all subsequent formulae arising from this result are correct). Notice that this transport equation differs from the standard form ((2.4) with n = 0 and A −1 (x, y) ≡ 0) because of the inclusion of the final term involving ∇v · ∇v; this is solely due to the O(k 1/2 ) term in the exponent in (2.13); only for precisely this power does this happen -any other power will leave the original leading-order transport equation unaffected, further strengthening the claim that the O(k −1/2 ) scale of the boundary perturbations considered here are indeed canonical. We end this section by noting that since ∇u = (p 0 (s), q 0 (s)) = ( dx dτ , dy dτ ) along a ray (along which s is constant and τ is arc-length measured from the boundary), we see that ∇u · ∇v = 0 implies dv dτ = 0, so that v is invariant along the rays i.e. v = v 0 (s), where v 0 (s) is the value of v on ∂D. Also, a straightforward calculation using the ray equations yields the useful derivative relationships for general F, allowing ∇ 2 u = ∂p0(s) ∂x + ∂q0(s) ∂y and ∇v = v 0 (s)∇s to be evaluated. The leadingorder transport equation (2.18) for A 0 (s, τ) along each ray then follows as where a(s) is given by (2.11) and This can be solved to give the general formula for the leading-order amplitude in our interpretation of a Friedlander-Keller ray expansion with α = 1/2 in the form .
This generic theory plays a role when considering reflection at perturbed boundaries of arbitrary underlying curvature; for those which are near-planar, we nonetheless deliberately adopt a partial differential equation approach to illustrate an additional, alternative means of solution available in that case.

Formulation of the boundary value problems
Writing the total potential φ T W as the superposition of an incident field φ I W and reflected field φ R W (W = P , C for plane and cylindrical wave incidence, respectively) being the analogous relation for Dirichlet data) on ∂D and a 'radiation condition' where ∂ ∂n denotes the normal derivative at the rigid boundary ∂D. The condition (3.4) will be dealt with in each of the two cases (P and C) separately; it essentially means that the field represented by φ R W must propagate away from, and not towards, the boundary ∂D. Regarding the functional forms of φ I W (α = P , C), we have already noted that the general form of an incoming plane wave propagating towards the x-axis of a Cartesian coordinate system has a phase which gives a leading-order form φ I P (x, y) ∼ A I P (x, y)e ik(x cos θ−y sin θ) , (3.5)

R. H. Tew
where θ is the angle of incidence measured from the x-axis. Of course, the incident phase u I P (x, y) = x cos θ − y sin θ satisfies the eikonal equation ( so that A I P (x, y) = F I P (x + y cot θ) for some function F I P of a single variable. We assume that F I P is given, so that the incident plane wave has the general, variable-amplitude form at leading order. For the case of an incident cylindrical wave φ I C , we suppose that its source is located at the remote point (0, h), and so has an associated phase function which of course also satisfies (2.3). Writing the incident field in this case as (2.4) now implies the leading-order transport equation which has general solution for any arbitrary function F I C of the single variable y−h x . As with F I P previously, we take F I C to be given and so we maintain as general as possible leading-order amplitude variation in both cases. Notice that, in this second case, y−h x = tan θ, where now θ is defined to be the polar angle measured from the source location and is not related to the angle of incidence defined in φ I P . This allows us to write where the (known) function D I C (θ) is the leading-order directivity of φ I C which, in turn, can now be expressed in the more recognisable form where R = [x 2 + (y − h) 2 ] 1/2 is radial distance measured from the source. Our final comments on the problem formulation concern the satisfaction of the Neumann boundary condition (3.2).
First, in doing so, we must simultaneously balance (i) the O (k) and (ii) the O(k 1/2 ) terms within the various exponents that arise, as well as (iii) the O(1) terms within the multiplicative amplitude, and this contains some subtle features. For example, if we consider the near-planar boundary y = k −1/2 f(x), then a typical exponent (whether in φ I W or φ R W ) evaluated on this perturbed boundary is of the form ∂v ∂y (x, 0) (3.14) (with an exactly similar argument available for the other more general class of boundaries considered). The point is that we 'linearise' the exponent onto the unperturbed boundary (y = 0 in this case) before proceeding with the ray treatment, and we see that derivatives of u alone appear within the O(k 1/2 ) terms, whilst derivatives of both u and v arise in the final O(1) terms. That is to say, the solution for u, which is found independently of that for either v or the leading-order amplitude, feeds into that for v via the boundary condition, whilst the solutions for both u and v play a role in providing part of the boundary data (via the final two O(1) terms, or their analogues, on the right-hand side of (3.14)) for the leading-order amplitude. Our second point regarding (3.2) concerns its mathematical implementation. Of course, ∂ ∂n = n · ∇ where n is the outward-pointing unit normal to the interface, and in the twodimensional problems that we consider here we have either n ∝ (−k −1/2 f (x), 1) for the near-planar case or n ∝ (−(y 0 (s) + k −1/2ŷ 0 (s)), x 0 (s) + k −1/2x 0 (s)) for the generally curved examples, allowing us to replace (3.2) by either on ∂D, or else on ∂D, respectively. The analogue of both (3.15) and (3.16) for Dirichlet data is encapsulated in the single condition (3.3). Notice also that (3.16) appears to mix the Cartesian (x, y) and curvilinear (s, n) coordinate systems; this is justified since x and y derivatives of φ T W necessarily involve ∂uW ∂x , ∂uW ∂y and ∂vW ∂x , ∂vW ∂y when we consider Friedlander-Keller exponentials e ikuW (x,y)+ik 1/2 vW (x,y) , and these derivatives arise naturally (via ∇u, ∇v) in the subsequent ray analysis.
Strictly speaking, both sides of (3.16) should be multiplied by ds dŝ , whereŝ is arc-length along the perturbed boundary ∂D; the equation relating s toŝ is (1.3).
We now have all of the mathematical machinery that we need, and the solution strategy is clear: having specified the incident field, we pose an ansatz and then use the boundary condition (3.2) to establish boundary data for the various u, v and A 0 that arise, and then use it to solve (2.3), (2.14) and (2.18), in that order, for these quantities to be determined explicitly. When all three of these quantities have been found, we will say that the leading-order solution for the reflected field has been constructed.

Plane wave incidence
Guided by the previous discussions leading to (3.7), we write down the leading-order expression for the total field in the form in which A R P (x.y) is the amplitude of the reflected field to leading order. Of course, the overall potential φ T P must satisfy (3.15), and we have already noted that this will require that the exponents arising in each of the two terms on the right-hand side of (4.1) must agree on ∂D : y = k −1/2 f(x); this in turn gives two further equations (one at O(k), the other at O(k 1/2 )) providing boundary data for u P and, subsequently, v P , respectively.
Hence, the problem for u P (x, y) is to solve the eikonal equation (2.3) subject to data together with the 'radiation condition' that these reflected rays propagate away from the boundary and into the region y > 0. Equation (2.9) and the ray analysis around it then give u P (x, y) = x cos θ + y sin θ, (4.3) so that (2.14) now in turn gives cos θ ∂v P ∂x + sin θ ∂v P ∂y = 0, (4.4) so that v P (x, y) = v P (ξ) is a function of the single variable ξ = x − y cot θ. The argument surrounding (3.14) now provides the data for v P in the form from all of which we conclude that v P (x, y) = −2 sin θf (x − y cot θ) . Hence, the solutions for u P and v P are now both fully determined, with only the latter depending explicitly upon the boundary profile f(x) and the former essentially seeing no difference from the totally rigid flat boundary case.
In order to calculate the leading-order amplitude A R P (x, y), we substitute the results so far into (2.18), to obtain the boundary value problem where the boundary condition (4.8) follows from the leading-order contribution to (3.15). Introducing a second independent variable ζ = x+y tan θ, the partial differential equation This equation is easily solved by separating variables and doing so, applying the boundary condition and then reverting back to the original Cartesian coordinates yields the solution for A R P in the final form and this completes the solution for the reflected field for the case of variable-amplitude plane wave incidence subject to Neumann data. Dirichlet data is accommodated instead by simply negating the expressions on the right-hand sides of (4.8) and (4.10).

Cylindrical wave incidence
With the total field the methodology is as before and as far as the 'phase functions' u C , v C (in an obvious notation) are concerned it is straightforward to obtain the two boundary relations arising at orders k and k 1/2 , respectively, within the exponents as and (4.13) The eikonal problem for u C is straightforward to solve, the 'outgoing' solution being u C (x, y) = x 2 + (y + h) 2 1/2 , (4.14) corresponding to an outgoing (in y > 0) cylindrical wave emanating from the image source
Since R(x, ±y) both satisfy the eikonal equation and the boundary condition, u C must be one or the other. Taking the positive sign yields a field incoming towards the boundary and is therefore rejected since it violates the radiation condition. Hence, u C (x, y) = R(x, −y) and (4.14) emerges. Given this solution, we find from (2.14) that v C satisfies so that v C has the self-similar form v C (x, y) = g (η), η = y+h x = tan Θ, for some function to be determined g. Notice that v C depends only upon the polar angle Θ measured from the image source and so this term within the exponent can be regarded as a k-dependent modification to the final directivity associated with the reflected field.
Combining (4.13) and (4.14) provides the refined boundary condition v C (x, 0) = − 2h is immediate, and v C (x, y) = g( y+h x ) is determined; all that remains to be done is to compute the leading-order amplitude A R C . We begin by substituting our solutions for u C and v C into (2.18), which reveals that and, guided by our construction of the incident field (which is the main reason for including that discussion), we now set so that is a known function of the single variable η.

Equation (4.20) admits an exact solution
for a function F R C (η) to be determined, and piecing all of this information together we deduce that and we can identify F R C as another function dependent solely on the polar angle Θ measured from the image source, and therefore (like v C ) closely allied to the directivity of the reflected field. We shall be more precise about this shortly, linking it directly to the prescribed incoming directivity F I C as prescribed in the given incident field. We also note that Λ is pure imaginary, of O(1) modulus and vanishes if g, and hence f, does. It therefore describes an O(1) shift in the phase due to the presence of the boundary undulations described by f(x).
The determination of the reflected field in this case is complete once the 'directivity' F R C has been found, and this is done by appealing to the one remaining condition to be satisfied, namely (3.15). Feeding into that condition all that has been found so far then gives, after considerable algebra, the closed-form expression this now determines the leading-order amplitude A R C of the reflected field explicitly and the reflection problem is solved, noting that the equivalent relation to (4.24) appropriate to Dirichlet data is obtained by negating this result.

Perturbed boundaries of general curvature
In this section, we replace u W and v W (W = P , C) by μ C and ν C , respectively, to avoid possible confusion with the near-planar boundary analysis just considered. This is purely notational, all underlying field equations and boundary conditions remain unchanged.

Cylindrical wave incidence
With cylindrically spreading wavefronts emanating from the source location (0, h), the eikonal boundary data for μ C is now and if we define c 0 (s), s 0 (s) and ψ 0 (s) through the relations c 0 (s) = cos ψ 0 (s) = x 0 (s) R 0 (s) , s 0 (s) = sin ψ 0 (s) = y 0 (s) − h R 0 (s) , (5.16) so that (c 0 (s)) 2 + (s 0 (s)) 2 = 1, then (2.7) becomes p 0 (s)x 0 (s) + q 0 (s)y 0 (s) = c 0 (s)x 0 (s) + s 0 (s)y 0 (s) (5.17) and p 0 (s) then satisfies the quadratic equation It is useful to note that in this case, (5.11) must be replaced by p 0 (s) q 0 (s) = ψ 0 (s) − 2κ 0 (s). (5.21) With these, we of course now have the ray solution μ C (s, τ) = R 0 (s) + τ (5.22) valid along the rays given by (2.8). The same general method as before is used to compute ν C and the upshot is the compactly factorised expression ν C (s) = 2 R 0 (s) x 0 (s)y 0 (s) − (y 0 (s) − h) x 0 (s) y 0 (s)x 0 (s) − x 0 (s)ŷ 0 (s) . (5.23) This now determines completely the k-dependent terms within the exponent of the reflected field, and we are also able to write down expressions for a(s) and b(s) relevant to this case in the forms where for completeness, we also note that ψ 0 (s) defined through (5.16) satisfies Referring once more to (2.22), we now need to findÃ R C (s, 0) to finalise the solution and, as before, this utilises (3.16) and so must take account of the O(1) contributions in all phase terms in reflected and incident fields. After some particularly involved algebra, the final result turns out to bẽ and so the solution is now complete for the Neumann data case, simply negating the final overall result covers Dirichlet data. We end this section by commenting that for either type of incoming wave, the latter class of perturbed boundaries of general curvature can, in principle at least, encapsulate the former, of a near-planar nature, by setting (x 0 (s), y 0 (s)) = (s, 0) and (x 0 (s),ŷ 0 (s)) = (0, f(s)) in the notation of the analysis presented.
If we do so, then lengthy calculation does indeed reveal that the results of Section 4 are reproduced in every detail from those in Section 5. We do not present the details, but remark instead that this is to be interpreted in the current context as an independent check on the accuracy of the results presented, rather than as an alternative means of arriving at the 'near-planar' results without the need to perform that analysis separately.

Discussion and concluding remarks
Our first observation is that the presence of the boundary perturbation does not in itself induce amplifications in the amplitudes (by way of focusing or caustic formation, for example) of the reflected fields for either type of wave incidence. Any amplification that is present is primarily due to the geometry of the unperturbed boundary, and is located parametrically at focal points or caustic curves in the physical domain satisfying the equation τ + a(s) = 0 (in the notation of, say, equation (2.22)). However, such instances are now singularities of the phase distortion in the perturbed cases because of the corrections to the phase via the exponential factor in (2.22) (noting that a(s) and b(s) are both real-valued) that the perturbation undulations induce.
So although no new caustic curves (or focal points) are induced within the reflected field because of these perturbing undulations -those that exist would have been present anyway even in their absence -it is the case that the field variation across those that do exist, classically described by Airy functions in the case of caustics ( [15,16]), will now need some adjustment to account for this modified singular phase variation across the caustic and this is currently under investigation.
A special class of problems that can arise, not discussed at all here, pertains to incoming plane waves for which the angle of incidence θ (as defined earlier within the main text of the paper) is small i.e. the incoming plane wave is at, or is close to, grazing incidence and when it would meet the boundary perturbation broadside on. More specifically, if θ = O k −1/2 , then the incoming plane wave φ I P (x, y) can be approximated, having first set θ = k −1/2θ for |θ| = O(1) and fixed, by where the error in the exponent, for O (1) values of x and y, is of O k −1/2 and would constitute a higher order correction to the amplitude, and not the phase (e.g. the omitted exponential is e ik −1/2θ3 y/6 ∼ 1 − 1 6 ik −1/2θ3 y if |y| = O(1)). We see from (7.1) that once more an ansatz of the form (2.13) is required, but for completely different reasons to before -these originally being motivated strongly by the geometry of the boundary ∂D : y = k −1/2 f(x) rather than the form of the incoming field itself. Indeed, to make matters even more interesting, when the modified incident field (7.1) valid away from the boundary is actually evaluated upon it, then propagating exponentials such as e ikx−i f(x)θ− 1 2θ 2 x emerge, and rather than containing O k 1/2 terms within the exponent to justify (2.13), it does not actually contain any. A completely separate analysis is required for this class of problems, and this is also being pursued elsewhere.
Another extension currently under investigation include reflection of elastic waves at both near-planar and perturbed but otherwise generally curved free surfaces. A new feature within this class of problems is mode conversion, whereby an elastic wave of one type (such as longitudinal 'P' or vertically polarised shear 'SV') can, and generally will, give rise to a reflected wave involving the other. This not only brings about the possibility of total internal reflection but also that of exciting a Rayleigh surface wave. In either case, propagation along the perturbed boundary is a key characteristic, allowing these waves to accrue the cumulative effects of the boundary undulations; this is in stark contrast to the reflection-only problems considered here for which the interaction of rays -incoming or reflected -occurs in a pointwise fashion.
Finally, we remark that the Friedlander-Keller expansions that we have considered all involve exponentials of the form e iku+ik 1/2 v , considering the exponent as a sequence of terms with decreasing powers of k 1/2 we see immediately that this is the most general possible, it being impossible to 'fit' any more terms in between the 'u' and 'v' terms (and we note that we can disregard any terms in k 0 , since these lead to terms that can be absorbed into amplitude considerations if we choose to).
Suppose instead that the relevant power of k was one-third, rather than one-half, as occurs in creeping and whispering gallery wave propagation (referred to earlier). In those examples, it is well known that full descriptions can be obtained using Friedlander-Keller expansions with exponents of the form iku + ik 1/3 v. However, there appears to be no a priori reason why there should not also be an intermediate term proportional to ik 2/3 , prompting a more general Friedlander-Keller exponent of the form iku(x, y) + ik 2/3 v(x, y) + ik 1/3 w(x, y) exhibiting all intermediate powers of k 1/3 up to and including k.
This strongly motivates a completely separate study of a yet further generalised for any integer m > 1. A specific example is wave scattering by a perturbed circle of profile r = r 0 + k −1/3 F (θ), in terms of plane polar coordinates r and θ. In that case, a full ansatz of the form (7.2) with m = 3 is required, with all three terms in the exponent being needed. Similar issues arise in scattering by near-planar boundaries of the more general form y = k −1/n f n (x), and all of these aspects are currently under investigation and review.