### 物理代写|理论力学代写theoretical mechanics代考|MATH4022

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## 物理代写|理论力学代写theoretical mechanics代考|The Study of the Problem in the Local Formulation

Let a circular monochromatic high-frequency wave fall from the point $x_{0}$ of the infinite elastic plane to the boundary contour $l$ of an obstacle or a system of obstacles in it. The wave is generated by the force $Q e^{i \omega t}$ located at point $x_{0}$, where $\omega$ is the oscillation frequency. In this case, the displacements at the point $y$ of the elastic plane are determined by the Kupradze matrix [7].

The aim is to study the amplitude characteristics of the scattered field by the contours of obstacles in the through-transmitted elastic wave.

In the directions $\mathbf{q}{1}$ and $\mathbf{q}{2}$ we have asymptotic representations of the amplitudes of displacements in the incident wave
$$\begin{gathered} \mathbf{u}{\mathbf{q}}^{(p)}(y)=\frac{Q{\mathrm{q}}}{4 \mu} \mathbf{q} i \frac{k_{p}^{2}}{k_{s}^{2}} \sqrt{\frac{2}{\pi k_{p}}} \mathrm{e}^{-i \frac{\pi}{4}} \frac{\mathrm{e}^{i k_{p} R_{0}}}{\sqrt{R_{0}}}\left[1+\mathrm{O}\left(\left(\frac{1}{k_{p} R_{0}}\right)\right)\right], \quad Q_{\mathrm{q}}=(\mathrm{Q}, \mathrm{q}), \ \mathbf{u}{\mathrm{q}{1}}^{(s)}(y)=\frac{Q_{\mathrm{q}{1}}}{4 \mu} \mathbf{q}{1} i \sqrt{\frac{2}{\pi k_{s}}} \mathrm{e}^{-i \frac{\pi}{4}} \frac{\mathrm{e}^{i k_{s} R_{0}}}{\sqrt{R_{0}}}\left[1+\mathrm{O}\left(\frac{1}{k_{s} R_{0}}\right)\right], \quad Q_{\mathbf{q}{1}}=\left(\mathrm{Q}, \mathrm{q}{1}\right) . \end{gathered}$$
Here the tangential direction $\mathbf{q}{1}$ is perpendicular to $\mathbf{q}{\mathbf{1}} Q_{\mathbf{q}}$ and $Q_{\mathbf{q}{1}}$ are the projections of the force $\mathbf{Q}$ on the directions $\mathbf{q}$ and $\mathbf{q}{1}$. Here $\rho$ is the mass density, $\lambda, \mu$ are the Lamè coefficients, $k_{p}=\omega / c_{p}, k_{s}=\omega / c_{s}, c_{p}$ and $c_{S}$ are the wave numbers and the velocities of the longitudinal and transverse waves. The components of the displacement vector in the reflected wave from the free boundary contour at the point $x$ of the elastic plane are determined by the following integral [8]
$$\begin{gathered} u_{k}(x)=\int_{l} \mathbf{T}{y}\left[\mathbf{U}^{(k)}(y, x)\right] \cdot \mathbf{u}(y) d l, \quad k=1,2 \ \mathbf{T}{y}\left[\mathbf{U}^{(k)}(y, x)\right]=2 \mu \frac{\partial \mathbf{U}^{(k)}}{\partial n}+\lambda \mathbf{n} \operatorname{div}\left(\mathbf{U}^{(k)}\right)+\mu\left(\mathbf{n} \times \operatorname{rot}\left(\mathbf{U}^{(k)}\right)\right) \end{gathered}$$
where the Kupradze matrix $\mathbf{U}^{(k)}(y, x)$ is obtained from the matrix $\mathbf{U}^{(k)}\left(y, x_{0}\right)$ by replacing $x_{0}$ by $x$ and $R_{0}$ by $R=|y-x|$. $\mathbf{T}_{y}$ is the force vector at the point $y$, $\mathbf{u}(y)$ is the vector of the total displacement field on the boundary surface, $\mathbf{n}$ is the outer unit normal to the contour $l$, directed toward the elastic medium.

## 物理代写|理论力学代写theoretical mechanics代考|Two-Fold Reflection of Elastic Waves on the Plane

This section is devoted to the development of the ray diffraction theory with respect to arbitrary (nonconvex) smooth two-dimensional obstacles in an elastic medium. Double re-reflection of the high-frequency wave, taking into account possible transformations, can be formed both within the contour of one obstacle (Fig. 1) and two different obstacles (Fig. 2). Numerical investigation of the problems of highfrequency scattering of elastic waves is considerably complex if the wavelength is much smaller than the average size of the scatterer. There are some known numerical methods-the finite element method, the method of boundary elements, all require in this case a large number of nodes on the grid. This leads to instability of the calculation. To calculate the displacement amplitude in a multiply re-reflected wave, it is possible to use the Keller geometric theory of diffraction (GTD) [11], based on the use of divergence coefficients, which is rather cumbersome. If we investigate the problem of the reflection of a high-frequency wave from an obstacle contour in an elastic medium with various possible wave transformations of an arbitrary finite number of times $N$, then it is more convenient to start from the estimate of the $N$-fold multiple diffraction integral by the multidimensional stationary phase method. The basis for the investigation of the general case of an arbitrary number of re-reflections is the solution of the problem of double reflection (Figs. 1 and 2), to which we turn.
The direct usage of the integral representation (3) over the entire “light” zone for reflected waves is impossible [9], since it does not describe multiply reflected waves. If one substitutes to the Green’s formula (3) the solution of [12] for local problems (8) and $(10$ ) and as the primary field takes the total field $u(y)$, then the integral formula (3) gives only a single-reflected wave. A doubly reflected wave is obtained only when the values of $u(y)$ include both the primary field and its single reflection. To solve the problem of double re-reflection, we start from the modification [9] of the integral formula (3). Following this modification, the doubly reflected waves will be found by integrating along the neighborhood $l_{2}^{}$ of the second mirror reflection point $y_{2}^{}$ the rays obtained upon single reflection from the neighborhood $l_{1}^{}$ of the first mirror reflection point $y_{1}^{}$. Such a modification means that when finding the leading term of the asymptotics of the double diffraction integral, we stay within the framework of the calculation of the displacement amplitude in a doubly reflected wave in accordance with the GTD.

## 物理代写|理论力学代写theoretical mechanics代考|Multiple Reflections with All Possible Transformations

The geometry of the boundary contours of the obstacles in the elastic medium and their arrangement can form such trajectories of the rays $x_{0}-y_{1}^{}-y_{2}^{}-\cdots-y_{N}^{}-x_{N+1}$ which lead to any possible sequence of reflections and wave transformations at the points of specular reflection. Suppose that for any $N$ times re-reflected ray, in a certain order, $p-p$ and $s-s$ reflections have been realized at the mirror reflection points $y_{1}^{}, y_{2}^{}, \ldots, y_{N-1}^{}, y_{N}^{}$, respectively $N_{1}$ and $N_{3}$ times, and $p-s$, and $s-p$, transformations-respectively $N_{2}$ and $N_{4}$ times. At the receiving point $x_{N+1}$, both the longitudinal wave $u\left(x_{N+1}\right)=u_{r}^{(p)}\left(x_{N+1}\right)$ and the transverse one $u\left(x_{N+1}\right)=u_{\theta}^{(s)}\left(x_{N+1}\right)$ may be received. In this case, the amplitude of the radial or tangential displacement of the $N$ times reflected ray at the point $x_{N+1}$ relatively the local polar coordinate system $r, \theta$ at the point $y_{N}^{}$ of the boundary contour of the obstacle is represented by the multiple Kirchhoff integral, which is formed according to the same laws as the diffraction integral (11), by taking into account reflections and transformations of the propagating ray at the points of mirror reflection:
$u_{r}^{(p)}\left(x_{N+1}\right)=B(-1)^{N} \mathrm{e}^{-i \frac{\pi}{4}}\left(\frac{k_{p}}{2 \pi}\right)^{\frac{N_{1}+N_{2}}{2}}\left(\frac{k_{s}}{2 \pi}\right)^{\frac{N_{3}+N_{4}}{2}} \frac{1}{\sqrt{L_{0}}} \prod_{n=1}^{N} \frac{\cos \gamma_{n}^{(2)}}{\sqrt{L_{n}}} V\left(y_{n}^{}\right)$ $\times \int_{l_{N}} \int_{l_{N-1}} \ldots \int_{l_{2}^{}} \int_{l_{i}^{}} \mathrm{e}^{i k_{P \psi}} d l_{N} d l_{N-1} \ldots d l_{2} d l_{1}$ $\varphi=k_{p}^{-1}\left(k_{1}\left|x_{0}-y_{1}\right|+\sum_{n=1}^{N-1} k_{n}\left|y_{n}-y_{n+1}\right|+k_{N}\left|y_{N}-x_{N+1}\right|\right)$ $L_{0}=\left|x_{0}-y_{1}^{}\right|, L_{n}=\left|y_{n}^{}-y_{n+1}^{}\right|, L_{N}=\left|y_{N}^{*}-x_{N+1}\right|, \quad n=1,2, \ldots, N-1 .$

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