# Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

# Ray Expansion Theory

• Yury A. Rossikhin
• Marina V. Shitikova
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_97-1

## 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

Let the surface Σ(t), the equation of which can be written in the explicit form as follows:
\displaystyle \begin{aligned} x_i=x_i(\theta ^\alpha,t), \end{aligned}
(1)
move in an elastic continuum, where xi (i = 1, 2, 3) are the Cartesian coordinates and θα (α = 1, 2) are the curvilinear coordinates on the surface Σ(t).
The covariant components of the metric tensor for the surface (1) are defined using the equalities
\displaystyle \begin{aligned} A_{\alpha\beta}=x_{i,\alpha}x_{i,\beta}, \end{aligned}
(2)
where xi,α = ∂xi∂θα are the components of the vectors tangent to the surface. The metric tensor contravariant components Aαβ can be obtained from the covariant components in the usual fashion (McConnell, 1957), such that
\displaystyle \begin{aligned} A_{\alpha\sigma} A^{\sigma\beta}=\delta_\alpha^\beta,\qquad \delta_\alpha^\beta=\left\{\begin{array}{cccc} \displaystyle 1\quad \alpha=\beta\\ \displaystyle 0\quad \alpha\ne\beta \end{array}\right. \end{aligned}
(3)
where $$\delta _\alpha ^\beta$$ is the Kronecker symbol.

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).

Introducing the outward unit normal n to the wave surface Σ(t), the following equalities are valid for its components ni:
\displaystyle \begin{aligned} n_i n_i=1,\qquad \quad x_{i,\alpha} n_i=0. \end{aligned}
(4)
Differentiating the first relationship of (4) with respect to θα yields
\displaystyle \begin{aligned} n_{i,\alpha} n_i=0. \end{aligned}
(5)
Let us introduce the tensor (covariant) derivative of the covariant Xα, Yαβ and contravariant Xα, Yαβ 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)
\displaystyle \begin{aligned} X_{\alpha,\beta}&=\frac{\partial X_\alpha}{\partial \theta ^\beta} -\varGamma_{\alpha\beta}^\sigma X_\sigma,\\ Y_{\alpha\beta,\gamma}&=\frac{\partial Y_{\alpha\beta}}{\partial \theta ^\gamma} -\varGamma_{\alpha\gamma}^\sigma Y_{\sigma\beta} -\varGamma_{\beta\gamma}^\sigma Y_{\alpha\sigma}, \end{aligned}
(6)
\displaystyle \begin{aligned} X_{,\beta}^\alpha&=\frac{\partial X^\alpha}{\partial \theta ^\beta} +\varGamma_{\sigma\beta}^\alpha X^\sigma,\\ Y_{,\gamma}^{\alpha\beta}&=\frac{\partial Y^{\alpha\beta}}{\partial \theta ^\gamma} +\varGamma_{\sigma\gamma}^\alpha Y^{\sigma\beta} +\varGamma_{\sigma\gamma}^\beta Y^{\alpha\sigma}. \end{aligned}
(7)
Here
\displaystyle \begin{aligned} \varGamma_{\sigma\gamma}^\alpha=\frac 12 A^{\alpha\beta}\left(\frac{\partial A_{\sigma\beta}}{\partial \theta ^\gamma} +\frac{\partial A_{\gamma\beta}}{\partial \theta ^\sigma}-\frac{\partial A_{\sigma\gamma}}{\partial \theta ^\beta}\right) \end{aligned}
(8)
are the Christoffel symbols on the surface, Xα = AαβXβ, Yαβ = AασAβγYσγ, 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.

Taking the tensor derivative of equality (2) with respect to θγ and considering (6), (7), and (8) yields
\displaystyle \begin{aligned} 0=A_{\alpha\beta,\gamma}=x_{i,\alpha\gamma} x_{i,\beta}+x_{i,\alpha} x_{i,\beta\gamma}, \end{aligned}
and then deriving from this relationship the other two by the circular rearrangement of the indices α, β, and γ, one can find by adding the two of these relations and by subsequent subtraction of the third one that
\displaystyle \begin{aligned} x_{i,\alpha\beta}x_{i,\gamma}=0. \end{aligned}
In other words, xi,αβ is directed along the normal n to the surface Σ(t), that is,
\displaystyle \begin{aligned} x_{i,\alpha\beta}=B_{\alpha\beta}n_i. \end{aligned}
(9)
Here Bαβ are the coefficients of the second fundamental quadratic form of the surface which are defined from the equality
\displaystyle \begin{aligned} x_{i,\alpha\beta}n_i=B_{\alpha\beta}. \end{aligned}
(10)
Point to the identity
\displaystyle \begin{aligned} n_{i,\alpha}=-A^{\beta\gamma}B_{\beta\alpha}x_{i,\gamma} \end{aligned}
(11)
the validity of which can be checked by projecting it onto xi,σ and ni. Really, taking (2), (9), and (10) into account, we have
\displaystyle \begin{aligned} n_{i,\alpha}x_{i,\sigma}&=(n_i x_{i,\sigma})_{,\alpha} -n_i x_{i,\sigma\alpha}\quad =-B_{\sigma\alpha}, \\ &-A^{\beta\gamma}B_{\beta\alpha}x_{i,\gamma}x_{i,\sigma}=-A^{\beta\gamma} B_{\beta\alpha}A_{\gamma\sigma}\\ &=-B_{\beta\alpha}\delta_\sigma^\beta=-B_{\sigma\alpha}, \end{aligned}
and
\displaystyle \begin{aligned} n_{i,\alpha} n_i=-A^{\beta\gamma}B_{\beta\alpha}x_{i,\gamma} n_i=0, \end{aligned}
whence it follows the validity of the identity (11).
Differentiating (11) with respect to θβ and considering (9) yields
\displaystyle \begin{aligned} n_{i,\alpha\beta}=-C_{\alpha\beta} n_i-A^{\sigma\tau}B_{\sigma\alpha,\beta} x_{i,\tau}. \end{aligned}
(12)
Here
\displaystyle \begin{aligned} C_{\alpha\beta}=A^{\rho\gamma}B_{\rho\alpha}B_{\gamma\beta} \end{aligned}
(13)
are the coefficients of the third fundamental quadratic form of the surface Σ(t).
Let us take the tensor derivative of (9) with respect to θβ, 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
\displaystyle \begin{aligned} x_{i,\alpha\beta\gamma}&-x_{i,\alpha\gamma\beta}=(B_{\alpha\beta,\gamma} -B_{\alpha\gamma,\beta})n_i\\ &-A^{\sigma\tau}(B_{\alpha\beta}B_{\sigma\gamma} -B_{\alpha\gamma}B_{\sigma\beta})x_{i,\tau}. \end{aligned}
(14)
Multiplying the equality (14) by xi,ρAργ and considering that the Riemann–Christoffel tensor components are defined as
\displaystyle \begin{aligned} R_{\rho\alpha\beta\gamma}=(x_{i,\alpha\beta\gamma}-x_{i,\alpha\gamma\beta})x_{i,\rho} \end{aligned}
(15)
yield
\displaystyle \begin{aligned} R_{\rho\alpha\beta\gamma}A^{\rho \gamma }=-2\varOmega B_{\alpha \beta }+C_{\alpha \beta }, \end{aligned}
(16)
or with due account for
\displaystyle \begin{aligned} R_{\rho\alpha\beta\gamma}A^{\rho \gamma }=-K A_{\alpha \beta }, \end{aligned}
from (14), it follows that
\displaystyle \begin{aligned} K A_{\alpha \beta }-2\varOmega B_{\alpha \beta }+C_{\alpha \beta }=0. \end{aligned}
(17)
Here
\displaystyle \begin{aligned} 2\varOmega=A^{\rho\gamma}b_{\rho\gamma}, \end{aligned}
(18)
\displaystyle \begin{aligned} 2K=-R_{\rho\alpha\beta\gamma}A^{\rho\gamma}A^{\alpha\beta}, \end{aligned}
(19)
where Ω and K are the mean and Gaussian curvatures of the surface.
Multiplying (17) by Aαβ yields
\displaystyle \begin{aligned} C_{\alpha\beta}A^{\alpha\beta}=4\varOmega^2-2K , \end{aligned}
(20)
or
\displaystyle \begin{aligned} B^{\alpha\beta}B_{\alpha\beta}=4\varOmega^2-2K , \end{aligned}
(21)
since according to (13)
\displaystyle \begin{aligned} C_{\alpha\beta}A^{\alpha\beta}=B^{\alpha\beta}B_{\alpha\beta}. \end{aligned}
(22)

Relationships (2)–(22) are valid at any instant of the time in the curvilinear coordinate system θα chosen arbitrarily on the surface.

To derive the succeeding expressions useful for applications, the specialized coordinate system θα could be introduced for simplicity. For this purpose it could be assumed that the vector field of the unit vector n (with the components ni(xj)) 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 θα which are equal to the coordinates of its initial point xi = xi(θα, 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 δδt (Thomas, 1961a). Then the following formulas are true:
\displaystyle \begin{aligned} \frac{\partial x_i(u^\alpha,t)}{\partial t}= \frac{\delta x_i(u^\alpha,t)}{\delta t}=\frac{d x_i}{ds}\;\frac{ds}{dt}=G n_i, \end{aligned}
(23)
where G = dsdt 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 xi = xi(θα, 0) (Fig. 1).
Let us find the variation of the normal to the surface during its motion along the normal trajectory. From (4) and (23), it follows
\displaystyle \begin{aligned} \begin{array}{rcl} {} x_{i,\alpha}\,\frac{\delta n_i}{\delta t}&\displaystyle =&\displaystyle -n_i\,\frac{\delta x_{i,\alpha}}{\delta t}= -n_i \left(\frac{\delta x_i}{\delta t}\right)_{,\alpha}\\ &\displaystyle =&\displaystyle -n_i(G n_i)_{,\alpha}, \end{array} \end{aligned}
(24)
After the convolution with Aαβxj,β, relationships (24) take the form
\displaystyle \begin{aligned} A^{\alpha\beta}x_{i,\alpha}x_{j,\beta}(\delta n_i/\delta t)= -A^{\alpha\beta}x_{j,\beta} n_i (G n_i)_{,\alpha}. \end{aligned}
(25)
Using the identity
\displaystyle \begin{aligned} A^{\alpha\beta}x_{i,\alpha}x_{j,\beta}&=\delta_{ij}-n_i n_j,\\ \delta_{ij}&=\left\{\begin{array}{cccc} \displaystyle 1,\quad i=j\\ \displaystyle 0,\quad i\ne j \end{array}\right.\end{aligned}
the validity of which is easily verified by projecting it on xi,γ and ni, as well as considering equalities (4) and (11), the following formula could be found from (25)
\displaystyle \begin{aligned} \frac{\delta n_i}{\delta t}=-A^{\alpha\beta}G_{,\alpha}x_{i,\beta} . \end{aligned}
(26)

If an elastic medium is isotropic, then G = const, and it follows from (26) that δniδt = 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 xi,α, Aαβ, Aαβ, Bαβ, Ω, and K with respect to time.

Using formulas (23) and (26) yields
\displaystyle \begin{aligned} \begin{array}{rcl} {} \frac{\delta x_{i,\alpha}}{\delta t}&\displaystyle =&\displaystyle \frac{\partial}{\partial \theta ^\alpha} \left(\frac{\delta x_i}{\delta t}\right)=\frac{\partial}{\partial \theta ^\alpha} (G n_i)\\ &\displaystyle =&\displaystyle G_{,\alpha} n_i-GA^{\beta\gamma}B_{\beta\alpha}x_{i,\gamma} . \end{array} \end{aligned}
(27)
Considering (2) and (27), it follows that
\displaystyle \begin{aligned} \frac{\delta A_{\alpha\beta}}{\delta t}=\frac{\delta x_{i,\alpha}}{\delta t}\, x_{i,\beta}+\frac{\delta x_{i,\beta}}{\delta t} \,x_{i,\alpha}=-2GB_{\alpha\beta} . \end{aligned}
(28)
Differentiating (3) with respect to time t and multiplying the resulted relation by Aαγ, with allowance made for (28), yield
\displaystyle \begin{aligned} \frac{\delta A^{\gamma\beta}}{\delta t}=2G B^{\gamma\beta} . \end{aligned}
(29)
To determine δBαβδt, it is sufficient to differentiate (10) with respect to t and take account for (9) and (26). Then
\displaystyle \begin{aligned} \frac{\delta B_{\alpha\beta}}{\delta t}=\frac{\delta x_{i,\alpha\beta}}{\delta t}\, n_i . \end{aligned}
(30)
Using the definitions of the tensor derivative (6)–(7) and considering (4), (11), (23), and (27) yield
\displaystyle \begin{aligned} \begin{array}{rcl} \frac{\delta x_{i,\alpha\beta}}{\delta t} \, n_i&\displaystyle =&\displaystyle \frac{\delta}{\delta t}\left(\frac{\partial^2 x_i}{\partial \theta ^\alpha\partial \theta ^\beta}\right) n_i -\frac{\delta}{\delta t}\left(x_{i,\sigma}\varGamma_{\alpha\beta}^\sigma\right) n_i \\ &\displaystyle =&\displaystyle \frac{\partial^2(G n_i)}{\partial \theta ^\alpha\partial \theta ^\beta}\, n_i -G_{,\sigma}\varGamma_{\alpha\beta}^\sigma=\frac{\partial^2G}{\partial \theta ^\alpha\partial \theta ^\beta}\\ &\displaystyle &\displaystyle -G_{,\sigma}\varGamma_{\alpha\beta}^\sigma-\frac{\partial^2 n_i}{\partial \theta ^\alpha\partial \theta ^\beta} \, n_iG . \end{array} \end{aligned}
Since according to (6) the first two terms of this relationships produce the tensor derivative of G,α with respect to θβ and with due account for (12)
\displaystyle \begin{aligned} n_{i,\alpha\beta} n_i&=\frac{\partial^2 n_i}{\partial \theta ^\alpha\partial \theta ^\beta} \, n_i-n_{i,\sigma}\varGamma_{\alpha\beta}^\sigma n_i\\ &=\frac{\partial^2 n_i}{\partial \theta ^\alpha\partial \theta ^\beta}\, n_i=-C_{\alpha\beta}, \end{aligned}
then it could be conclusively obtained
\displaystyle \begin{aligned} \frac{\delta B_{\alpha\beta}}{\delta t}=G_{,\alpha\beta}-G C_{\alpha\beta} . \end{aligned}
(31)
Taking δδt of the left and right parts of (13) yields
\displaystyle \begin{aligned} \frac{\delta C_{\alpha\beta}}{\delta t}&=\frac{\delta }{\delta t}\left( A^{\sigma \tau } B_{\alpha \sigma } B_{\beta \tau } \right)\\ &=A^{\sigma \tau }\left( B_{\alpha \sigma } G,_{\beta \tau } +B_{\beta \tau } G,_{\alpha \sigma }\right) . \end{aligned}
(32)
Now taking the δ-derivative of (18) with respect to t and considering expressions (20), (29), and (31) result in
\displaystyle \begin{aligned} \frac{\delta\varOmega}{\delta t}=2G \varOmega^2-GK+\frac 12A^{\alpha\beta}G_{,\alpha\beta} . \end{aligned}
(33)
To obtain the differential equation determining the value K, let us differentiate the right and left parts of (20) with respect to δδt and take (32), (33), and (29) into account
\displaystyle \begin{aligned} \frac{\delta K}{\delta t}=2\varOmega KG+ G_{,\alpha\beta}(2\varOmega A^{\alpha\beta}-B^{\alpha\beta}) \end{aligned}
(34)
If an elastic medium is isotropic, then G,αβ = 0, and from (33) and (34), it follows that
\displaystyle \begin{aligned} \frac{\delta\varOmega}{\delta t}=2G\varOmega^2-GK , \end{aligned}
(35)
\displaystyle \begin{aligned} \frac{\delta K}{\delta t}=2\varOmega KG . \end{aligned}
(36)
In order to integrate the set of Eqs. (35) and (36), let us first multiply them, respectively, by K and Ω, and then subtract the obtained equations, resulting in
\displaystyle \begin{aligned} \frac{\delta}{\delta t}\left( \frac{\varOmega}{K}\right)=-G, \end{aligned}
(37)
or after integration
\displaystyle \begin{aligned} \varOmega = K (-Gt+\varOmega ^* {K^*}^{-1}), \end{aligned}
(38)
where Ω and K are the median and Gaussian curvatures of the surface Σ(0), respectively.
Thus, the solution of Eqs. (35) and (36) has the form
\displaystyle \begin{aligned} \varOmega=\frac{\varOmega^*-K^*Gt}{1-2\varOmega^*Gt+K^*(Gt)^2}, \end{aligned}
(39)
\displaystyle \begin{aligned} K=\frac{K^*}{1-2\varOmega^*Gt+K^*(Gt)^2} . \end{aligned}
(40)

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 GL, 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

Assume that the function f dependent on the Cartesian coordinates xi and the time t is defined in the space xi. Then on the surface Σ(t) where the coordinates θα are determined in the same manner as it has been done above.
The partial derivative of the function f with respect to time has the form
\displaystyle \begin{aligned} \delta f/\delta t=\dot f + f_{,i}\,\delta x_i/\delta t, \end{aligned}
where a dot over a value denotes its time derivative, i.e., $$\dot f={\partial f}/{\partial t}$$.
Substituting δxiδt from (23) yields
\displaystyle \begin{aligned} \delta f/\delta t=\dot f + f_{,i}\, n_iG. \end{aligned}
(41)
The partial derivatives of the function f with respect to the coordinates of the space xi are connected with the derivatives with respect to the curvilinear coordinates of the surface Σ(t) by the relationships
\displaystyle \begin{aligned} f_{,i}=f_{,j}\, n_j n_i + A^{\alpha\beta}f_{,\alpha}\,x_{i,\beta}. \end{aligned}
(42)

The validity of the equality (42) could be demonstrated by its projection on ni and xi,γ.

Let the function f(xi, 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.

Extend the function 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.,
\displaystyle \begin{aligned} f_{,i}^+&=f_{,j}^+\, n_j n_i+A^{\alpha\beta}f_{,\alpha}^+x_{i,\beta},\\ \delta f^+/\delta t&=\dot f^++f_{,j}^+\, n_j G , \end{aligned}
(43)
\displaystyle \begin{aligned} f_{,i}^-&=f_{,j}^-\, n_j n_i+A^{\alpha\beta}f_{,\alpha}^-x_{i,\beta},\\ \delta f^-/\delta t&=\dot f^-+f_{,j}^-\, n_j G .{} \end{aligned}
(44)
Subtracting relations (44) from (43) yields
\displaystyle \begin{aligned}{}[f_{,i}] &=[f_{,j}] n_j n_i+A^{\alpha\beta} [f ]_{,\alpha}\,x_{i,\beta}, \end{aligned}
(45)
\displaystyle \begin{aligned}{}[\dot f ]&=- [ f_{,j}] n_j G+\delta [f ]/\delta t , \end{aligned}
(46)
where $$[ f_{,i} ]=f_{,i}^+-f_{,i}^-$$, $$[ \dot f ]=\dot f^+-\dot f^-$$, and [f] = f+ − f.

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

Eliminating the value [f,j]nj from (45) and (46) yields
\displaystyle \begin{aligned} G [f_{,i} ]=- [\dot f ] n_i+\delta [f ]/\delta t\, n_i+G A^{\alpha\beta} [f ]_{,\alpha}\,x_{i,\beta}. \end{aligned}
(47)
Substituting the function f for f,(k) = k f∂tk (k = 1, 2, 3, …) in (47) results in
\displaystyle \begin{aligned} G [f_{,(k)i} ]&=- [f_{,(k+1)} ] n_i+\frac{\delta [f_{,(k)} ]}{\delta t}\, n_i\\ &+G A^{\alpha\beta} [f_{(k)} ]_{,\alpha}\,x_{i,\beta}. \end{aligned}
(48)
Let us now derive the compatibility conditions for the second-order derivatives of a certain function. Assume that the function 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
\displaystyle \begin{aligned}{}[ \varphi_{,ij}]&= [(\varphi_{,i})_n] n_j+A^{\alpha\beta} [\varphi_{,i}]_{,\alpha} x_{j,\beta}, \end{aligned}
(49)
\displaystyle \begin{aligned}{}[(\dot\varphi)_i ]&= [\dot\varphi_n ] n_i+A^{\alpha\beta} [\dot\varphi ]_{,\alpha} x_{j,\beta} , \end{aligned}
(50)
\displaystyle \begin{aligned}{}[\dot\varphi_{,i} ]&=- [(\varphi_{,i})_n ]G+\frac{\delta [\varphi_{,i} ]}{\delta t}, \end{aligned}
(51)
\displaystyle \begin{aligned}{}[\ddot\varphi ]&=- [\dot\varphi_n ] G+\frac{\delta [\dot\varphi ]}{\delta t} . \end{aligned}
(52)
Since the function φ is continuous on the surface Σ(t), then from relations (45) and (46), it follows
\displaystyle \begin{aligned}{}[ \varphi_{,i} ]=[\varphi_n ] n_i,\qquad [\dot\varphi ]=- [\varphi_n ] G . \end{aligned}
(53)
Substituting the magnitude [φ,i] in (49) and considering (11) yield
\displaystyle \begin{aligned}{}[\varphi_{,ij}]&=[(\varphi_{,i})_n ] n_j+A^{\alpha\beta}[\varphi_n ]_{,\alpha} n_ix_{j,\beta}\\ &\quad -A^{\alpha\beta}A^{\sigma\tau}B_{\sigma\alpha}[\varphi_n ] x_{i,\tau}x_{j,\beta} . \end{aligned}
(54)
Since the values [φ,ij] and AαβAστ Bσα[φn]xi,τxj,β are symmetric about the indices i and j, then the following equality should be held:
\displaystyle \begin{aligned}{}[(\varphi_{,i})_n ] n_j&+A^{\alpha\beta} [\varphi_n ]_{,\alpha} n_i \, x_{j,\beta} =[ (\varphi_{,j})_n ] n_i\\ &+A^{\alpha\beta} [\varphi_n ]_{,\alpha} n_j\, x_{i,\beta}. \end{aligned}
(55)
The convolution of (55) with nj yields
\displaystyle \begin{aligned}{}[(\varphi_{,i})_n ]= [(\varphi_{,j})_n ] n_j n_i+A^{\alpha\beta} [\varphi_n ]_{,\alpha} x_{i,\beta} . \end{aligned}
(56)
Substituting the value [(φ,i)n] from (56) into (54) gives where [φnn] = [(φ,j)n]nj = [φ,ij]ninj.

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

If the first-order derivatives of the function φ are continuous, then [φn] = 0, and from (57), it follows that
\displaystyle \begin{aligned}{}[\varphi_{,ij} ]=[\varphi_{nn} ] n_i n_j . \end{aligned}
(58)
Substituting the values [(φ,i)n] from (56) in (51) yields
\displaystyle \begin{aligned}{}[\dot\varphi_{,i} ]&=- [\varphi_{nn} ]G n_i+\frac{\delta [\varphi_n ]}{\delta t} \, n_i\\ &\quad -G A^{\alpha\beta}[\varphi_n ]_{,\alpha}x_{j,\beta}+[\varphi_n ]\,\frac{\delta n_i}{\delta t} . \end{aligned}
(59)
In view of (26), relations (59) could be rewritten as
\displaystyle \begin{aligned}{}[\dot\varphi_{,i} ]&=- [\varphi_{nn} ]G n_i+\frac{\delta [\varphi_n ]}{\delta t}\, n_i\\ &\quad -A^{\alpha\beta}\left( [\varphi_n ]G\right)_{,\alpha}x_{i,\beta}. \end{aligned}
(60)
Substituting the value $$[\dot \varphi ]$$ from (53) in (50) yields
\displaystyle \begin{aligned}{}[\dot\varphi_{,i} ]=[\dot\varphi_n ] n_i-A^{\alpha\beta}\left([\varphi_n ]G\right)_{,\alpha} x_{i,\beta}. \end{aligned}
(61)
From relations (60) and (61), one can determine the value $$[\dot \varphi _n ]$$
\displaystyle \begin{aligned}{}[\dot\varphi_n ]=- [\varphi_{nn} ]G+\frac{\delta [\varphi_n ]}{\delta t}. \end{aligned}
(62)
Substituting the value $$[ \dot \varphi _n ]$$ from (62) in (52) yields
\displaystyle \begin{aligned}{}[\ddot\varphi ]=G\left([\varphi_{nn} ]G-\frac{\delta [\varphi_n ]}{\delta t}\right) -\frac{\delta( [\varphi_n ]G)}{\delta t}. \end{aligned}
(63)

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 σij and displacement velocity vector vi, 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 dfdn = f,ini 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 ddn 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 xi 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 xi, although there exists invariance with respect to arbitrary differentiable transformations of the curvilinear coordinates θα of the surface Σ(t). Consider the question of writing the conditions of compatibility in arbitrary coordinates xi, which can be derived by the transformation from the Cartesian coordinates.

Relative to these general coordinates, the surface Σ(t) will be also defined by equations of the type (1) or as
\displaystyle \begin{aligned} F\left[ x^i(\theta ^\alpha ,t) \right]=0 \end{aligned}
(64)
In this case the covariant and contravariant components of the normal vector n to the surface are the following, respectively:
\displaystyle \begin{aligned} n_i=\frac{\partial F/\partial x^i}{\sqrt{A^{km}\frac{\partial F}{\partial x^k} \frac{\partial F}{\partial x^m}}},\qquad n^i=\frac{A^{ij} \partial F/\partial x^j}{\sqrt{A^{km}\frac{\partial F}{\partial x^k} \frac{\partial F}{\partial x^m}}}. \end{aligned}
(65)
Differentiating (64) with respect to t and considering formulas (23) and (65) yield
\displaystyle \begin{aligned} \frac{\partial F}{\partial t}+\frac{\partial F}{\partial x^i}\,\frac{A^{ij} \partial F/\partial x^j}{\sqrt{A^{km} \frac{\partial F}{\partial x^k} \frac{\partial F}{\partial x^m}}}\,G=0, \end{aligned}
(66)
or
\displaystyle \begin{aligned} G=-\, \frac{\partial F/\partial t}{\sqrt{A^{km}\frac{\partial F}{\partial x^k}\, \frac{\partial F}{\partial x^m}}}. \end{aligned}
(67)

The contravariant Aij and covariant Aij components of the metric tensor of a space in the coordinate system xi can be obtained from the corresponding components δij and δij of this tensor in the Cartesian system with the help of ordinary transformations.

Besides, the values
\displaystyle \begin{aligned} x_{,\alpha}^i=\frac{\partial x^i}{\partial \theta ^\alpha} \end{aligned}
(68)
are the components of the spatial vector Xi contravariant with respect to differentiable transformations of the coordinates xi, but relatively to differentiable transformations of the curvilinear coordinates of the surface θα, these values are the components of the covariant surface vector Xα. As in the previous section, it may be noted that both vectors Xα are tangent to the corresponding coordinate lines θα of the surface Σ(t).
The components of the metric tensor of the surface Σ(t) will be designated by Aαβ and Aαβ as before, in so doing
\displaystyle \begin{aligned} A_{\alpha\beta}=A_{mn}x_{,\alpha}^m\,x_{,\beta}^n. \end{aligned}
(69)

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.

Let f(xi, 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 xi take the form
\displaystyle \begin{aligned}{}[f_{,i} ]=[f_{,j} ] n^j n_i+A^{\alpha\beta}A_{ij} [f ]_{,\alpha}x_{,\beta}^j, \end{aligned}
(70)
where a comma in the right part denotes covariant differentiation of the spatial value [f] with respect to the surface coordinates θα. 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 xi are the Cartesian ones.
Let us illustrate the construction of the kinematic conditions of compatibility in arbitrary coordinates for the case when f = Xi, where Xi are the covariant components of a certain vector. From the transformation formulas of the components Xi, it readily follows that during transition from the coordinates xi to the new coordinates $$\overline {x^i}$$, the discontinuities [Xi] are transformed by the formulas
\displaystyle \begin{aligned}{}[ \overline{X_i} ]= [ X_k ]\, \frac{\partial x^k}{\partial\overline{x}^i}. \end{aligned}
(71)
Applying the procedure of δ-differentiation with respect to time to the both sides of the equality (71) yields
\displaystyle \begin{aligned} \frac{\delta [\overline{X_i} ]}{\delta t}=\frac{\delta [ X_k ]}{\delta t}\, \frac{\partial x^k}{\partial\overline{x_i}} + [X_k ] \frac{\partial^2x^k}{\partial\overline{x^i}\partial\overline{x^j}}\, \frac{\delta x^j}{\delta t}. \end{aligned}
(72)
Considering relation (23) which in the system of coordinates $$\overline {x^i}$$ takes the form
\displaystyle \begin{aligned} \frac{\delta\overline{x^j}}{\delta t}=G\overline{n^j}, \end{aligned}
(73)
as well as the connection between the spatial Christoffel symbols $$\varGamma _{ij}^k$$ in the old xi and new $$\overline {x^i}$$ systems of coordinates (McConnell, 1957)
\displaystyle \begin{aligned} \frac{\partial^2x^k}{\partial\overline{x^i}\partial\overline{x^j}}=\overline{\varGamma_{ij}^m}\frac{\partial x^k}{\partial\overline{x^m}} - \varGamma_{ab}^k\frac{\partial x^a}{\partial\overline{x^i}}\,\frac{\partial x^b}{\partial\overline{x^j}}, \end{aligned}
(74)
from (72) it follows
\displaystyle \begin{aligned} \frac{D [\overline{X}_i ]}{Dt}=\frac{D [ X_k ]}{Dt}\,\frac{\partial x^k}{\partial\overline{x^i}}, \end{aligned}
(75)
where
\displaystyle \begin{aligned} \frac{D [X_k ]}{Dt}=\frac{\delta [X_k ]}{\delta t}-G [ X_m ]\varGamma_{ka}^m \, n^a. \end{aligned}
(76)

Equalities (75) express the fact that the values D[Xk]∕Dt are the components of the covariant vector. Moreover, it is evident that D[Xk]∕Dt are reduced to δ[Xk]∕δt in the Cartesian coordinate system. The value possessing the components D[Xk]∕Dt will be denoted as the absolute or invariant derivative of the discontinuity [Xk] 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.

Denote the components of the covariant derivative of the covariant components of the vector Xi with respect to the spatial coordinates xk by Xi,k, and consider the relation
\displaystyle \begin{aligned}{}[ \dot X_i ]=-G [X_{i,j} ] n^j+\frac{D [ X_i ]}{Dt}. \end{aligned}
(77)

Relations (77) are invariant with respect to common transformations of the coordinates xi. It follows from (75) and from the fact that the values $$[ \dot X_i ]$$ and [Xi,j]nj 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 Xi in the general coordinates.

In the general case, the first-order kinematic conditions of compatibility can be written similarly to the conditions (70) in the symbolic form
\displaystyle \begin{aligned}{}[ \dot f ]=-G [ f_{,j} ] n^j+\frac{D [f ]}{Dt}. \end{aligned}
(78)

Conditions (78) are invariant both with respect to arbitrary differentiable transformations of the spatial coordinates xi and with respect to transformations of the curvilinear surface coordinates θi.

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.

Eliminating the value [f,j]nj from (70) and (78) yields
\displaystyle \begin{aligned} G [f_{,i} ]=-\, [\dot f ] n_i+\frac{D [f ]}{Dt}\, n_i+G A^{\alpha\beta}A_{ij} [f ]_{,\alpha}x_{,\beta}^j. \end{aligned}
(79)
In closing of this section, the formula essential for applications is added
\displaystyle \begin{aligned} \frac{D n^i}{Dt}=-A^{\alpha\beta} G_{,\alpha}x_{,\beta}^i , \end{aligned}
(80)
the validity of which is easily verified, if one notes that it is invariant with respect to arbitrary differentiable transformations of the coordinates xi and, moreover, it is reduced to (26) in the case when the coordinates xi 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.

Let consider the case essential for applications, when the spatial coordinates x1 and x2 coincide at each instant of the time, respectively, with the coordinates θ1 and θ2 on the surface Σ(t) moving in the direction of the coordinate lines x3, i.e., θ1, θ2, x3 are the ray coordinated (Fig. 2). In this case, n1 = n2 = 0, n3 = 1, and $$x^3=G t+x_0^3$$. Calculations will be carried out for the covariant components of the vector Xi on the assumption that the spatial coordinates are orthogonal. Putting f = Xi in (79) and considering relations (76), as well the following formula (McConnell, 1957)
\displaystyle \begin{aligned}{}[ X_{i,j} ]= \left[\frac{\partial X_i}{\partial x^j} \right] -\varGamma_{ij}^m [X_m ], \end{aligned}
(81)
yield
\displaystyle \begin{aligned} G \left[\frac{\partial X_j}{\partial x^i} \right] &=- [\dot X_j ] n_i +\frac{\delta [X_j ]}{\delta t}\, n_i\\ &\quad - G [X_m ]\varGamma_{ja}^m n^a n_i + G [ X_m ]\varGamma_{ji}^m\\ &\quad + G A^{\alpha\beta}A_{ik} [ X_j ]_{,\alpha}x_{,\beta}^k. \end{aligned}
(82)
Putting i = 3 in (82) yields
\displaystyle \begin{aligned} G \left[\frac{\partial X_j}{\partial x^3} \right]&=- \left[\frac{\partial X_j}{\partial t} \right]+\frac{\delta [X_j ]}{\delta t}\\ &\quad -G[X_m ]\varGamma_{j3}^m+G[X_m ]\varGamma_{j3}^m\\ &\quad +G A^{\alpha\beta}A_{33} [ X_j ]_{,\alpha}x_{,\beta}^3 , \end{aligned}
whence it follows
\displaystyle \begin{aligned} G \left[\frac{\partial X_j}{\partial x^3} \right]=- \left[\frac{\partial X_j}{\partial t} \right]+\frac{\delta [X_j ]}{\delta t}. \end{aligned}
(83)
Putting i = α (α = 1, 2) in (82), it follows that
\displaystyle \begin{aligned} G \left[\frac{\partial X_j}{\partial x^\alpha } \right]=G[X_m ]\varGamma_{j\alpha }^m+G [ X_j ]_{,\alpha}, \end{aligned}
or
\displaystyle \begin{aligned} \left[\frac{\partial X_j}{\partial x^\alpha } \right]=\frac{\partial [X_j]}{\partial x^\alpha }. \end{aligned}
(84)
The last equality is the identity.
Substituting the covariant components of the second-rank tensor Xst instead of f in (79) and considering that
\displaystyle \begin{aligned}{}[ X_{st,i} ]=\left[\frac{\partial X_{st}}{\partial x^i } \right]-\varGamma_{si }^m [X_{mt} ]-\varGamma_{ti }^m [X_{sm} ] , \end{aligned}
(85)
it could be obtained the following: Putting i = 3 in (87) provides
\displaystyle \begin{aligned} G \left[\frac{\partial X_{st}}{\partial x^3 } \right] =- \left[ \frac{\partial X_{st}}{\partial t}\right] +\frac{\delta [X_{st}]}{\delta t} , \end{aligned}
(88)
but when i = α (α = 1, 2), then the identity could be obtained
\displaystyle \begin{aligned} \left[\frac{\partial X_{st}}{\partial x^\alpha } \right] =\frac{\partial [X_{st}]}{\partial x^\alpha }. \end{aligned}
(89)

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

In practically important cases, one could deal with 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:
\displaystyle \begin{aligned} X_i&=H_iX_i^{ph}, \qquad X_{st}=H_s H_t X_{st}^{ph}, \end{aligned}
(90)
\displaystyle \begin{aligned} X_i&=H_i^{-1}X^i_{ph}, \qquad X^{st}=H_s^{-1} H_t^{-1} X^{st}_{ph}, \\ X^s_t&=H_s^{-1} H_t (X^s_t)_{ph}, \end{aligned}
(91)
where $$H_i=\sqrt {A_{ii}}$$ are Lamé’s constants and the summation over the i, s, t indices is not performed.
Substituting (90) into (83) and (88) and considering that
\displaystyle \begin{aligned} \frac{\delta H_j}{\delta t}=\frac{\partial H_j}{\partial x^3}\, \frac{\delta x^3}{\delta t}=\frac{\partial H_j}{\partial x^3}\,G n^3, \end{aligned}
(92)
the conditions of compatibility for the physical components could be found
$$\displaystyle \begin{gathered} {} G \left[\frac{\partial X_j^{ph}}{\partial x^3}\right]=-\left[\frac{\partial X_j^{ph}}{\partial t}\right]+\frac{\delta [X_j^{ph} ]}{\delta t}, \end{gathered}$$
(93)
$$\displaystyle \begin{gathered} {} G \left[\frac{\partial X_{st}^{ph}}{\partial x^3}\right]=-\left[\frac{\partial X_{st}^{ph}}{\partial t}\right] +\frac{\delta [X_{st}^{ph} ]}{\delta t}. \end{gathered}$$
(94)

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 nth-order differential equations, which require for their analysis the utilization of the conditions of compatibility for the nth-order derivatives of the desired values.

Such a condition of compatibility, which is an immediate generalization of relationships (93) and (94), has the form
\displaystyle \begin{aligned} G^{n} \left[\frac{\partial ^{n} Z}{\partial z^{n}}\right]=\sum_{m=0}^{n-1} (-1)^m \frac{(n)!}{m! (n-m)!}\; \frac{\delta ^{n-m} [Z_{,(m)}]}{\delta t^{n-m}}, \end{aligned}
(95)
where 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

The combination of the ray theory with the theory of discontinuities suggested in Thomas (1961a) enables one to construct the ray series as (Achenbach, 1973; Rossikhin and Shitikova, 1995a)
\displaystyle \begin{aligned} \begin{array}{rcl} {} f(s,t)&\displaystyle =&\displaystyle \sum_{k=0}^\infty \frac{1}{k!} [f_{,(k)}] \bigg|{}_{t=\tau(s)} \left(t-\tau(s) \right)^k\\ &\displaystyle &\displaystyle \times H\left(t-\tau(s)\right), \end{array} \end{aligned}
(96)
where [f,(k)] = [kf∂tk] are the jumps in the kth-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 kth-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 θ1 and θ2, which are found, in their turn, from the boundary conditions (Rossikhin and Shitikova, 1995a).

As an illustration it is considered a thin semi-infinite linearly viscoelastic homogeneous rod which is assumed to be initially undisturbed but subjected to a constant velocity at one end or subjected to a constant stress impact at the initial instant of time, resulting in the propagation of surfaces of discontinuity. The given problem was considered in (Achenbach and Reddy, 1977), wherein the stress σ(x, t) (0 ≤x ≺) behind the front of the strong discontinuity surface was obtained using the ray series (96) in the form
\displaystyle \begin{aligned} \sigma (x,t)=\sum _{n=0}^{\infty }\frac{1}{n!} \left(t-\frac{x}{G} \right)^{n} \left[\frac{\partial ^{n} \sigma }{\partial t^{n} } \right]_{t=x/G}, \end{aligned}
(97)

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 J0 = J and J(t) is the uniaxial creep function.

The coefficients of the ray series (97) were determined from the recurrent equation
\displaystyle \begin{aligned} &{\frac{d}{dt} \left[\frac{\partial ^{n} \sigma }{\partial t^{n} } \right] a_{1} \left[\frac{\partial ^{n} \sigma }{\partial t^{n} } \right]=F_{n-1} }, \\ &\quad {F_{n-1} (t)=\frac{1}{2} \frac{d^{2} }{dt^{2} } \left[\frac{\partial ^{n-1} \sigma }{\partial t^{n-1} } \right]}\\ &\quad -\sum\nolimits_{i=1}^{n} a_{i+1} \left[\frac{\partial ^{n-i} \sigma }{\partial t^{n-i} } \right], a_{i} =\frac{J_{0}^{(i)} }{2J_{0} } , \end{aligned}
(98)
which was obtained from the equation of motion and the constitutive equation for linear viscoelastic material behavior by employing the geometric and kinematic condition of compatibility (95). In Eq. (98), $$J_{0}^{(i)} =J^{(i)}$$, and Open image in new window.
Sun (1970) applied Eqs. (97) and (98) to a viscoelastic rod of linear Maxwell material subjected to a particle velocity v0 for t > 0 and a rod of a standard linear solid subjected to a constant stress σ0 for t > 0. In both examples, the relations for the dimensionless stress Z can be obtained as
\displaystyle \begin{aligned} Z=\exp (-\alpha \xi )\sum _{k=0}^{5}z_{k} \left(\xi \right)\left(\tau -\xi \right)^{k},\end{aligned}
(99)
where $$Z=\sigma ^{*} =-\sigma \left (\rho Gv_{0} \right )^{-1}$$ and $$Z=\sigma _{*} =\sigma \sigma _{0}^{-1}$$ for the first and second examples, respectively; zk(ξ) are the polynomials of degree k in the variable ξ which harbor k arbitrary constants to be determined from the initial conditions, $$\xi =x\left (2G\tau _{0} \right )^{-1}$$, τ0 is the relaxation time, $$\tau =t\left (2\tau _{0} \right )^{-1}$$; and α is a certain constant.
Sun (1970) has compared the results due to the formulas (99) with the exact solutions obtained in Lee and Kanter (1953) and Morrison (1956) by employing the Laplace transform technique to the first and second problems, respectively. It has been shown that for short-time range elapsed after the moment of the load application, it has been observed that there is good agreement between the approximate and exact solutions in the whole domain of the wave motion existence. But as time increases, the ray expansions show characteristics of forward-area asymptotics, i.e.,
\displaystyle \begin{aligned} Z=\exp \left(-\alpha \xi \right)\sum _{k=0}^{5}z_{k} \left(\xi \right) \varepsilon ^{k}, \end{aligned}
(100)
where 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.

To adjust the ray expansions and thereby expand the region of their uniform fitness, it could be shown (Rossikhin and Shitikova, 1991, 1994) that the expression (100) satisfies the equation
\displaystyle \begin{aligned} \frac{dZ}{dt} +\alpha _{1} Z=\frac{1}{2} \varepsilon \frac{d^{2} Z}{dt^{2} } -\sum _{i=1}^{n}\alpha _{i+1} \varepsilon ^{i} Z. \end{aligned}
(101)
To prove (101), it is necessary to substitute (100) into Eq. (101) and equate coefficients of like powers of 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
\displaystyle \begin{aligned} \begin{array}{rcl} {} Z&\displaystyle =&\displaystyle Z_{0} \left(T_{0}, T_{1}, T_{2},..\right)+\varepsilon Z_{1} \left(T_{0}, T_{1}, T_{2},..\right)\\ &\displaystyle &\displaystyle +\varepsilon ^{2} Z_{2} \left(T_{0}, T_{1}, T_{2},..\right)+.. \end{array} \end{aligned}
(102)

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

Substituting (102) into Eq. (101), equating the coefficients of like powers of ε, and eliminating secular terms yield
\displaystyle \begin{aligned} Z&=\left[a_{0} +\varepsilon a_{1} +\ldots+\varepsilon ^{n} a_{n} \right]\\ &\quad \times \exp \left[\left(b_{0} +\varepsilon b_{0} +\ldots +\varepsilon ^{n} b_{n} \right)t\right], \end{aligned}
(103)
where an are constants dependent on boundary conditions, b0 = a1, and
\displaystyle \begin{aligned} b_{n} =\left\{\begin{array}{l} {-a_{n+1} +\frac{1}{2} b_{(n-1)/2}^{2} +\sum _{i=0}^{(n-3)/2}b_{i} b_{n-i-1} \text{ for an even n} } \\ {} {-a_{n+1} +\sum _{i=0}^{(n-2)/2}b_{i} b_{n-i-1} \text{ for an uneven n} } \end{array}\right. \end{aligned}

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 εn 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).

Figures 3 and 4 present the dimensionless stress distribution for a rod of a Maxwell material and for a rod of a standard linear solid due to (103), respectively, for several values of 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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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

## Authors and Affiliations

1. 1.Voronezh State Technical UniversityVoronezhRussia
2. 2.Research Center on Dynamics of Solids and StructuresVoronezh State Technical UniversityVoronezhRussia

## Section editors and affiliations

• Marina V. Shitikova
• 1
1. 1.Research Center on Dynamics of Solids and StructuresVoronezh State Technical UniversityVoronezhRussia