# Ray Expansion Theory

**DOI:**https://doi.org/10.1007/978-3-662-53605-6_97-1

## Synonyms

## Definitions

Ray series are power series, the coefficients of which are the discontinuities in arbitrary order partial time derivatives of the desired functions, while the time of arrival of the wave front is the independent variable; in so doing the order of the partial time derivative coincides with the power exponent of the independent variable.

## Preliminary Remarks

In solving problems of the propagation and attenuation of transient waves carrying the jumps in the field parameters on the wave front, the methods utilizing *ray expansions* are most efficient. A zeroth term of a ray series exactly describes the changes in the field parameter discontinuity along the ray, but the rest of the terms within the radius of the series convergence reveal the changes in the field behind the wave front.

The questions of the ray series application to transient wave problems have been considered by many investigators, and yet until recently the region of its practicality, which is largely determined by a possibility to calculate a sufficient number of the ray series coefficients and by a radius of the ray series convergence, has remained scantily known. As in practice to construct the solution, one is led to restrict to finite truncated ray series, and then in place of the question on the convergence, it is more common to solve the problem on the uniform validity of the decided truncated ray series in the domain of the wave motion existence.

The majority of the studies devoted to the ray methods on the basis of the ray series deals with the volume wave investigation, waves propagating along a free surface and along the interface of two media, as well as in thin bodies (Rossikhin and Shitikova, 1995a; Podil’chuk and Rubtsov, 1996).

## Basic Insights into the Differential Geometry of Moving Surfaces

*Σ*(

*t*), the equation of which can be written in the explicit form as follows:

*x*

_{i}(

*i*= 1, 2, 3) are the Cartesian coordinates and

*θ*

^{α}(

*α*= 1, 2) are the curvilinear coordinates on the surface

*Σ*(

*t*).

*x*

_{i,α}=

*∂x*

_{i}∕

*∂θ*

^{α}are the components of the vectors tangent to the surface. The metric tensor contravariant components

*A*

^{αβ}can be obtained from the covariant components in the usual fashion (McConnell, 1957), such that

From hereafter the Latin indices take on the values 1, 2, and 3, but the Greek indices take on the values 1 and 2, subscripts and superscripts denote the covariant and contravariant properties of the corresponding magnitudes, and the summation is carried out over two repeated indices (except the cases which will be discussed separately).

**n**to the wave surface

*Σ*(

*t*), the following equalities are valid for its components

*n*

_{i}:

*θ*

^{α}yields

*X*

_{α},

*Y*

_{αβ}and contravariant

*X*

^{α},

*Y*

^{αβ}components of an arbitrary vector and arbitrary tensor on the surface

*Σ*(

*t*). These derivatives are invariant with respect to the transformation of the curvilinear coordinates on the wave surface and have the form (McConnell, 1957)

*X*

_{α}=

*A*

_{αβ}

*X*

^{β},

*Y*

_{αβ}=

*A*

_{ασ}

*A*

_{βγ}

*Y*

^{σγ}, and an index after a point denotes tensor (covariant) differentiation with respect to the corresponding coordinate.

Note that distinctions between the covariant and contravariant components disappear in the Cartesian system of coordinates, and the tensor derivatives go over into the partial derivatives with respect to coordinates.

*θ*

^{γ}and considering (6), (7), and (8) yields

*α*,

*β*, and

*γ*, one can find by adding the two of these relations and by subsequent subtraction of the third one that

*x*

_{i,αβ}is directed along the normal

**n**to the surface

*Σ*(

*t*), that is,

*B*

_{αβ}are the coefficients of the second fundamental quadratic form of the surface which are defined from the equality

*x*

_{i,σ}and

*n*

_{i}. Really, taking (2), (9), and (10) into account, we have

*θ*

^{β}and considering (9) yields

*Σ*(

*t*).

*θ*

^{β}, interchange the indices

*β*and

*γ*in the found expression, and then subtract one relation from the other. Having regard for (11), as a result, it could be found that

*x*

_{i,ρ}

*A*

^{ργ}and considering that the Riemann–Christoffel tensor components are defined as

*Ω*and

*K*are the mean and Gaussian curvatures of the surface.

*A*

^{αβ}yields

Relationships (2)–(22) are valid at any instant of the time in the curvilinear coordinate system *θ*^{α} chosen arbitrarily on the surface.

*θ*

^{α}could be introduced for simplicity. For this purpose it could be assumed that the vector field of the unit vector

**n**(with the components

*n*

_{i}(

*x*

_{j})) normal to the moving surface

*Σ*(

*t*) is specified in that domain of the space through which the surface

*Σ*(

*t*) has already gone. Let this field be such that only one integral curve can be drawn through each point of the surface

*Σ*(0), in so doing the vector

**n**is directed along a tangent to this curve. In the subsequent discussion, the integral curve will be named as

*the normal trajectory*. The points of the surface lying along one normal trajectory possess the same curvilinear coordinates

*θ*

^{α}which are equal to the coordinates of its initial point

*x*

_{i}=

*x*

_{i}(

*θ*

^{α}, 0) (Fig. 1). Under such a decision on the coordinate system, the partial derivative of a function with respect to time is equal to the time derivative along the normal to the surface

*Σ*(

*t*) and is designated by

*δ*∕

*δt*(Thomas, 1961a). Then the following formulas are true:

*G*=

*ds*∕

*dt*is the velocity of propagation of the surface

*Σ*(

*t*) in the normal direction and

*s*is the arc length measured along the normal trajectory from the point with the coordinates

*x*

_{i}=

*x*

_{i}(

*θ*

^{α}, 0) (Fig. 1).

*A*

^{αβ}

*x*

_{j,β}, relationships (24) take the form

*x*

_{i,γ}and

*n*

_{i}, as well as considering equalities (4) and (11), the following formula could be found from (25)

If an elastic medium is isotropic, then *G* = const, and it follows from (26) that *δn*_{i}∕*δt* = 0. This condition implies that the normal trajectories of the spatial surfaces *Σ*(*t*) are straight lines, and hence the surfaces *Σ*(*t*) form a family of parallel surfaces.

Let us next calculate the *δ*-derivatives of the values *x*_{i,α}, *A*_{αβ}, *A*^{αβ}, *B*_{αβ}, *Ω*, and *K* with respect to time.

*t*and multiplying the resulted relation by

*A*

^{αγ}, with allowance made for (28), yield

*δB*

_{αβ}∕

*δt*, it is sufficient to differentiate (10) with respect to

*t*and take account for (9) and (26). Then

*G*

_{,α}with respect to

*θ*

^{β}and with due account for (12)

*δ*∕

*δt*of the left and right parts of (13) yields

*δ*-derivative of (18) with respect to

*t*and considering expressions (20), (29), and (31) result in

*K*, let us differentiate the right and left parts of (20) with respect to

*δ*∕

*δt*and take (32), (33), and (29) into account

*G*

_{,αβ}= 0, and from (33) and (34), it follows that

*K*and

*Ω*, and then subtract the obtained equations, resulting in

*Ω*

^{∗}and

*K*

^{∗}are the median and Gaussian curvatures of the surface

*Σ*(0), respectively.

Formulas (39) and (40) were derived for the first time in Thomas (1961b) but using the more complicated procedure.

It should be remembered that the normal trajectories may be inconsistent with the lines along which disturbances propagate with the velocities *G*_{L}, as is the case for anisotropic bodies, as an example. These lines are said to be *the rays*. However, wherever it would not be a matter of concern, one could credit *the normal trajectories as the rays*.

## Conditions of Compatibility

*f*dependent on the Cartesian coordinates

*x*

_{i}and the time

*t*is defined in the space

*x*

_{i}. Then on the surface

*Σ*(

*t*) where the coordinates

*θ*

^{α}are determined in the same manner as it has been done above.

*f*with respect to time has the form

*δx*

_{i}∕

*δt*from (23) yields

*f*with respect to the coordinates of the space

*x*

_{i}are connected with the derivatives with respect to the curvilinear coordinates of the surface

*Σ*(

*t*) by the relationships

The validity of the equality (42) could be demonstrated by its projection on *n*_{i} and *x*_{i,γ}.

Let the function *f*(*x*_{i}, *t*) be continuous and differentiable in each of the domains *V*^{+} and *V*^{−} adjacent to the surface *Σ*(*t*) on two sides (Fig. 1). On the surface *Σ*(*t*) itself, the function *f* and its partial derivatives may have a discontinuity. Such a surface will be named as a *wave surface* or, simply, *a wave*. Let *f*^{+} and *f*^{−} be the magnitudes of *f* on the wave surface *Σ*(*t*) when approaching *Σ*(*t*) by the points of the domains *V*^{+} and *V*^{−}, respectively.

*f*

^{+}into the domain

*V*

^{−}and the function

*f*

^{−}into the domain

*V*

^{+}such that each will be continuous and differentiable in the domain

*V*

^{+}+

*V*

^{−}. Then relationships (41) and (42) are fulfilled on the wave surface

*Σ*(

*t*) for

*f*

^{+}and

*f*

^{−}, i.e.,

*f*] =

*f*

^{+}−

*f*

^{−}.

Relationships (45) are said to be *the geometric conditions of compatibility*, but relationships (46) designate *the kinematic conditions of compatibility*.

*f*

_{,j}]

*n*

_{j}from (45) and (46) yields

*f*for

*f*

_{,(k)}=

*∂*

^{k}

*f*∕

*∂t*

^{k}(

*k*= 1, 2, 3, …) in (47) results in

*φ*is continuous on the surface

*Σ*(

*t*) and in the domains

*V*

^{+}and

*V*

^{−}, in so doing the function

*φ*has continuous first and second derivatives in each domain, which, generally speaking, may have a discontinuity on the surface

*Σ*(

*t*). Then replacing the function

*f*by the derivatives of the function

*φ*, from relations (45) and (46), it follows that

*φ*is continuous on the surface

*Σ*(

*t*), then from relations (45) and (46), it follows

*φ*

_{,i}] in (49) and considering (11) yield

*φ*

_{,ij}] and

*A*

^{αβ}

*A*

^{στ}

*B*

_{σα}[

*φ*

_{n}]

*x*

_{i,τ}

*x*

_{j,β}are symmetric about the indices

*i*and

*j*, then the following equality should be held:

*n*

_{j}yields

*φ*

_{,i})

_{n}] from (56) into (54) gives where [

*φ*

_{nn}] = [(

*φ*

_{,j})

_{n}]

*n*

_{j}= [

*φ*

_{,ij}]

*n*

_{i}

*n*

_{j}.

Relationships (57) are *the geometric conditions of compatibility for the second-order derivatives of the continuous function* *φ*.

*φ*are continuous, then [

*φ*

_{n}] = 0, and from (57), it follows that

*φ*

_{,i})

_{n}] from (56) in (51) yields

Relationships (60) and (63) are said to be *the kinematic conditions of compatibility for the second-order derivatives of the continuous function* *φ*.

Besides the geometric and kinematic conditions of compatibility, there exist in addition the dynamic conditions of compatibility which are defined by governing dynamic equations describing the behavior of a material medium. With these conditions, propagation velocities of various types of waves are determined.

To derive the dynamic conditions of compatibility, the following method is used: the surface of discontinuity *Σ* is interpreted as the limiting layer of the thickness *h* at *h* → 0, within which the desired values, the components of the stress tensor *σ*_{ij} and displacement velocity vector *v*_{i}, as an example, change from the magnitudes \(\sigma _{ij}^+, v_i^+\) to the magnitudes \(\sigma _{ij}^-, v_i^-\) monotonically and uninterruptedly. Considering that within the wave layer relations (41) and (42) are fulfilled, and noting that in these relations at *h* → 0 the derivatives with respect to the normal to the layer *df*∕*dn* = *f*_{,i}*n*_{i} are large in comparison with other derivatives, from the governing system of dynamic equations, one can obtain the system of ordinary differential equations of the first order in the derivatives *d*∕*dn* of the functions to be found. Integrating the resulting set of equations with respect to the normal to the wave layer from − *h*∕2 to *h*∕2 and going to the limit at *h* → 0, the dynamic conditions of compatibility for the chosen material medium could be obtained.

### Conditions of Compatibility in Arbitrary Coordinates

In the deduction of the compatibility conditions in the previous section, it has been assumed that the coordinates *x*_{i} are the Cartesian coordinates. Therefore the form obtained for the compatibility conditions is invariant, generally speaking, only with respect to the orthogonal transformation of the coordinates *x*_{i}, although there exists invariance with respect to arbitrary differentiable transformations of the curvilinear coordinates *θ*^{α} of the surface *Σ*(*t*). Consider the question of writing the conditions of compatibility in arbitrary coordinates *x*^{i}, which can be derived by the transformation from the Cartesian coordinates.

*Σ*(

*t*) will be also defined by equations of the type (1) or as

**n**to the surface are the following, respectively:

*t*and considering formulas (23) and (65) yield

The contravariant *A*^{ij} and covariant *A*_{ij} components of the metric tensor of a space in the coordinate system *x*^{i} can be obtained from the corresponding components *δ*^{ij} and *δ*_{ij} of this tensor in the Cartesian system with the help of ordinary transformations.

**X**

^{i}contravariant with respect to differentiable transformations of the coordinates

*x*

^{i}, but relatively to differentiable transformations of the curvilinear coordinates of the surface

*θ*

^{α}, these values are the components of the covariant surface vector

**X**

_{α}. As in the previous section, it may be noted that both vectors

**X**

_{α}are tangent to the corresponding coordinate lines

*θ*

^{α}of the surface

*Σ*(

*t*).

*Σ*(

*t*) will be designated by

*A*

^{αβ}and

*A*

_{αβ}as before, in so doing

This does not lead to confusion in notation, since the components of the metric tensors of the space and of the surface can be distinguished by their Greek and Latin indices, respectively.

*f*(

*x*

^{i},

*t*) denote a scalar or some component of a vector or tensor in the space. It is easily seen that the geometric conditions of compatibility (45) of the first order for

*f*in the arbitrary coordinates

*x*

^{i}take the form

*f*] with respect to the surface coordinates

*θ*

^{α}. Really, relations (70) are invariant with respect to the differentiable spatial and surface coordinates, and, moreover, they reduce to the conditions of compatibility (45) if the coordinates

*x*

^{i}are the Cartesian ones.

*f*=

*X*

_{i}, where

*X*

_{i}are the covariant components of a certain vector. From the transformation formulas of the components

*X*

_{i}, it readily follows that during transition from the coordinates

*x*

^{i}to the new coordinates \(\overline {x^i}\), the discontinuities [

*X*

_{i}] are transformed by the formulas

*δ*-differentiation with respect to time to the both sides of the equality (71) yields

*x*

^{i}and new \(\overline {x^i}\) systems of coordinates (McConnell, 1957)

Equalities (75) express the fact that the values *D*[*X*_{k}]∕*Dt* are the components of the covariant vector. Moreover, it is evident that *D*[*X*_{k}]∕*Dt* are reduced to *δ*[*X*_{k}]∕*δt* in the Cartesian coordinate system. The value possessing the components *D*[*X*_{k}]∕*Dt* will be denoted as *the absolute or invariant derivative of the discontinuity* [*X*_{k}] *with respect to time*. Application of precisely these values instead of the *δ*-derivatives with respect to time allows one thereafter to write the governing relationships in the form invariant with respect to common transformations of coordinates.

**X**

_{i}with respect to the spatial coordinates

*x*

^{k}by

*X*

_{i,k}, and consider the relation

Relations (77) are invariant with respect to common transformations of the coordinates *x*^{i}. It follows from (75) and from the fact that the values \([ \dot X_i ]\) and [*X*_{i,j}]*n*^{j} are defined by tensor transformations as well. Since relationships (77) in the Cartesian coordinates are reduced to the conditions of compatibility (46), then relations (77) represent the record of *the kinematic conditions of compatibility of the first order for* **X**_{i} *in the general coordinates*.

Conditions (78) are invariant both with respect to arbitrary differentiable transformations of the spatial coordinates *x*^{i} and with respect to transformations of the curvilinear surface coordinates *θ*^{i}.

It is evident that in a similar way one can obtain the geometric and kinematic conditions of compatibility of the second and higher orders in the general invariant form.

*f*

_{,j}]

*n*

^{j}from (70) and (78) yields

*x*

^{i}and, moreover, it is reduced to (26) in the case when the coordinates

*x*

^{i}are the Cartesian coordinates.

### The Case When the Spatial Coordinates Coincide with the Ray Coordinates

The set of three orthogonal coordinates, two of which locate on the wave surface while the third one is directed along the normal trajectory of the moving surface, is named as the *ray coordinate system*.

*x*

^{1}and

*x*

^{2}coincide at each instant of the time, respectively, with the coordinates

*θ*

^{1}and

*θ*

^{2}on the surface

*Σ*(

*t*) moving in the direction of the coordinate lines

*x*

^{3}, i.e.,

*θ*

^{1},

*θ*

^{2},

*x*

^{3}are the ray coordinated (Fig. 2). In this case,

*n*

_{1}=

*n*

_{2}= 0,

*n*

_{3}= 1, and \(x^3=G t+x_0^3\). Calculations will be carried out for the covariant components of the vector

**X**

_{i}on the assumption that the spatial coordinates are orthogonal. Putting

*f*=

*X*

_{i}in (79) and considering relations (76), as well the following formula (McConnell, 1957)

*i*= 3 in (82) yields

*i*=

*α*(

*α*= 1, 2) in (82), it follows that

*X*

_{st}instead of

*f*in (79) and considering that

*i*= 3 in (87) provides

*i*=

*α*(

*α*= 1, 2), then the identity could be obtained

Formulas similar to (83), (84), (88), and (89) could be obtained for the contravariant components of a vector and a tensor, as well as for the tensor mixed components.

### Conditions of Compatibility for the Physical Components

*the physical components*of a vector \(X_i^{ph}\) or a tensor \(X_{st}^{ph}\), which are connected with vector’s or tensor’s covariant and contravariant components by the following relationships:

*i*,

*s*,

*t*indices is not performed.

Similar formulas could be obtained for the values \(X^i_{ph}\), \(X^{st}_{ph}\), and \((X^s_t)_{ph}\), if one uses (91) in the condition of compatibility.

### Conditions of Compatibility for Rods and Beams

When solving particular boundary-value problems connected with the transient wave propagation in rods and beams, one deals with systems of the *n*th-order differential equations, which require for their analysis the utilization of the conditions of compatibility for the *n*th-order derivatives of the desired values.

*Z*is a physical value to be found and

*z*is the coordinate along which transient wave propagates.

The validity of formula (95) could be proved by the method of mathematical induction.

The compatibility condition (95) can be used for investigating transient processes in solid rods and thin-walled beams (see entries “Boundary-Value Dynamic Problems of Thin Bodies, Ray Expansion Approach”), in so doing the *z*-coordinate is directed along the axis of a rod or a beam.

## Ray Series and Their Application for Solving Boundary-Value Dynamic Problems

*f*

_{,(k)}] = [

*∂*

^{k}

*f*∕

*∂t*

^{k}] are the jumps in the

*k*th-order time derivatives of the function to be found,

*s*is the arc length measured along the ray from a certain initial wave surface,

*t*is the time,

*H*(

*t*) is the unit Heaviside function, \(\tau (s)=\int \limits _0^s\frac {ds}{G(s)}\), and

*G*(

*s*) is the normal speed of the wave surface.

The series (96) is used for solving boundary-value problems dealing with the propagation of wave surfaces of strong and weak discontinuities. To determine the coefficients of the ray series of the type (96) for the desired functions, it is necessary to differentiate the governing equations describing the dynamic behavior of a material medium *k* times with respect to time, take their difference on the different sides of the wave surface *Σ*(*t*), and apply the compatibility condition (48) for the (*k* + 1)th-order discontinuities of the desired functions. As a result of the procedure described, one is led to the closed set of recurrent first-order partial differential equations and algebraic equations with respect to the *k*th-order discontinuities of the desired values. As this takes place, the main characteristics defining the type of a wave fulfill the differential equations, but the “admixed” characteristics pointing to the association of this wave with other types of waves are found from the algebraic equations. The functions to be found are determined from the set of recurrent equations with an accuracy of arbitrary functions dependent on the surface coordinates *θ*^{1} and *θ*^{2}, which are found, in their turn, from the boundary conditions (Rossikhin and Shitikova, 1995a).

*σ*(

*x*,

*t*) (0 ≤x ≺

*∞*) behind the front of the strong discontinuity surface was obtained using the ray series (96) in the form

where Open image in new window are the discontinuities in the *n*-the order time derivatives of the stress, \(G=\left (\rho J_{0} \right )^{-1/2} \) is the propagation velocity of the discontinuity surface, and *J*_{0} = *J* and *J*(*t*) is the uniaxial creep function.

*v*

_{0}for

*t*> 0 and a rod of a standard linear solid subjected to a constant stress

*σ*

_{0}for

*t*> 0. In both examples, the relations for the dimensionless stress

*Z*can be obtained as

*z*

_{k}(

*ξ*) are the polynomials of degree

*k*in the variable

*ξ*which harbor

*k*arbitrary constants to be determined from the initial conditions, \(\xi =x\left (2G\tau _{0} \right )^{-1} \),

*τ*

_{0}is the relaxation time, \(\tau =t\left (2\tau _{0} \right )^{-1} \); and

*α*is a certain constant.

*ε*is a small value.

However, for a rod of significant extent, the near-front domains, wherein the exact solutions are closely approximated by the truncated ray series (100), are rapidly narrowed down contracting to the wave front with increase in *ξ*. The reason is that though the ray series (97) is a convergent series, its convergence is slow, and one fails to represent the solution for all *ξ* by a finite number of terms of (100). Slow convergence is governed by the fact that for the expansion (100), the ratio of the following term to the preceding term increases without bound with *ξ* →*∞*, i.e., the condition for uniform validity (Nayfeh, 1973) is violated for this expansion, and the further the wave surface gets from the boundary *ξ* = 0.

*ε*(Rossikhin and Shitikova, 1994). As a result, the recurrent Eq. (98) could be obtained. Equation (101) could be solved by the method of multiple scales (Nayfeh, 1973), in so doing the value

*Z*is represented as

where \(T_{n} =\varepsilon ^{n} t\left (n=0,1,2,\ldots \right )\) are new independent variables.

*ε*, and eliminating secular terms yield

*a*

_{n}are constants dependent on boundary conditions,

*b*

_{0}=

*a*

_{1}, and

For the expansion (103) to be treated as a uniformly applicable forward-area asymptotic, it is necessary to replace *τ* with *ξ* and the value *ε*^{n} by Open image in new window assuming that *ξ* varies from 0 to *τ*.

The method resulting in the expansions (103) has been named as the method of “forward-area regularization” (Rossikhin and Shitikova, 1994).

*τ*. It is seen that the expressions (103) give us forward-area asymptotics stable with reference to the wave front movement.

Thus, a formal increase in the number of ray series terms may not give the wanted result in the improvement of the solution approximation if the wave front is at a considerable distance from a boundary surface. To eliminate such a contradiction, it is coincidently necessary, with the increase in the number of terms of a ray series, to eliminate singular items entering into these terms due to the method of forward-area regularization.

## Concluding Remarks

In the present entry, it has been shown that for solving problems of the propagation and attenuation of transient waves carrying the jumps in the field parameters on the wave front, the methods utilizing ray expansions are most efficient. A zeroth term of a ray series exactly describes the changes in the field parameter discontinuity along the ray, but the rest of the terms within the radius of the series convergence reveal the changes in the field behind the wave front.

The application of one-term and multiple-term ray expansions in solving various problems of wave dynamics in elastic, thermoelastic, anisotropic, and elasto-visco-plastic media as well as the methods for improvement of these expansions could be found in Rossikhin and Shitikova (1995a,b), Podil’chuk and Rubtsov (1996), Rossikhin and Shitikova (1999, 2000, 2007, 2008, 2010), as well as are illustrated in entries “Transient Waves in Cosserat Beams: Ray Expansion Approach”; “Discontinuity Surfaces in Elasto-Visco-Plastic Medium”; “Boundary-Value Dynamic Problems of Thin Bodies, Ray Expansion Approach”; “Plane Transient Waves in Anisotropic Layer, Ray Expansion Approach”; “Ray Expansions in Dynamic Contact Problems”; “Ray Expansions in Impact Interaction Problems”.

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