10. Supersonic green's tensor and its antiderivative with respect to z

From Eq. (25), we get the regular representation of <sup>U</sup><sup>b</sup> <sup>j</sup> <sup>i</sup> in the supersonic case which has the form

$$\begin{aligned} 2\pi l I\_1^1 &= \frac{\theta\_2}{V\_2} + \frac{z^2 \mathbf{x}\_1^2}{r^4 M\_2^2} \left(\frac{\theta\_1}{V\_1} - \frac{\theta\_2}{V\_2}\right) - \frac{\mathbf{x}\_2^2}{r^4 M\_2^2} \ (\theta\_1 V\_1 - \theta\_2 V\_2), \\ 2\pi l I\_2^2 &= \frac{\theta\_2}{V\_2} + \frac{z^2 \mathbf{x}\_2^2}{r^4 M\_2^2} \left(\frac{\theta\_1}{V\_1} - \frac{\theta\_2}{V\_2}\right) - \frac{\mathbf{x}\_1^2}{r^4 M\_2^2} \ (\theta\_1 V\_1 - \theta\_2 V\_2), \end{aligned} \tag{46}$$

$$2\pi l I\_1^2 = \frac{\mathbf{x}\_1 \mathbf{x}\_2}{r^4} \left(z^2 \left(\frac{\theta\_1}{V\_1} - \frac{\theta\_2}{V\_2}\right) + (\theta\_1 V\_1 - \theta\_2 V\_2)\right), 2\pi l I\_3^3 = \left(\frac{\theta\_1}{V\_1} + \frac{\theta\_2 m\_2^2}{V\_2}\right).$$

$$2\pi l I\_1^3 = -\frac{\mathbf{x}\_1 z}{r^2} \left(\frac{\theta\_1}{V\_1} - \frac{\theta\_2}{V\_2}\right), 2\pi l I\_2^3 = -\frac{\mathbf{x}\_2 z}{r^2} \left(\frac{\theta\_1}{V\_1} - \frac{\theta\_2}{V\_2}\right)$$

Here <sup>θ</sup><sup>j</sup> <sup>¼</sup> <sup>θ</sup> <sup>z</sup> � mj∥x<sup>∥</sup> � �, Vj � ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>z</sup><sup>2</sup> � <sup>m</sup><sup>2</sup> <sup>j</sup> <sup>∥</sup>x∥<sup>2</sup> <sup>q</sup> , mj ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Mj <sup>2</sup> � <sup>1</sup>: q It satisfies the radiation conditions:

$$\operatorname\*{supp}\_z \mathcal{U}(\mathbf{x}, z) \in \{z > 0\}, \mathcal{U}\_i^k \to 0, \mathcal{U}\_{i\_j'}^k \to 0 \text{ by } \mathbf{x}' \to \text{as}.\tag{47}$$

One can readily see that its components are zero outside the sonic cones:

$$K\_l^+ = \{ (\mathfrak{x}, z) : z > m\_l \| \mathfrak{x} \| \} , l = 1, 2.$$

On the surfaces of the cones, the components U<sup>3</sup> <sup>1</sup> have singularities of the type (z <sup>2</sup> � <sup>m</sup><sup>2</sup> jr 2 ) �1/2 . For solution of supersonic problems, we introduce the tensor W ci j ð Þ x; z , which is the antiderivative of <sup>U</sup><sup>b</sup> <sup>i</sup> <sup>j</sup> with respect to z:

$$
\widehat{\boldsymbol{W}^{i}\_{j}} = \sum\_{k=1}^{2} \widehat{\boldsymbol{W}^{i}\_{jk}} = \widehat{\boldsymbol{U}^{i}\_{j}} \ast \delta(\mathbf{x}\_{1}) \delta(\mathbf{x}\_{2}) \boldsymbol{\theta}(\boldsymbol{z}) = \widehat{\boldsymbol{U}^{i}\_{j}} \boldsymbol{\!}\_{\boldsymbol{z}} \boldsymbol{\mathsf{e}}(\boldsymbol{z}), \quad \widehat{\boldsymbol{W}^{i}\_{j}} \boldsymbol{z} = \widehat{\boldsymbol{U}^{i}\_{j}} \tag{48}
$$

Uj i

We introduce antiderivative stress tensor

presented in the following form:

properties around the Z-axis:

except for (i,j) = (1,3), (2,3), (3,1):

j

Hi j

Components H<sup>i</sup>

H3

Hj i

<sup>i</sup> ð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>H</sup><sup>3</sup>

then we can again single out two terms in H<sup>i</sup>

ð Þ¼ <sup>x</sup>; <sup>z</sup> <sup>H</sup>is

ð Þ¼� <sup>x</sup>; <sup>y</sup>; <sup>z</sup>; <sup>m</sup> <sup>H</sup><sup>j</sup>

<sup>j</sup> ð Þþ <sup>x</sup>; <sup>z</sup> <sup>H</sup>id

properties

ð Þ¼� <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>U</sup><sup>j</sup>

Σ~j <sup>i</sup><sup>3</sup> <sup>¼</sup> <sup>Σ</sup>b<sup>i</sup> j

Hb i <sup>j</sup> <sup>¼</sup> <sup>T</sup>b<sup>i</sup> j i

ð Þ <sup>y</sup>; <sup>x</sup>; <sup>z</sup> , W<sup>j</sup>

11. Fundamental supersonic antiderivative stress tensor Hb and its

∗θð Þz δð Þx

<sup>∗</sup>θð Þ<sup>z</sup> <sup>δ</sup>ð Þ¼ <sup>x</sup> <sup>T</sup>b<sup>i</sup>

i

j ∗ z

This tensor can be obtained in a different way, by analogy with T, using Hooke's law, except that the Green tensor should be replaced with its antiderivative W. By using the presentation of the basic functions of Green's tensor construction (Eq. (25)) in the supersonic case, it can be

Obviously, for z < τ, all the introduced shifted tensors are zero. It has the following symmetry

ð Þ¼� <sup>y</sup>; <sup>x</sup>; <sup>z</sup>; <sup>m</sup> <sup>H</sup><sup>j</sup>

<sup>i</sup> ð Þ¼ <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>H</sup><sup>3</sup>

ð Þ <sup>x</sup>; <sup>z</sup> have weak singularities on the fronts of the type <sup>z</sup><sup>2</sup> � <sup>m</sup><sup>2</sup>

more stronger singularity of the type of ∥x∥�<sup>1</sup> on the axis Z. If we put Eq. (51) in Hook's law,

2

k¼1

j ð Þ x; z :

<sup>j</sup> ð Þ¼ <sup>x</sup>; <sup>z</sup> <sup>X</sup>

i

<sup>θ</sup>kð Þ <sup>z</sup> � mkr <sup>H</sup>is

ð Þ x; y; z; �m

<sup>i</sup> ð Þ y; x; z , i ¼ 1, 2:

jkð Þþ <sup>x</sup> <sup>H</sup>id

jk ð Þ x; z � � (55)

j∥x∥<sup>2</sup> � ��1=<sup>2</sup>

, but

i

<sup>i</sup> ð Þ <sup>y</sup>; <sup>x</sup>; <sup>z</sup> , H<sup>3</sup>

<sup>θ</sup>ð Þ<sup>z</sup> , <sup>H</sup><sup>b</sup> <sup>i</sup> j , <sup>z</sup> <sup>¼</sup> <sup>T</sup>b<sup>i</sup> j

ð Þ¼� <sup>x</sup>; <sup>y</sup>; <sup>z</sup> <sup>W</sup><sup>j</sup>

General Functions Method in Transport Boundary Value Problems of Elasticity Theory

i

ð Þ y; x; z (53)

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

(54)

145

They are also fundamental solutions of Eq. (14) for the mass forces of the corresponding Fj ∗ <sup>z</sup> θð Þz . After calculation, we define its components as:

$$2\pi W\_1^1 = \frac{z}{r^4} \left(\mathbf{x}\_1^2 - \mathbf{x}\_2^2\right) \left(\theta\_1 V\_1 - \theta\_2 V\_2\right) + 0.5 m\_1^2 \theta\_1 \ln \frac{z + V\_1}{m\_1 r} + \left(M\_2^2 - 0.5 m\_2^2\right) \theta\_2 \ln \frac{z + V\_2}{m\_2 r} \tag{49}$$

$$2\pi W\_2^2 = -\frac{z}{r^4} \left(\mathbf{x}\_1^2 - \mathbf{x}\_2^2\right) \left(\theta\_1 V\_1 - \theta\_2 V\_2\right) + 0.5 m\_1^2 \theta\_1 \ln \frac{z + V\_1}{m\_1 r} + \left(M\_2^2 - 0.5 m\_2^2\right) \theta\_2 \ln \frac{z + V\_2}{m\_2 r} \tag{40}$$

$$2\pi W\_3^3 = \theta\_1 \ln \frac{z + V\_1}{m\_1 r} + m\_2^2 \theta\_2 \ln \frac{z + V\_2}{m\_2 r}, \quad 2\pi W\_2^3 = -\mathbf{x}\_2 r^{-2} (\theta\_1 V\_1 - \theta\_2 V\_2)$$

$$2\pi W\_1^2 = \pi \mathbf{x}\_1 \mathbf{x}\_2 r^{-4} (\theta\_1 V\_1 - \theta\_2 V\_2), \quad 2\pi W\_1^3 = -\mathbf{x}\_1 r^{-2} (\theta\_1 V\_1 - \theta\_2 V\_2)$$

Tensor W ci <sup>j</sup> has the same support as <sup>U</sup><sup>b</sup> <sup>i</sup> <sup>j</sup> but as at the cone Kj

$$\dot{m}\_{jl} \parallel \dot{x} \parallel \dot{z} \parallel \dot{z} \parallel z \parallel \dot{\theta} \parallel V\_{jl}^{-1}(\dot{x}, z) \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\theta} \parallel \dot{\$$

it continues on fronts Kj. Wj i ð Þ x; z has weak singularity by x' = 0 and weak logarithmic singularity on Z with respect to ∥x∥ by x = 0. To single out these singularities, we decompose it into the terms:

$$\begin{split} \mathcal{W}\_{\rangle}^{i}(\mathbf{x},z) &= \mathcal{W}\_{\rangle}^{is}(\mathbf{x},z) + \mathcal{W}\_{\rangle}^{id}(\mathbf{x},z) = \sum\_{k=1}^{2} \theta\_{k}(z - m\_{k}r) \ \mathcal{W}\_{\neq k}^{is}(\mathbf{x}) + \mathcal{W}\_{\neq}^{id}(\mathbf{x},z), \\ & \quad 2\pi c^{2} \mathcal{W}\_{\neq 1}^{is}(\mathbf{x}) = -\left\{\delta\_{i3}\delta\_{\overline{3}} + 0, 5m\_{1}^{2}(1 - \delta\_{i3})\delta\_{i\overline{j}}\right\} \text{lnm}\_{1}r, \\ & \quad 2\pi c^{2} \mathcal{W}\_{\neq 2}^{is}(\mathbf{x}) = (\delta\_{i3}\delta\_{\overline{3}} + \delta\_{\overline{i}\overline{j}}(0, 5m\_{1}^{2}(1 - \delta\_{i3}) - M\_{2}^{2})\text{lnm}\_{2}r \end{split} \tag{51}$$

The tensors Wis <sup>j</sup> of diagonal form are independent of z inside the sonic cones Kl(l = 1, 2) and have a logarithmic singularity with respect to ∥x∥ on the Z-axis. Unlike the generating tensor Wis <sup>j</sup> , Wid <sup>j</sup> has bounded jumps on the Kl. One can readily see that the tensor shifts

$$\mathcal{U}\_i^j(\mathbf{x}, y, z) = \widehat{\mathcal{U}}\_i^j(\mathbf{x} - y, z), \\ \mathcal{W}\_i^j(\mathbf{x}, y, z) = \widehat{\mathcal{W}}\_i^j(\mathbf{x} - y, z)$$

have the following symmetry properties around the Z-axis:

$$\mathcal{U}\_i^j(\mathbf{x}, y, z) = \mathcal{U}\_i^j(y, \mathbf{x}, z) = \mathcal{U}\_j^i(\mathbf{x}, y, z), \quad \mathcal{W}\_i^j(\mathbf{x}, y, z) = \mathcal{W}\_i^j(y, \mathbf{x}, z) = \mathcal{W}\_j^i(\mathbf{x}, y, z), \text{ i.j} = 1, 2 \tag{52}$$

But for the components with indices (i,j) = (1,3), (3,1), (2,3), (3,2)

$$\mathcal{U}\_{i}^{j}(\mathbf{x},y,z) = -\mathcal{U}\_{i}^{j}(y,\mathbf{x},z), \quad \mathcal{W}\_{i}^{j}(\mathbf{x},y,z) = -\mathcal{W}\_{i}^{j}(y,\mathbf{x},z) \tag{53}$$
