Transition Zone in High-Frequency Diffraction on Impedance Contour with Jumping Curvature. Kirchhoff’s Method and Boundary Layer Method

Abstract

The wavefield is investigated in a two-dimensional problem of nontangential incidence of high-frequency wave from a point source on an impedance contour with a jump in curvature. The field in the vicinity of the limit ray is described in detail. It is shown that expressions found within Kirchhoff’s method agree with results of the boundary layer method. Suitability areas of the expressions obtained are described, which is based upon a detailed study of geometry of the problem.

APPENDIX

ANALYSIS OF THE RAY FORMULA FOR REFLECTED WAVE

Let us analyze the classical (see, e.g., [19]) ray formula for the reflected wave ${{u}^{{{\text{ref}}}}}$ in the vicinity of the limit ray at moderate ($\kappa {{\rho }} \sim 1$) and small ($\kappa {{\rho }} \ll 1$) distances from the nonsmoothness point $O$. This will allow us to determine the applicability areas of formulas obtained above.

A.1. Ray Method Formula

In the case of a smooth boundary, the ray method [19] gives the following formula for a specularly reflected wave:

(A.1)

Here, $l_{{0,1}}^{{\text{*}}}$ are the distances from the source point ${{M}_{0}}$ and from the observation point ${{M}_{1}}$ to the point of specular reflection $N{\text{*}}$, respectively, ${{\tau {\text{*}}}} = l_{0}^{{\text{*}}} + l_{1}^{{\text{*}}}$ is the value of the eikonal at ${{M}_{1}}$ (that is, the geometrical travel time), ${{\alpha {\text{*}}}}$ is the angle of incidence at $N{\text{*}}$, is the value of curvature at $N{\text{*}}$, $R{\text{*}}$ is reflection coefficient at $N{\text{*}}$ (see Fig. 3). For a contour whose curvature has a jump at the point $O$, expression (A.1) correctly describes waves specularly reflected from smooth parts of the contour to the right and left of $O$. However, expression (A.1) has a discontinuity on the limit ray and therefore is unsuitable in its narrow vicinity⁴.

Fig. 3.

Specular reflection.

Let us simplify the ray formula (A.1) for observation points ${{M}_{1}}$ close to the limit ray, $\left| {{{\delta \varphi }}} \right| \ll 1$, similarly to how it was done in [8, 20] for the case of a plane wave incidence. We will separately consider moderate and small distances.

A.2. Moderate Distance

First, let $\kappa {{\rho }} \sim 1$. We will obtain an expression for the arc length $s{\text{*}}$ corresponding to the reflection point $N{\text{*}}$ (see Fig. 3). It is clear that for observation points from a narrow neighborhood of the limit ray ($\left| {{{\delta \varphi }}} \right| \ll 1$) reflection points lie near $O$, that is, $s{\text{*}}$ is small.

The law of specular reflection reads

$${{\varphi }}_{0}^{*} + {{\psi{\text{*}}}} = {{\varphi }}_{1}^{*} - {{\psi{\text{*}}}},$$

(A.2)

where ${{\varphi }}_{{0,1}}^{{\text{*}}} = {{{{\varphi }}}_{{0,1}}}(s*)$ are the angles between the axis $Ox$ and lines ${{N}_{{\text{*}}}}{{M}_{{0,1}}}$, ${{\psi{\text{*}}}} = {{\psi }}\left( {s{\kern 1pt}{\text{*}}} \right)$ is the tangential angle of the contour at the point $N{\text{*}}$. It follows from (A.2) that ${{\tan\varphi }}_{0}^{{\text{*}}} = {\text{tan}}\left( {{{\varphi }}_{1}^{{\text{*}}} - 2{{\psi*{ *}}}} \right)$. This is written in terms of coordinates as

$$\begin{gathered} - T\left( {{{x}_{0}},{{y}_{0}}} \right) = \frac{{T\left( {{{x}_{1}},{{y}_{1}}} \right) - {\text{tan}}2{{\psi{\text{*}}}}}}{{1 + T\left( {{{x}_{1}},{{y}_{1}}} \right){\text{tan}}2{{\psi{\text{*}}}}}}, \\ T\left( {{{x}_{{0,1}}},{{y}_{{0,1}}}} \right) = \frac{{{{x}_{{0,1}}} - X{\text{*}}}}{{{{y}_{{0,1}}} - Y{\text{*}}}}. \\ \end{gathered} $$

(A.3)

Here, $X{\text{*}} = X\left( {s{\text{*}}} \right)$ and $Y{\text{*}} = Y\left( {s{\text{*}}} \right)$ are Cartesian coordinates of the point $N{\text{*}}$.

The contour consists of two arcs of circles (see (3)), hence

(A.4)

Substituting (A.4) into (A.3) and using the smallness of $s{\text{*}}$, we straightforwardly derive

(A.5)

Considering $\left| {{{\delta \varphi }}} \right| \ll 1$ and employing (44), after some algebra, from (A.5) we get

$$\begin{gathered} s{\text{*}} = - \frac{{{{r}_{0}}{{r}_{1}}{{\delta \varphi }}}}{{\sin {{{{\varphi }}}_{0}}}} \\ \times \,\,\left( {\frac{1}{{{{\mathcal{J}}_{ - }}}} + {{\theta }}\left( { - {{\delta \varphi }}} \right)\left( {\frac{1}{{{{\mathcal{J}}_{ + }}}} - \frac{1}{{{{\mathcal{J}}_{ - }}}}} \right)} \right)\left( {1 + O\left( {{{\delta \varphi }}} \right)} \right). \\ \end{gathered} $$

(A.6)

The expression (3) for the curvature ${\text{of}}$ the contour is used here.

Now let us simplify the phase of exponential function in the ray formula (A.1). For points $N{\text{*}}$ close to $O$, the value of eikonal ${{\tau{\text{*}}}} = l_{0}^{{\text{*}}} + l_{1}^{{\text{*}}}$ at the point ${{M}_{1}}$ is written quite similarly to (27):

$$\begin{gathered} {{\tau {\text{*}}}} = {{r}_{0}} + {{r}_{1}} + s{\text{*}}\sin {{{{\varphi }}}_{0}}{{\delta \varphi }} \\ + \,\,\frac{{{{{\left( {s{\text{*}}} \right)}}^{2}}}}{{2{{r}_{0}}{{r}_{1}}}}{{\sin }^{2}}{{{{\varphi }}}_{0}}\left( {{{\mathcal{J}}_{ - }} + {{\theta }}\left( {s{\text{*}}} \right)\left( {{{\mathcal{J}}_{ + }} - {{\mathcal{J}}_{ - }}} \right)} \right) \\ + \,\,O\left( {s{\text{*}}{{{\left( {{{\delta \varphi }}} \right)}}^{2}}} \right) + O\left( {{{{\left( {s{\text{*}}} \right)}}^{2}}{{{{\delta \varphi }}} \mathord{\left/ {\vphantom {{{{\delta \varphi }}} {{\rho }}}} \right. \kern-0em} {{\rho }}}} \right) + O\left( {{{{{{\left( {s{\text{*}}} \right)}}^{3}}} \mathord{\left/ {\vphantom {{{{{\left( {s{\text{*}}} \right)}}^{3}}} {{{{{\rho }}}^{2}}}}} \right. \kern-0em} {{{{{\rho }}}^{2}}}}} \right). \\ \end{gathered} $$

(A.7)

Substituting here (A.6) and again taking into account the smallness of ${{\delta \varphi }}$, we obtain an expression for the phase in (A.1):

$$\begin{gathered} ik{{\tau {\text{*}}}} = ik\left( {{{r}_{1}} + {{r}_{0}}} \right) - ik\frac{{{{r}_{0}}{{r}_{1}}}}{2}{{\left( {{{\delta \varphi }}} \right)}^{2}} \\ \times \,\,\left( {\frac{1}{{{{\mathcal{J}}_{ - }}}} + {{\theta }}\left( { - {{\delta \varphi }}} \right)\left( {\frac{1}{{{{\mathcal{J}}_{ + }}}} - \frac{1}{{{{\mathcal{J}}_{ - }}}}} \right)} \right) + O\left( {k{{\rho }}{{{\left( {{{\delta \varphi }}} \right)}}^{3}}} \right). \\ \end{gathered} $$

(A.8)

The correction term is small under condition (48).

Similarly, we transform the expression in the denominator of the formula (A.1):

$$\mathcal{J}{\text{*}} = \left( {{{\mathcal{J}}_{ - }} + {{\theta }}\left( { - {{\delta \varphi }}} \right)\left( {{{\mathcal{J}}_{ + }} - {{\mathcal{J}}_{ - }}} \right)} \right)\left( {1 + O\left( {{{\delta \varphi }}} \right)} \right).$$

We transfer the last term in (A.8) to the amplitude and obtain a simplified form of the ray formula for $\kappa {{\rho }} \sim 1$ and $\left| {{{\delta \varphi }}} \right| \ll 1$:

$${{u}^{{{\text{ref}}}}} = \frac{{R\left( 0 \right)}}{{2\sqrt {2{{\pi }}} }}\exp \left( {ik\left( {{{r}_{1}} + {{r}_{0}}} \right) + i\frac{{{\pi }}}{4}} \right)\left( {{{\mathcal{A}}_{ - }} + {{\theta }}\left( { - {{\delta \varphi }}} \right)\left( {{{\mathcal{A}}_{ + }} - {{\mathcal{A}}_{ - }}} \right)} \right)\left( {1 + O\left( {k{{\rho }}{{{\left( {{{\delta \varphi }}} \right)}}^{3}}} \right)} \right),$$

(A.9)

see (46). Formula (45) agrees with (A.9) in the area where inequalities (47) and (48) are satisfied.

A.3. Small Distance

Address now the case where the observation point ${{M}_{1}}$ is close to $O$, $\kappa {{\rho }} \ll 1$, but $k{{\rho }} \gg 1$ (see (29)). Let us transform the formula (A.9). Obviously,

(A.10)

and then expression (A.9) takes the form

(A.11)

Formula (A.11) was obtained under the assumption that inequalities (48) and (57) hold.⁵ In the main approximation (with respect to the parameter $\kappa {{\rho }} \ll 1$) the reflected wave coincides with a wave from an imaginary source corresponding to a flat boundary. The terms linear in curvature are of smaller order.

From (A.11) it is clear that the terms in (53) corresponding to ${{U}_{0}}$ and ${{U}_{1}}$, in the area of their suitability, match with the specularly reflected wave.