3. Lame transport equations and Mach numbers

Suppose that the force affecting the medium moves at a constant velocity c along the X<sup>3</sup> axis (for convenience, in its negative direction) and, in a moving coordinate system x<sup>0</sup> ¼ ð Þ x1; x2; z ¼ x<sup>3</sup> þ ct it does not depend on t:

$$\mathbf{G}(\mathbf{x}, \mathbf{z}) = \mathbf{G}\_{\rangle}(\mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3 + ct)e\_{\rangle} \tag{12}$$

Transport solutions are solutions of Eq. (4) with the same structure:

$$\mathbf{u} = \mathbf{u}(\mathbf{x}\_1, \mathbf{x}\_2, \mathbf{x}\_3 + \varepsilon t) = \mathbf{u}(\mathbf{x}, \mathbf{z}) \tag{13}$$

The speed of transport loads is called subsonic if c<c2,transonic if c<sup>2</sup> <c<c1, and supersonic if c>c1. A speed is called the first or second sound speed if c = cj, j = 1, 2, respectively.

In the new variables, the equations of motion are brought to the form

$$L\_j^i \left(\frac{\partial}{\partial x'}\right) u\_i = \left\{ \left(M\_1^{-2} - M\_2^{-2}\right) \frac{\partial^2}{\partial x\_i' \partial^t x\_j} + \left(M\_2^{-2} \Delta - \frac{\partial^2}{\partial x\_3^2}\right) \delta\_j^i \right\} u\_i + g\_j = 0 \tag{14}$$

Here gj <sup>¼</sup> <sup>r</sup>c<sup>2</sup> �<sup>1</sup> Gj; Mj ¼ c=cj are Mach numbers: (M<sup>1</sup> < M2).

As Mj < 1(j = 1, 2) the load is subsonic and the system of equations is elliptic. If the load is supersonic, i.e., Mj > l, j = 1, 2, then the system becomes hyperbolic. In the case of transonic speeds, i.e., M1 < 1 and M<sup>2</sup> > 1, the equations are hyperbolic-elliptic. In the case of sound speeds, the equations are parabolic-elliptic if M<sup>2</sup> = 1 and parabolic-hyperbolic if M<sup>1</sup> = 1. We will show this later when considering fundamental solutions of Eq. (14).

Since the original system is hyperbolic, Eq. (14) can also have discontinuous solutions. Let F be a discontinuity surface in the space of variables x 0 such that it is stationary in this space and moves at one of the sound velocities V = c1,c<sup>2</sup> in the space of (x1,x2,x3). It follows from Eq. (7) that V = cn3, where n = (n1,n2,n3) is the unit normal to F in R<sup>3</sup> . Therefore, since c = cj/n<sup>3</sup> and |n3| ≤ 1, such surfaces can arise only at supersonic speeds: c ≥ cj.

(the continuity of the tangent derivatives on Ft). Additionally, Eq. (4) implies dynamical consistency conditions for solutions at the wave front, which are equivalent to the momentum

Definition. A wave is called a shock wave if the jump in the stresses across the wave front is finite: eimj[σij]Ft = 0̸. If mj[σij]Ft = 0, then this is a weak shock wave. If mj[σij]Ft = ∞, then this is a

Velocity suffers a jump discontinuity across a shock front. At fronts of weak shock waves, the velocities are continuous, but the second derivatives of solutions are not. Strong shock waves (in the sense of the aforementioned definition) do not occur in actual media, since, at large stress jumps, the medium is destroyed and ceases to be elastic. However, strong shock waves in elastic media play an important theoretical role in the construction of solutions, specifically,

Suppose that the force affecting the medium moves at a constant velocity c along the X<sup>3</sup> axis (for convenience, in its negative direction) and, in a moving coordinate system

The speed of transport loads is called subsonic if c<c2,transonic if c<sup>2</sup> <c<c1, and supersonic if

As Mj < 1(j = 1, 2) the load is subsonic and the system of equations is elliptic. If the load is supersonic, i.e., Mj > l, j = 1, 2, then the system becomes hyperbolic. In the case of transonic speeds, i.e., M1 < 1 and M<sup>2</sup> > 1, the equations are hyperbolic-elliptic. In the case of sound speeds, the equations are parabolic-elliptic if M<sup>2</sup> = 1 and parabolic-hyperbolic if M<sup>1</sup> = 1. We

c>c1. A speed is called the first or second sound speed if c = cj, j = 1, 2, respectively.

Gj; Mj ¼ c=cj are Mach numbers: (M<sup>1</sup> < M2).

will show this later when considering fundamental solutions of Eq. (14).

G xð Þ¼ ; z Gjð Þ x1; x2; x<sup>3</sup> þ ct ej (12)

u ¼ uðx1; x2; x<sup>3</sup> þ ctÞ ¼ u xð Þ ; z (13)

ð11Þ

ð14Þ

conservation law in its neighborhood:

132 Differential Equations - Theory and Current Research

fundamental solutions of Eq. (4).

3. Lame transport equations and Mach numbers

Transport solutions are solutions of Eq. (4) with the same structure:

In the new variables, the equations of motion are brought to the form

x<sup>0</sup> ¼ ð Þ x1; x2; z ¼ x<sup>3</sup> þ ct it does not depend on t:

strong shock wave.

Here gj <sup>¼</sup> <sup>r</sup>c<sup>2</sup> �<sup>1</sup>

It follows from Eqs. (9) to (11) and Eq. (13) that the kinematic and dynamical consistency conditions for solutions at discontinuities in the mobile coordinate system have the form:

$$[u(\mathbf{x}, \mathbf{z})]\_F = \mathbf{0} \quad \Rightarrow \qquad \left[n\_z u\_{i\cdot j} - n\_j u\_{i\cdot z}\right]\_F = \mathbf{0};\tag{15}$$

$$
\sigma\_{\left[\sigma\_{\dot{\imath}}\right]} \mathfrak{u}\_{\dot{\jmath}} = -\rho \mathfrak{c}\_{k} \mathfrak{c} [\mathfrak{u}\_{\dot{\imath}\cdot z}]\_{\mathbb{R}^{\flat}} ; \quad \mathfrak{u}\_{z} = -\mathfrak{c}\_{k}/\mathfrak{c} , \quad \text{ for } \mathfrak{c} \ge \mathfrak{c}\_{k} \tag{16}
$$

n ¼ f g n1; n2; nz ¼ n<sup>3</sup> is a wave vector, k = 1 for shock dilatational waves, k = 2 for shock shear waves. Here and hereafter, the derivative with respect to xj is denoted by the index j after a comma in the function notation or by the variable itself.

Definition. If c>c2, the solution of the system in Eq. (14) is called classical if it is continuous and twice differentiable everywhere, except for, possibly, wave fronts. The number of fronts is finite at any fixed t and the conditions on the gaps, Eqs. (15) and (16), are satisfied on the wave fronts.

At first, we construct the solutions of the transport Lame equation using methods of generalized functions theory.
