4. Shock waves as generalized solutions of transport Lame equations: conditions on wave front

Consider Eq. (14) and its solutions on the space of generalized vector functions D'<sup>3</sup> (R<sup>3</sup> ) with components being generalized functions from D<sup>0</sup> (R<sup>3</sup> ) (see [3]). Obviously, if u is a solution of Eq. (14) that is twice differentiable, then it is also a generalized solution of Eq. (14). If a vector function u satisfies Eq. (14) in the classical sense almost everywhere, except for some surfaces, on which its derivatives are discontinuous, then, generally speaking, u is not a generalized solution of Eq. (14).

Let u(x,z) be a shock wave (x = (x1,x2)), i.e., a classical solution of the Lame transport equations, Eq. (14), that satisfies conditions Eqs. (15) and (16) at the front <sup>F</sup>. Let <sup>b</sup>u xð Þ ; <sup>z</sup> denote the corresponding regular generalized function.

Theorem 4.1. The shock wave <sup>b</sup>u xð Þ ; <sup>z</sup> is a generalized solution of the Lame equation in D'<sup>3</sup> (R<sup>3</sup> ).

Proof. Using the rules for differentiating generalized functions with derivatives having jump discontinuities across some surfaces (see [3]), for the equations of motion in D'<sup>3</sup> (R<sup>3</sup> ), we obtain

$$\begin{aligned} \frac{\partial \widehat{\boldsymbol{w}}\_{\dot{\boldsymbol{\eta}}}}{\partial \boldsymbol{\omega}\_{\dot{\boldsymbol{\eta}}}'} - \rho c^2 \frac{\partial^2 \widehat{\boldsymbol{u}}\_{\boldsymbol{i}}}{\partial \mathbf{x}\_{\boldsymbol{z}}^2} + \boldsymbol{G}\_{\boldsymbol{i}} &= \left[ \ \boldsymbol{\sigma}\_{\dot{\boldsymbol{\eta}}} \boldsymbol{h}\_{\boldsymbol{\eta}} - \rho c^2 \ \boldsymbol{h}\_{\boldsymbol{z}} \frac{\partial \boldsymbol{u}\_{\boldsymbol{i}}}{\partial \mathbf{z}} \right]\_{\boldsymbol{F}} \delta\_{\boldsymbol{F}} + \\ &+ \frac{\partial}{\partial \mathbf{x}\_{\boldsymbol{j}}'} \{ \left[ \boldsymbol{\lambda} \ \boldsymbol{u}\_{\boldsymbol{k}} \boldsymbol{h}\_{\boldsymbol{k}} \delta\_{\boldsymbol{ij}} + \mu (\boldsymbol{u}\_{\boldsymbol{i}} \boldsymbol{h}\_{\boldsymbol{j}} + \boldsymbol{u}\_{\boldsymbol{j}} \boldsymbol{h}\_{\boldsymbol{i}}) \right]\_{\boldsymbol{F}} \delta\_{\boldsymbol{F}} \right] - \frac{\partial}{\partial \mathbf{z}} \{ \left[ \boldsymbol{u}\_{\boldsymbol{i}} \ \ \boldsymbol{l}\_{\boldsymbol{F}} \delta\_{\boldsymbol{k}} \delta\_{\boldsymbol{F}} \right]\_{\boldsymbol{i}} \end{aligned} \tag{17}$$

as ½ � u <sup>F</sup> ¼ 0: For the gaps of these functions, the theorem has been proved on the basis of classic

General Functions Method in Transport Boundary Value Problems of Elasticity Theory

http://dx.doi.org/10.5772/intechopen.74538

5. Fundamental Green's tensors and generalized solutions of transport

The matrix of fundamental solutions U x b ð Þ ; z satisfies Eq. (14) with a delta function in the mass

This matrix is called Green's tensor for the transport Lame equations if it satisfies the decay

For a fixed k, its components describe the displacements of the elastic medium under a concentrated force moving at the velocity c along the axis Z = X<sup>3</sup> and acting in the Xk direction. Green's tensor can be obtained by taking the Fourier transform of Eq. (17) and solving the corresponding system of linear algebraic equations for the Fourier transforms U(ξ1,ξ2,ξ3). It is

It can be seen that U x b ð Þ ; z has no classical inverse Fourier transform since it has non-integrable singularities in its denominators. This is associated with the fact that the matrix of fundamental solutions is defined, generally speaking, up to solutions of the homogeneous system of

are of crucial importance in the construction of the original Green's tensor. It is easy to see that

This equation is similar to the elliptic Laplace equation at subsonic speeds if Mk < 1 and to the wave equation at supersonic speeds if Mk > 1. At the sound speed (Mk ¼ 1), the variable z

<sup>0</sup><sup>m</sup> is the Fourier transform of the fundamental solution to the equation

ð21Þ

135

ð22Þ

ð23Þ

ð24Þ

methods (see [4, 5]). For full proof of this theorem, see [6].

Lame equations

conditions at infinity

reduced to the form (see [4]).

equations. The functions

f ¯

force:

Here, the right-hand side involves singular generalized functions, namely, single layers δ<sup>F</sup> (x,z) and double layers on F. By virtue of conditions Eqs. (15) and (16), the densities of these layers are equal to zero, so the right-hand side of Eq. (17) vanishes; i.e., the shock wave satisfies the same equations, Eq. (14), but in the generalized sense.

As a result, we obtain a simple formal method for deriving conditions at jumps in solutions and their derivatives across the shock fronts in hyperbolic equations. Namely, these equations are written in the space of generalized functions and the densities of the singular functions corresponding to single, double, etc., layers are set to zero.

Define as follows the kinetic energy density

$$\mathbf{K} = 0.5\rho \|\boldsymbol{\mu}, \boldsymbol{\mu}\|^2 = 0.5\rho \boldsymbol{c}^2 \|\boldsymbol{\mu}, \boldsymbol{\mu}\|^2 \tag{18}$$

and elastic potential

$$\mathcal{W} = 0.5 \sigma\_{\vec{\eta}} u\_{i \cdot j} = 0.5 \sigma\_{i \vec{\eta}} \varepsilon\_{\vec{\eta}} \tag{19}$$

Consider the following functions: the energy density E = K + W of elastic deformations and the Lagrangian Λ = K � W.

Theorem 4.2. If G is continuous, then the Lagrangian Λ is continuous at the shock waves fronts.

(½ � Λ <sup>F</sup> ¼ 0) and the jump in the energy density satisfies the relation

$$\|h\_z[E]\_{\!\!F} = \left[ \left( \sigma\_{\vec{\eta}} h\_{\vec{\}} \right) \mu\_{\nu, z} \right]\_{\!\!F'} \tag{20}$$

First formula is equivalent to the equality:

$$[E]\_{F\_{\mathbb{Q}}} = -\frac{c\_k}{c} h\_{\rangle}^k \left[ \begin{array}{c} \sigma\_{i\dot{\jmath}} u\_{i\cdot z} \end{array} \right]\_{F\_{\mathbb{Q}\_k}} \quad k = 1, 2.$$

where ck is the sound velocity corresponding to front F, h<sup>k</sup> <sup>j</sup> is the components of the wave vector to F.

The last formula may be easy to get if we write the equation for E in D<sup>0</sup> <sup>3</sup> <sup>R</sup><sup>3</sup> � � in the form

$$\begin{aligned} \widehat{\mathbf{E}}\_{\cdot z} &= \mathbf{E}\_{\cdot z} + [\mathbf{E}]\_{\mathbf{F}} \mathbf{h}\_{\mathbf{z}} \delta\_{\mathbf{F}} = \left( \sigma\_{\overline{\mathbf{j}}} \mu\_{\mathbf{i} \cdot \mathbf{z}} \right)\_{\mathbf{j}} + \rho(\mathbf{G}, \boldsymbol{\mu}\_{\cdot z}) + \left[ \sigma\_{\overline{\mathbf{j}}} \mu\_{\mathbf{i} \cdot \mathbf{z}} \right]\_{\mathbf{F}} \delta\_{\mathbf{F}} \mathbf{h}\_{\mathbf{j}} + \rho\left( \mathbf{G}, [\mathbf{u}]\_{\mathbf{F}} \right) \mathbf{h}\_{\mathbf{z}} \delta\_{\mathbf{F}} & \Rightarrow\\ [\mathbf{E}]\_{\mathbf{f}} \mathbf{h}\_{\mathbf{z}} &= \left[ \sigma\_{\overline{\mathbf{j}}} \mu\_{\mathbf{i} \cdot \mathbf{z}} \right]\_{\mathbf{f}} \mathbf{h}\_{\mathbf{j}} \end{aligned}$$

as ½ � u <sup>F</sup> ¼ 0: For the gaps of these functions, the theorem has been proved on the basis of classic methods (see [4, 5]). For full proof of this theorem, see [6].

∂bσij ∂x<sup>0</sup> j

134 Differential Equations - Theory and Current Research

þ ∂ ∂x<sup>0</sup> j

Define as follows the kinetic energy density

First formula is equivalent to the equality:

Eb, <sup>z</sup> ¼ E, <sup>z</sup> þ ½ � E <sup>F</sup>hzδ<sup>F</sup> ¼ σijui, <sup>z</sup>

Fhj

� �

½ � E <sup>F</sup>hz ¼ σijui, <sup>z</sup>

and elastic potential

Lagrangian Λ = K � W.

vector to F.

� <sup>r</sup>c<sup>2</sup> <sup>∂</sup><sup>2</sup>bui ∂x<sup>2</sup> z

same equations, Eq. (14), but in the generalized sense.

corresponding to single, double, etc., layers are set to zero.

(½ � Λ <sup>F</sup> ¼ 0) and the jump in the energy density satisfies the relation

½ � <sup>E</sup> Fc<sup>k</sup>

where ck is the sound velocity corresponding to front F, h<sup>k</sup>

þ Gi ¼ σijhj � r c

� � � <sup>∂</sup>

Here, the right-hand side involves singular generalized functions, namely, single layers δ<sup>F</sup> (x,z) and double layers on F. By virtue of conditions Eqs. (15) and (16), the densities of these layers are equal to zero, so the right-hand side of Eq. (17) vanishes; i.e., the shock wave satisfies the

As a result, we obtain a simple formal method for deriving conditions at jumps in solutions and their derivatives across the shock fronts in hyperbolic equations. Namely, these equations are written in the space of generalized functions and the densities of the singular functions

<sup>K</sup> <sup>¼</sup> <sup>0</sup>:5<sup>r</sup> u, k k<sup>t</sup> <sup>2</sup> <sup>¼</sup> <sup>0</sup>:5r<sup>c</sup>

Consider the following functions: the energy density E = K + W of elastic deformations and the

� �ui, <sup>z</sup> � �

Fck

, k ¼ 1, 2

� �

Theorem 4.2. If G is continuous, then the Lagrangian Λ is continuous at the shock waves fronts.

hz½ � E <sup>F</sup> ¼ σijhj

� �, <sup>j</sup> <sup>þ</sup> <sup>r</sup> <sup>G</sup>; u, ð Þþ <sup>z</sup> <sup>σ</sup>ijui, <sup>z</sup>

¼ � ck c hk <sup>j</sup> σijui, <sup>z</sup> � �

The last formula may be easy to get if we write the equation for E in D<sup>0</sup>

λ ukhkδij þ μðuihj þ ujhi � � <sup>2</sup> hz ∂ui ∂z

<sup>F</sup>δ<sup>F</sup>

F δFþ

ui ½ �Fhzδ<sup>F</sup> � �,

<sup>2</sup> u, k k<sup>z</sup> <sup>2</sup> (18)

<sup>F</sup>, (20)

<sup>j</sup> is the components of the wave

<sup>F</sup>δFhj þ r G; ½ � u <sup>F</sup>

<sup>3</sup> <sup>R</sup><sup>3</sup> � � in the form

� �hzδ<sup>F</sup> )

W ¼ 0:5σijui, <sup>j</sup> ¼ 0:5σijεij (19)

(17)

∂z

� �
