Reports of the Academy of Sciences of the USSR
1962. Volume 146, No. 3

MATHEMATICAL PHYSICS

V. M. BABICH

ON THE SHORT-WAVE ASYMPTOTICS OF THE GREEN FUNCTION FOR THE EXTERIOR OF A BOUNDED CONVEX DOMAIN

(Presented by Academician V. I. Smirnov on 10 V 1962)

Let the function \(u(x,y,x_0,y_0,k)\) be the solution of the following problem:

\[ \left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}+k^2\right)u = -\delta(x-x_0,y-y_0); \]

\[ (x_0,y_0)\ \text{outside } S;\qquad \left.\frac{\partial u}{\partial n}\right|_{S}=0;\qquad \sqrt r\left(\frac{\partial u}{\partial r}-iku\right)_{r=\sqrt{x^2+y^2}\to\infty}\longrightarrow 0. \tag{1} \]

Here \(S\) is a sufficiently smooth closed convex contour, outside which the solution of problem (1) is sought; \(\delta\) is the Dirac delta function. The paper studies the behavior of the function \(u(x,y,x_0,y_0,k)\) as \(k\to\infty\).

\(1^\circ\). The asymptotics of \(u\) are different in different regions of the \(x,y\)-plane. Behind the contour \(S\) a shadow region is formed, separated from the illuminated region by two rays \(l_1\) and \(l_2\), which are tangent to the contour \(S\) at certain points \(A\) and \(B\) and go off to infinity. The continuations of \(l_1\) and \(l_2\) into the illuminated region pass through the source \(M_0\).

We shall denote the illuminated part of the contour by \(AmB\), and the shadow part by \(AnB\). The asymptotics of \(u\) are essentially different in the illuminated region, in the shadow region, and in the neighborhood of the limiting rays \(l_1\) and \(l_2\).

We shall first find the asymptotics for \(u\) when \(x,y\in S\). Using the methods of work \((^1)\), one can construct a function \(L(x,y,s_0,k)\) \((s_0\in S)\) having the following properties:

\[ (\Delta+k^2)L=0;\qquad \left.\frac{\partial L}{\partial n}\right|_{x,y\in S} = \delta(s-s_0)+K(s,s_0,k); \tag{2} \]

\[ |K|\le C_1\left(\exp\left(-Ck^{1/3}|s-s_0|\right)+\exp\left(-Ck^{1/7}\right)\right). \tag{3} \]

Here \(s\) and \(s_0\) are points on \(S\); \(|s-s_0|\) is the length of the arc connecting \(s\) and \(s_0\). The function \(L(x,y,s_0,k)\) is expressed in a rather complicated way in terms of the solution of problem (1) for the exterior of a circle. In estimate (3), and in the subsequent estimates, constants \(C,C_1,C_2\), etc., occur. Unless the contrary is specifically stated, the \(C_i\) are completely determined by the contour \(S\) and by the position of the source \(M_0\).

Applying Green’s formula to the desired function \(u\) and to the function \(L\), we obtain without difficulty, for

\[ u(s,k)=u(x,y,x_0,y_0,k)\big|_{x,y\in S}, \]

the integral equation

\[ u(s_0,k)+\int_S K(s,s_0,k)u(s,k)\,ds = L(x_0,y_0,s_0,k), \tag{4} \]

whose kernel, by virtue of estimates (3), will be small.

Construct the circle of curvature for the contour \(S\) corresponding to the point \(s_0\). Denote the solution of problem (1) for the exterior of this circle of curvature by \(\Phi(x,y,s_0,k)\). Using the explicit expression for \(L\), it is not difficult to show that, up to a term of order \(\exp(-Ck^{-1/7})\), the right-hand side in the inte-

in the integral equation (4) may be replaced by \(\Phi(x(s_0), y(s_0), s_0, k)\) \((x(s_0), y(s_0)\) are the Cartesian coordinates of the point \(s_0 \in S)\).

The integral equation (4) can be solved by the method of successive approximations:

\[ u(s_0)=\Phi(x(s_0),y(s_0),s_0,k)+K\Phi+K^2\Phi+\cdots+O(\exp(-Ck^{1/7})); \]

\[ K\Phi=\int K(s,s_0,k)\Phi(\ldots)\,ds;\qquad K^n\Phi=K(K^{n-1}\Phi). \tag{5} \]

Estimate (3) and formula (5) give

\[ u(s_0)=\Phi(x(s_0),y(s_0),s_0,k)+R, \tag{6} \]

where

\[ |R|\leq C_2\max_{s_0\in S}|\Phi(x(s_0),y(s_0),s_0,k)|\,\frac{1}{k^{1/8}}. \tag{7} \]

Using the explicit form of \(\Phi\) and of the kernel \(K\), the estimate for the remainder term \(R\) can be considerably improved. This improved estimate has the form:

\[ |R|\leq \begin{cases} C_3 k^{-2-1/6}, & s\in A m B,\quad \min(|s-s_A|,|s-s_B|)\geq \alpha,\\ C_4 k^{-1/2-1/3}, & s\in A m B,\quad \min(|s-s_A|,|s-s_B|)\leq \alpha,\\ C_5 k^{-1/2-1/3}\bigl(\exp(-Ck^{1/3}|s-s_A|)+\exp(-Ck^{1/3}|s-s_B|)+{}\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad +\exp(-Ck^{1/7})\bigr). \end{cases} \tag{8} \]

Here \(\alpha\) is some positive constant independent of \(k\); \(C_3\) depends on \(\alpha\). We note that the function \(\Phi\) on the arc \(A n B\) has the estimate

\[ |\Phi|\leq C_6 k^{-1/2}\bigl(\exp(-Ck^{1/3}|s-s_A|)+\exp(-Ck^{1/3}|s-s_B|)\bigr). \]

It follows from this formula that in the region of deep shadow (i.e., for \(s\in A n B\), \(\min(|s_A-s|,|s_B-s|)\geq \alpha\), where \(\alpha\) does not depend on \(k\)) the function majorizing \(R\) exceeds \(|\Phi|\). Thus, in the deep shadow we obtain not an asymptotic formula for \(u\), but only the estimate

\[ |u|\leq C_7\exp(-Ck^{1/7}),\qquad s\in A n B,\qquad \min(|s-s_A|,|s-s_B|)\geq \alpha. \]

\(2^\circ\). In the illuminated region, for the function \(u\) it is easy to construct (formally) an asymptotic expansion. Represent \(u\) in the form

\[ u=\frac{i}{4}H_0^{(1)}(kr)+g, \]

where \(g\) is the reflected part of the Green’s function; \(H_0^{(1)}\) is the Hankel function. Replace the function \(H_0^{(1)}(kr)\) by its asymptotic series, assuming that \(kr\gg 1\). We shall also seek the function \(g\) in the form

\[ g=\frac{e^{ik\tau(x,y)}}{\sqrt{k}}\sum_{l=0}^{\infty}\frac{v_l(x,y)}{k^l}. \tag{9} \]

We require that the series (9) formally satisfy the Helmholtz equation and that, on the surface \(S\), for the series (9) and the asymptotic series for

\[ \frac{i}{4}H_0^{(1)}(kr) \]

the boundary condition be fulfilled:

\[ \left.\frac{\partial}{\partial n}\left(\frac{i}{4}H_0^{(1)}+g\right)\right|_S=0. \]

The coefficients of the series (9) will be uniquely determined by these conditions if the point \(x,y\) is in the illuminated region. Such formal constructions have been carried out by many authors (an extensive bibliography is given in [ ? ]). It is natural to expect that the expansion (9) is an asymptotic series for \(g\).

\(3^\circ\). Using the explicit expression for \(\Phi\), it is not difficult to show that in the illuminated region for \(\Phi(x(s_0),y(s_0),s_0,k)\) there is an asymptotic expansion of the form (9), the first two terms of which coincide with the first two

terms of the expansion obtained by the method just described for the function

\[ \frac{i}{4} H_0^{(1)}(kr) + g\big|_S . \]

It follows from the estimates (8) that

\[ u\big|_S = e^{ikr(s)}\left(\frac{u_0(x,y)}{\sqrt{k}}+ \frac{u_1(x,y)}{k\sqrt{k}}\right)+O\left(k^{-2-\frac16}\right), \tag{10} \]

\[ s\in AmB,\qquad \min\bigl(|s-s_A|,\ |s-s_B|\bigr)\ge a, \]

\[ r(s)=\sqrt{(x_0-x(s))^2+(y_0-y(s))^2}. \]

Here \(u_0\) and \(u_1\) are obtained by means of the formal constructions of item \(2^\circ\).

\(4^\circ\). In the case when \(x,y\bar{\in} S\),

\[ u(x,y,x_0,y_0,k) = \frac{i}{4}H_0^{(1)}(kr) - \int_S \frac{i}{4}\frac{\partial}{\partial n}H_0^{(1)}(kr_1)\,u(s)\,ds, \tag{11} \]

\[ u(s)=u(x,y,\ldots)\big|_{x,y\in S}, \]

\[ r=\sqrt{(x-x_0)^2+(y-y_0)^2},\qquad r_1=\sqrt{(x-x(s))^2+(y-y(s))^2}. \]

Using formulas (5), (8), (10) and the explicit form of \(K\) and \(\Phi\), one can show that in the illuminated region

\[ u(x,y,x_0,y_0,K) = \frac{i}{4}H_0^{(1)}(kr) + \frac{e^{ik\tau(x,y)}}{\sqrt{k}} \left(v_0+\frac{v_1}{k}\right) + O\left(k^{-5/3}\right). \tag{12} \]

Here \(v_0\) and \(v_1\) are the same as in formula (9); the estimate \(O(k^{-5/3})\) is uniform for points \((x,y)\) lying in any finite closed subdomain of the illuminated region having no common points with the limiting rays.

If the point \((x,y)\) belongs to the shadow zone, then from formula (11) one can derive only the estimate

\[ u=O\left(k^{-5/3}\right). \tag{13} \]

Estimate (13) is uniform for points \((x,y)\) belonging to any finite closed domain, together with its boundary, lying in the shadow zone and having no common points with the limiting rays.

Leningrad State University
named after A. A. Zhdanov

Received
11 IV 1962

CITED LITERATURE

¹ F. Ursell, Proc. Cambridge Phil. Soc., 53, No. 1, 115 (1957). ² V. M. Babich, A. S. Alekseev, B. Ya. Gelchinskii, Problems of the Dynamic Theory of the Propagation of Seismic Waves, collection V, L., 1961, p. 3.