## Abstract

This paper is concerned with wave arrival singularities in the elastodynamic Green's functions of infinite anisotropic elastic solids, and their unfolding into smooth wave trains, known as quasi-arrivals, through spatial dispersion. The wave arrivals treated here are those occurring in (i) the displacement response to a suddenly applied point force or three-dimensional Green's function, , and (ii) the displacement response to an impulsive line force or two-dimensional Green's function, . These arrivals take on various analytical forms, including step function and logarithmic and power-law divergences. They travel outwards from the source at the group velocities in each direction, and their locus defines the three- and two-dimensional acoustic wave surfaces, respectively. The main focus of this paper is on the form of the wave arrivals in the neighbourhood of cuspidal points in the wave surfaces, and how these arrivals unfold into quasi-arrivals under the first onset of spatial dispersion. This regime of weak spatial dispersion, where the acoustic wavelength, *λ*, begins to approach the natural length scale, *l*, of the medium, is characterized by a correction to the phase velocity, which is quadratic in the wavevector, ** k**, and the presence of fourth-order spatial derivatives of the displacement field in the wave equation. Integral expressions are established for the quasi-arrivals near to cuspidal points, involving the Airy function in the case of and the Scorer function in the case of . Numerical results are presented, illustrating the oscillatory nature of the quasi-arrivals and the interference effects that occur near to cuspidal points in the wave surface.

## 1. Introduction

Wave arrivals are singular features that punctuate the space–time domain elastodynamic Green's functions of solids (e.g. Ben-Menahem & Singh 1981; Every & Kim 1994; Burridge 1995; Aki & Richards 2002). In an anisotropic solid, these arrivals travel outwards from the point of excitation at the acoustic group velocities in each direction, and their locus defines the acoustic wave surface. In the absence of dispersion, the stationary phase approximation dictates the various analytical forms that wave arrivals can take, which include the step, delta and ramp functions and logarithmic and power-law divergences (Every & Kim 1994). Most of the acoustic energy radiated by a localized event, such as an impulsive or suddenly applied point force, is concentrated at these arrivals, and in the far field, they are essentially all that remains of the wave field to be discerned. Hence, they feature importantly in any discussion of elastodynamics. The wave arrivals treated in this paper are those occurring in (i) the displacement response to a suddenly applied point force or three-dimensional Green's function, , and (ii) the displacement response to an impulsive line force or two-dimensional Green's function, . The latter is treated within the context of plane strain or two-dimensional elasticity, with displacements confined to the plane perpendicular to the line along which the force is distributed and independent of the position along that line.

Dispersion has the effect of unfolding wave arrival singularities, i.e. causing them to spread out into characteristic wave trains known as quasi-arrivals (e.g. Elices & Garcia-Moliner 1968; Christensen 1979; Kaplunov *et al.* 1998; Every 2005; Every *et al.* 2006). This paper is concerned with this wave arrival unfolding under the first onset of spatial dispersion as the wavelength, *λ*, approaches the natural length scale, *l*, of the medium (e.g. the lattice constant or range of interatomic forces in a crystal, or the repeat distance in a layered or fibrous solid etc.). In centrosymmetric solids, this domain of weak spatial dispersion is characterized by a correction to the phase velocity, which is quadratic in the wavevector, ** k**, and the presence of fourth-order spatial derivatives of the displacement field in the wave equation. In the more common situation of positive or downwardly curving dispersion relation,

*ω*(

**), the quasi-arrival wave field rapidly converges to the undispersed wave field in front of the arrival and oscillates around that field with increasing frequency and diminishing amplitude with distance behind the arrival. In the case of negative or upwardly curving dispersion relation,**

*k**ω*(

**), the order is reversed, with the ripples leading the arrival. The assumption of weak spatial dispersion goes hand in hand with confining one's attention to the central region of the quasi-arrival, encompassing a limited number of ripples, which is dominated by low spatial frequencies in the spectrum of the acoustic field. The ripple separation in this region, as we will see, is of the order (**

*k**l*

^{2}

*X*)

^{1/3}, where

*X*is the source–detector distance.

In a previous paper (Every *et al.* 2006), we have described the unfolding of wave arrivals to be associated with generic points in the three-dimensional wave surface of an anisotropic solid. The main focus of the present paper is on the unfolding of arrivals in the neighbourhood of cuspidal points in the two- and three-dimensional wave surfaces of anisotropic solids. Central to the discussion is the geometry of the acoustic slowness surface, with points of inflection in the two-dimensional slowness surface (curve) mapping onto cuspidal points in the two-dimensional wave surface (curve), and lines of vanishing Gaussian curvature in the three-dimensional slowness surface mapping onto lines of cuspidal points (cuspidal edges) in the three-dimensional wave surface, where this surface is folded in a crease. We establish integral expressions for the quasi-arrivals near cuspidal points, involving the Airy function, *Ai*(*z*), in the case of and the Scorer function, *Gi*(*z*), (Scorer 1950) in the case of . Numerical results are presented, displaying the oscillatory nature of the quasi-arrivals and the interference effects that arise from overlapping quasi-arrivals near to cuspidal points in the wave surface. Cuspidal edges in the three-dimensional wave surface can meet at isolated points near which the wave surface has the swallowtail shape (Berry 1976; Kravtsov & Orlov 1983; Burridge 1995). The unfolding of wave arrivals near to swallowtails will be the subject of a later publication.

The issues dealt with here, while pertaining to transient wave propagation in three-dimensional solids (e.g. Hao & Maris 2001), also bear on transient waves in wave guides, such as rods (Skalak 1957), strings (Ducasse 2005) and plates (e.g. Kaplunov *et al.* 1998; Kaplunov & Pichugin 2005), which are subject to geometric dispersion, in which the natural length scale is the diameter or thickness, *h*.

This paper draws on results from two earlier papers by the authors (Every 2005; Every *et al.* 2006): the first paper describes the unfolding of wave arrivals in two- and three-dimensional isotropic solids and the second one describes the unfolding of wave arrivals at generic points in the three-dimensional wave surface of anisotropic solids. Note that in those papers, the wave field near an arrival was displayed as a function of time for a fixed point in space, whereas in this paper we find it expedient to plot the results as a function of the position for a fixed value of time. By doing this, the quasi-arrivals for cusps are thereby represented as a function of two spatial variables, rather than one spatial variable and time. In §2 we present an integral formula for the singular part of the point force elastodynamic Green's functions *G*^{3}, and in §§3 and 4 we treat, respectively, the arrivals and their unfoldings for generic points in the three-dimensional wave surface and near cuspidal edges in the wave surface. In §5 we present an integral formula for the singular part of the line force elastodynamic Green's functions *G*^{2}, and in §§6 and 7 we treat, respectively, the arrivals and their unfoldings for generic points in the two-dimensional wave surface and near cuspidal points in the wave surface.

## 2. Point force Green's function : general formulation

For a centrosymmetric anisotropic solid, the first onset of spatial dispersion as the wavelength approaches the natural length scale, *l*, expresses itself through the presence of fourth-order derivatives of the displacement field ** u**=(

*u*

_{j}) in the elastic wave equation (e.g. Kröner 1968; DiVincenzo 1986; Lakes 1995; Every 2005; Every

*et al.*2006). The three-dimensional elastodynamic Green's function , defined as the

*x*

_{q}th component of the displacement response of the infinite continuum at point

**and time**

*X**t*to a unit point force in the

*x*

_{p}direction acting at the origin

**=0, and having unit step-function time dependence, is governed by the equation (Every**

*X**et al.*2006)(2.1)where

*ρ*is the mass density, and

_{qjkl}and

_{qjklmn}are the elastic modulus and elastic dispersion tensors, respectively. The formal solution of (2.1) may be obtained by integral transforms. We are specifically interested in the nature of the singular behaviour of the solution in the vicinity of a wave arrival associated with one of the three branches of the acoustic slowness surface, and how this unfolds under the influence of spatial dispersion, regardless of the particular component indices

*q*and

*p*. This is given by an integral of the form (Every

*et al.*2006)(2.2)where

**=**

*S***/**

*k**ω*is the phase slowness and d

*Ω*is the solid angle element in which

**lies;**

*S**Λ*is a dimensionless coupling coefficient; and

*γ*, which is of the order of magnitude (

*l*/2

*π*)

^{2}, is the coefficient in the dispersion relation(2.3)

*v*being the phase velocity in the long wavelength

**→0 limit. The condition of weak spatial dispersion may be expressed as**

*k**γk*

^{2}≪1 or, equivalently,

*l*≪

*λ*.

The singular behaviour at an observation point, *X*_{0}, arises in the integration over the neighbourhood of a point *S*_{0} in the slowness surface where the outward normal, the ray, points in the direction of *X*_{0}, rendering ** S**.

*X*_{0}stationary. The singularity has an arrival time

*S*_{0}.

*X*_{0}=

*t*

_{0}and propagates with group velocity

*V*_{0}=

*X*_{0}/

*t*

_{0}. The locus of points

*X*_{0}for all

*S*_{0}represents the wave surface. To determine the singular form of the arrival and its unfolding, i.e. the behaviour of

*G*

^{3}in the neighbourhood of

*X*_{0}at time

*t*

_{0}, we set

**=**

*S*

*S*_{0}+

*S*

_{0}

**and perform the angular integrations in (2.2) in a local coordinate frame with**

*s**s*

_{3}oriented along the outward normal and

*s*

_{1}and

*s*

_{2}in the directions of principal curvature of the slowness surface at

*S*_{0}. The equation for the slowness surface in terms of the dimensionless scaled variables,

*s*

_{3}=

*s*

_{3}(

*s*

_{1},

*s*

_{2}), is approximated locally by a power series containing quadratic and, where necessary, a small number of higher-order terms, ensuring

*inter alia*that |

*s*

_{3}|≪|

*s*

_{1}|, |

*s*

_{2}|. The location of a point in the neighbourhood of

*X*_{0}will be specified by the dimensionless scaled position vector

**=(**

*x***−**

*X*

*X*_{0})/

*X*

_{0}, in a coordinate frame with

*x*

_{3}in the direction of

*S*_{0}, which is normal to the wave surface, and

*x*

_{1}and

*x*

_{2}in the plane normal to

*S*_{0}, which is tangential to the wave surface at

*X*_{0}. The wave surface is the singularity set for certain integrals considered below. Near this surface, |

*x*

_{3}|≪|

*x*

_{1}|, |

*x*

_{2}|. The local coordinate systems for

**and**

*s***are not in general parallel, their relative orientation being determined by the set of directional cosines , where and are the unit vectors for the two coordinate systems.**

*x*Expressed in terms of the local coordinates introduced above, the solid angle differential is(2.4)(2.5)where(2.6)In arriving at (2.5) and (2.6), terms *s*_{3}*x*_{i}, *s*_{i}*x*_{3} and *s*_{3}*x*_{3}, being small in comparison with the other terms, have been dropped. With the substitutions of (2.4) and (2.5), and bearing in mind the limitation of the integration over ** S** in the near neighbourhood of

*S*_{0}, equation (2.2) becomes(2.7)where . On integrating with respect to

*k*, one obtains(2.8)where(2.9)is a small parameter setting the spatial scale for the unfolding of the arrival and(2.10)is the Airy function (Abramowitz & Stegun 1964). The dispersionless classical continuum limit for

*G*

^{3}as

*γ*→0, obtained using the identity(2.11)or directly from (2.7) by dropping the

*k*

^{3}term, is(2.12)The singularity set for the integral (2.12) is the set of points

**that simultaneously render(2.13)These represent a set of parametric equations for the wave surface. By eliminating the parameters**

*x**s*

_{1}and

*s*

_{2}, one arrives at the explicit equation for the wave surface,

*x*

_{3}=

*x*

_{3}(

*x*

_{1},

*x*

_{2}). The consideration of the variation of

*Χ*with

*s*

_{1}and

*s*

_{2}shows that this has no effect on the equation of the wave surface and analytical form of undispersed arrival.

## 3. Regular points on the slowness and wave surfaces

A generic or regular point *S*_{0} on the slowness surface is a point where both the principal curvatures are finite. By aligning *s*_{1} and *s*_{2} along the principal directions in the surface, the equation for the slowness surface is locally approximated by (e.g. Every & Kim 1994)(3.1)the coefficients *α* and *β* being dimensionless and of order unity, and so(3.2)The principal curvatures are 2*α*/*S*_{0} and 2*β*/*S*_{0}, and their product(3.3)is the Gaussian curvature of the slowness surface at *S*_{0}. Points of vanishing Gaussian curvature map onto cuspidal edges in the wave surface and are treated in §4. Not treated in this paper are acoustic axes, i.e. points of degeneracy in the slowness surface, where *αβ* is infinite or undefined, and conical points in the wave surface, onto which circles of points in the slowness surface map.

From (2.13), one derives the parametric equations for the wave surface, and elimination of *s*_{1} and *s*_{2} yields the local explicit equation for the wave surface(3.4)The cross term in this quadratic form can be eliminated by suitable rotation about . This places the - and -axes along the directions of principal curvature of the wave surface. It follows readily, using the fact that , that the principal curvatures of the wave surface are and , and that their product, the Gaussian curvature, is thus(3.5)Comparing (3.3) and (3.5), we see that the Gaussian curvatures of the slowness and wave surfaces bear the simple inverse relationship(3.6)Our main concern is the singular behaviour of *G* in the neighbourhood of the wave surface, and its unfolding under spatial dispersion. For regular points, this behaviour is a function of(3.7)the distance from the arrival in units of *ΧX*_{0}, and is independent of *x*_{1} and *x*_{2}, except insofar as they determine . Combining (2.12), (3.2) and (3.7), and implementing a change of integration variables to *u*=|*α*|^{1/2}*s*_{1} and *v*=|*β*|^{1/2}*s*_{2}, the undispersed arrival is given by(3.8)where .

Wave arrivals in *G*^{3} for regular points and their unfolding under spatial dispersion have been treated by Every *et al.* (2006), and we merely summarize the salient results here. One distinguishes between elliptic points, for which *αβ*>0 and both the slowness and wave surfaces are either convex (*α*, *β*>0) or concave (*α*, *β*<0), and hyperbolic points for which *αβ*<0 and the surfaces are saddle shaped, with the principal curvatures opposite in sign. For elliptic points, (2.12) yields a simple discontinuity for the wave arrival, which is given by(3.9)Its unfolding under dispersion, from (2.8), is given by(3.10)which can be expressed as the convolution integral(3.11)For hyperbolic points, an upper cut-off needs to be imposed on the integrals (2.12) and (2.8) in order to avoid an unphysical divergence. The position of the cut-off, *c*, merely determines an additive constant and has no influence on the singular behaviour at the arrival, which is in the form of the logarithmic divergence(3.12)Its unfolding under dispersion is given by(3.13)which is rather insensitive to *c*.

In front (or behind, if *γ*<0) of the arrival, the quasi-arrival converges rapidly to the undispersed waveform, while behind (or in front, if *γ*<0) the arrival, it oscillates with decreasing amplitude and period around the undispersed waveform (Every *et al.* 2006). The ripple separation near to the arrival in both cases is(3.14)

## 4. Parabolic line on the slowness surface: cuspidal edge in the wave surface

Generically, for an anisotropic solid, there can occur one-dimensional manifolds of points on the slowness surface, known as parabolic lines, along which one of the principal curvatures, say 2*α*/*S*_{0}, vanishes. These regions of positive and negative Gaussian curvatures of the slowness surface separate and map onto cuspidal edges in the wave surface, where that surface folds back on itself. To determine the local equation of the wave surface and describe wave arrivals at and near a cuspidal edge, one needs to consider the effect of cubic terms in the local equation for the slowness surface(4.1)all the coefficients of which are dimensionless and of order unity. The direction of *s*_{1} is chosen so that *ζ* is positive. For a parabolic line separating convex and saddle regions of the slowness surface, *β* is positive, while for the one separating concave and saddle regions, *β* is negative. For arbitrarily small *ϵ*, there will be a region sufficiently close to the local origin where all the terms in (4.1) are smaller than *ϵ*, and thus |*s*_{2}|<*ϵ*^{1/2} and |*s*_{3}|<*ϵ*^{1/3}. It follows that the cubic terms not shown in (4.1), namely , and , are in magnitude smaller than *ϵ*^{7/6}, *ϵ*^{4/3} and *ϵ*^{3/2}, respectively. Barring the vanishing of *β* and *ζ*, these terms can thus be neglected in comparison with and , and are therefore dropped. From (4.1),(4.2)

Elimination of *s*_{1} and *s*_{2} from the parametric equations (2.13) yields the local explicit equation for the wave surface(4.3)The result can be more succinctly expressed by not constraining and to be normal to each other, and using the angular parameter thereby freed to set . In this non-Cartesian frame (though with still normal to and , which are tangential to the wave surface),(4.4)From (4.4), it is evident that locally, the wave surface folds back on itself along the *x*_{2}-axis, and there are two sheets for this surface and hence two arrivals for *x*_{1}<0. For *x*_{1}>0, there is no real branch to the wave surface, but nevertheless for small *x*_{1}, there is a pronounced but non-singular feature or pseudo-arrival as we will see.

### (a) in the region of a cuspidal edge: absence of dispersion

The analytical form of the arrivals and their unfolding is independent of the value of *x*_{2}, and so we confine our attention below to the behaviour of *G*^{3} in the (*x*_{1}, *x*_{3}) plane, setting *x*_{2}=0 in (4.2) when evaluating (2.8) and (2.12). We first consider the case of *β*>0. In the absence of dispersion, equation (2.12), with the change of integration variables to *u*=*β*^{1/2}*s*_{2} and *v*=*ζ*^{1/3}*s*_{1}, yields(4.5)where *C*=*A*/(*β*^{1/2}*ζ*^{1/3}) and(4.6)On integrating (4.5) with respect to *u*, one obtains(4.7)For *x*_{3}=0, the integration of (4.7) with respect to *v* yields(4.8)while for , it yields(4.9)These two power-law dependencies have been discussed by Every & Kim (1994). Away from the - and *x*_{3}-axes, we proceed with a change of integration variable to , which renders (4.7) in the form(4.10)where(4.11)which can be expressed in terms of the complete elliptical integral (Hanyga & Seredynska 1991)(4.12)For and , i.e. within the folded region of the wave surface,(4.13)where *s*_{1}, *s*_{2} and *s*_{3} are respectively the smallest, medial and largest real roots of the cubic equation(4.14)Outside the folded region of the wave surface, i.e. for (i) and (ii) and , has only one real root, *s*_{1}, and(4.15)Asymptotically for large |*α*|,(4.16)from which one recovers the behaviour (4.9) of along the *x*_{3}-axis.

Figure 1*a* is a grey-scale representation of in the vicinity of a cuspidal edge in the wave surface for *β*>0. The outer sheet of the wave surface (in the upper left-hand part of figure 1*a*) propagates a logarithmic arrival and the inner sheet (in the lower left-hand part of figure 1*a*) a step arrival. Both are accompanied by a factor that diverges towards the cuspidal edge at . Beyond the cuspidal edge to the right for a short distance, there is a deep but non-singular minimum to , called a pseudo-arrival. It can be regarded as an arrival associated with a complex ray (Poncelet *et al.* 2000). For *β*<0, the situation is reversed, with the outer sheet of the wave surface propagating a step arrival and the inner sheet a logarithmic arrival.

### (b) in the region of a cuspidal edge: inclusion of dispersion

As in §4*a*, we will confine our attention to in the (*x*_{1}, *x*_{3}) plane section through the wave surface. To accommodate spatial dispersion, we proceed from equation (2.7), with *Φ* given by (4.2). With the change of integration variable to and , we obtain(4.17)This result can be expressed as the convolution integral of the Airy function with the dispersionless Green's function , thus(4.18)

Figure 1*b* is a grey-scale representation of in the neighbourhood of a cuspidal edge in the wave surface for *β*>0, *Χ*=0.1 and *γ*>0. In the region , where the wave surface is folded, each of the two arrivals is now transformed into a quasi-arrival, comprising a rounded main arrival trailed by Airy-type ripples with separation proportional to *Χ*. Where these ripples overlap in the region trailing the lower (step) arrival, an interference pattern results. The region between the two sheets of the wave surface is dominated by the quasi-arrival of the upper logarithmic arrival. For *γ*<0, the ripples precede the arrival and the interference takes place ahead of the first arrival. Figure 2*a*,*b* depicts two sets of waveforms corresponding to sections of figure 1 taken respectively at , which intersects the folded wave surface, and , which passes through the cusp where there is a single arrival. In each set, the behaviour of *G*^{3} with and without dispersion is contrasted. Well ahead of the first arrival, and further to the right of the cusp, the dispersed and undispersed arrivals become indistinguishable.

## 5. Line force Green's function : general formulation

The line force or two-dimensional elastodynamic Green's function, , is defined as the *x*_{q}th component of the displacement response of the infinite continuum at point ** X**=(

*x*

_{1},

*x*

_{3}) and time

*t*to an impulsive force acting in the

*x*

_{p}direction and distributed with unit line density along the

*x*

_{2}-axis. The (

*x*

_{1},

*x*

_{3}) plane is assumed to be a material's symmetry plane, and only the tensor components

*q*,

*p*=1, 3 of will be considered. This is a plane strain problem, with force and displacement confined to the (

*x*

_{1},

*x*

_{3}) plane and independent of

*x*

_{2}. The anti-plane strain problem involved in the calculation of will not be treated here, but it follows along similar lines. The equation of motion for is (Every 2005)(5.1)where the indices run over the values 1 and 3.

Again, we are specifically interested in the singular behaviour of the solution in the vicinity of a wave arrival, and how this unfolds under the influence of spatial dispersion, which can be expressed in the form (Every 2005)(5.2)

The singular behaviour at an observation point *X*_{0} arises in the integration over the neighbourhood of a point *S*_{0} on the slowness surface (the curve representing *S*(*θ*)), where the outward normal, the ray, points in the direction of *X*_{0}, rendering ** S**.

*X*_{0}stationary. The singularity has an arrival time

*S*_{0}.

*X*_{0}=

*t*

_{0}and propagates with group velocity

*V*_{0}=

*X*_{0}/

*t*

_{0}. The locus of points

*X*_{0}for all

*S*_{0}represents the wave surface (curve). As before, we set

**=**

*S*

*S*_{0}+

*S*

_{0}

**, and perform the integrations in (5.2) in a local coordinate frame, in this case with**

*s**s*

_{3}oriented along the outward normal and

*s*

_{1}tangential to the slowness surface at

*S*_{0}. The equation for the slowness surface

*s*

_{3}=

*s*

_{3}(

*s*

_{1}) is approximated locally by a single power of

*s*

_{1}. The location of a point in the neighbourhood of

*X*_{0}is specified by

**=(**

*x***−**

*X*

*X*_{0})/

*X*

_{0}, in a coordinate frame with

*x*

_{3}in the direction of

*S*_{0}, which is normal to the wave surface, and

*x*

_{1}tangential to the wave surface at

*X*_{0}. The local coordinate systems for

**and**

*s***are tilted with respect to each other, their relative orientation being described by the directional cosine . Expressed in terms of these local coordinates,(5.3)(5.4)where(5.5)With the substitutions of (5.3) and (5.4), equation (5.2) becomes(5.6)where . In the dispersionless limit**

*x**γ*→0, on integrating with respect to

*k*, one obtains(5.7)The parametric equations for the singularity set for this integral or wave surface are(5.8)Eliminating the parameter

*s*

_{1}yields the explicit equation for the wave surface,

*x*

_{3}=

*x*

_{3}(

*x*

_{1}).

## 6. Regular points on the slowness and wave surfaces

At a generic or regular point *S*_{0} on the slowness surface, the equation for the slowness surface can locally be approximated by(6.1)and so(6.2)the coefficient *α* being dimensionless and of order unity. The curvature is *L*_{S}=2*α*/*S*_{0} and is finite. Elimination of *s*_{1} from the parametric equations yields the explicit equation for the wave surface(6.3)the curvature of which is *L*_{X}=*η*^{2}/2*αX*_{0}. The curvatures of the slowness and wave surfaces bear the inverse relationship(6.4)

The analytical form of the undispersed arrival is obtained by substituting (6.2) in (5.7), and making use of (6.3), thus(6.5)where . Carrying out the integration yields(6.6)Thus, for convex slowness and wave surfaces (*α*>0), in front of the arrival and behind the arrival, while for concave slowness and wave surface (*α*<0), behind the arrival and in front of the arrival.

The dispersive unfolding of the arrival, or quasi-arrival, is given by(6.7)On integrating with respect to *s*_{1}, one obtains(6.8)With the change of variable to , where(6.9)this yields(6.10)where(6.11)The integral in (6.10) occurs also in the transient response of plates (Kaplunov & Pichugin 2005), and graphical illustrations of the resulting waveforms are provided in that paper. Alternatively, the integration with respect to *k* in (6.7) can be carried out first, leading to expressed as an integral over *s*_{1}, the integrand of which is in the form of a Scorer function. We follow this route for cuspidal points in §7, where it is advantageous. For *γ*<0, the ripples in the quasi-arrival lead the arrival and oscillate around the undispersed waveform, and behind the arrival the quasi-arrival converges rapidly to the undispersed waveform. The spatial separation between the ripples is(6.12)

## 7. Inflection point in slowness surface: cusp in the wave surface

At a point of inflection in the slowness surface *α* vanishes and locally, the equation for the slowness surface is approximated by(7.1)so that(7.2)The coefficient *ζ* is dimensionless and of order unity, and the direction of *s*_{1} is chosen so that it is positive. Elimination of *s*_{1} from the two parametric equations yields the explicit equation for the wave surface(7.3)Thus, locally, the wave surface folds back on itself at *x*_{1}=0, there being two arrivals for *x*_{1}<0 and no real arrival for *x*_{1}>0.

### (a) in the region of a cusp: absence of dispersion

In the absence of dispersion, substitution of (7.2) into (5.7) and changing the integration variable to yields(7.4)where *E*=*D*/ζ^{1/3} and(7.5)It is evident from (7.4) that is an odd function of *x*_{3}, and hence(7.6)while for *x*_{1}=0, integration of (7.4) yields(7.7)Away from the - and *x*_{3}-axes, a change of integration variable to renders (7.4) in the form(7.8)where(7.9)and *P* denotes the principal part of the integral. We evaluate this integral using the Cauchy residue theorem. For , the integrand has one pole at a point *s*_{+} on the real axis and a pair of poles at the complex conjugate points off the real axis. An integration path along the real axis, except for a small detour below the point *s*_{+}, and then completed in the upper half complex plane yields(7.10)For and points lying outside the cuspidal region, again, the integrand has one pole on the real axis, at a point *s*_{−}, and the Cauchy residue theorem yields(7.11)For points lying within the cuspidal region, there are three poles on the real axis, the residues of which cancel, and(7.12)Figure 3*a* is a grey-scale representation of in the vicinity of a cusp in the wave surface. The outer (upper) sheet of the wave surface propagates a −|*x*_{3}|^{−1/2} arrival and the inner (lower) sheet a +|*x*_{3}|^{−1/2} arrival. Both are accompanied by a factor that diverges towards the cusp at .

### (b) in the region of a cusp: inclusion of dispersion

To accommodate spatial dispersion, we proceed from (5.6) with the substitution of (7.2) and change of integration variables to *u*=*kΧ* and , where(7.13)yielding(7.14)where(7.15)is the Scorer function (Scorer 1950; Lee 1980; Gil *et al.* 2000).

Figure 3*b* is a grey-scale representation of in the neighbourhood of a cusp in the wave surface for *γ*>0 and *Χ*=0.1. In the region where the wave surface is folded, each of the two arrivals is transformed into a quasi-arrival, comprising a rounded main arrival trailed by Scorer-type ripples of separation proportional to *Χ*. The region between the two sheets of the wave surface is dominated by the quasi-arrival of the upper arrival. Where these ripples overlap in the region trailing the lower arrival, an interference pattern results. For *γ*<0, the ripples precede the arrival and the interference takes place ahead of the first arrival. Figure 4*a*,*b* depicts two sets of waveforms corresponding to sections of figure 3 taken respectively at , which intersects the folded wave surface, and , which passes through the cusp where there is a single arrival. In each set, the behaviour of *G*^{2} with and without dispersion is contrasted. Well ahead of the first arrival and further to the right of the cusp, the dispersed and undispersed arrivals become indistinguishable.

## Acknowledgements

This research was funded by a grant from the EPSRC. A.G.E. gratefully acknowledges the hospitality of Brunel University, where most of this work was carried out.

## Footnotes

- Received June 6, 2007.
- Accepted August 6, 2007.

- © 2007 The Royal Society