Abstract Generated abstract
This note formulates an inverse problem for scalar diffraction of a plane wave by a sphere whose surface admittance is expanded in Legendre polynomials with coefficients depending on the nondimensional parameter kr0. The objective is to choose these coefficients so that, in the low-frequency limit, the normalized pressure at a point on the sphere approximates an optimal angular function maximizing Sukharevsky’s functional under a prescribed uniqueness condition for its absolute maximum. The paper gives an exact solution for the simplest case N equals 1, deriving the required admittance functions and showing that they yield the corresponding optimal limiting angular distributions. Separate choices are presented for the cases where the maximum is imposed at one point or allowed at two points, giving functional values 4 and 6 respectively.
Full Text
UDC 534.88+517.946.9
MECHANICS
G. D. MALOZHYNETS
ON AN INVERSE PROBLEM IN DIFFRACTION THEORY
(Presented by Academician L. I. Sedov on 24 VI 1968)
The scalar problem of diffraction of a plane wave \(e^{ikr\cos\gamma}\) \((\cos\gamma=\cos\theta\cos\alpha+\sin\theta\sin\alpha\cos(\varphi-\beta);\ r,\theta,\varphi\) are spherical coordinates; \(\alpha,\beta\) are angles determining the direction of the incident wave) by a sphere of radius \(r_0\) with center at the point \(r=0\) and with a variable admittance \(g\), prescribed on the surface of the sphere, of the form
\[ g(\theta,\rho)=\sum_{l=0}^{N-1} g_l(\rho)P_l(\cos\theta)\qquad (\operatorname{Re} g\geqslant 0) \tag{1} \]
(\(N\) is a given positive integer; \(P_l(\cos\theta)\) are Legendre polynomials; \(g_l(\rho)\) are rational functions of the parameter \(\rho=kr_0\), which will subsequently be chosen in a special way) is formulated as follows: in the region \(r>r_0\) outside the sphere it is required to find a function \(p(r,\theta,\varphi,\alpha,\beta,N,k)\), meromorphically dependent on the parameter \(k\), satisfying, with respect to the variables \(r,\theta,\varphi\), the Helmholtz equation
\[ \Delta p+k^2p=0\qquad (p\sim e^{-i\omega t}) \tag{2} \]
and the absorption condition \({}^{(1-3)}\)
\[ \left|p-e^{ikr\cos\gamma}\right|<\infty\quad \text{for}\quad \operatorname{Im}k>|\operatorname{Re}k|, \tag{3} \]
and, on the surface of the sphere, the boundary condition
\[ \partial p/\partial r+ikgp=0\quad \text{for } r=r_0. \tag{4} \]
For given coefficients \(g_l(\rho)\) and number \(N\), the formulated direct diffraction problem (1), (2), (3), (4) has a unique \({}^{(2,3)}\) solution \(p(r,\theta,\varphi,\alpha,\beta,N,k)\).
Denote by \(p_N(\rho,\alpha)\) the value of the function \(p(r,\theta,\varphi,\alpha,\beta,N,k)\) at the point \(r=r_0,\ \theta=0\) of the surface of the sphere:
\[ p_N(\rho,\alpha)=p(r_0,0,\varphi,\alpha,\beta,N,k). \tag{5} \]
Owing to the axial symmetry of the domain and of the boundary condition, it does not depend on the angles \(\varphi,\beta\).
In the present note a formulation is proposed of the inverse diffraction problem, consisting in the determination of such coefficients \(g_l(\rho)\) entering into formula (1) that, as \(\rho\to 0\), the ratio
\[ f(\alpha,\rho)=p_N(\rho,\alpha)/p_N(\rho,0) \]
approximates as accurately as possible a certain optimal function \(f_N(\alpha)\).
As such a function we take
\[ f_N(\alpha)=(N+1)^{-2}\sum_{n=0}^{N}(2n+1)P_n(\cos\alpha), \tag{6} \]
having, at the single point of the closed interval \(0\leq \alpha \leq \pi\), an absolute maximum (in modulus) \(f_N(0)=1\). The coefficients in the right-hand side of (6) are chosen so that, for the given natural number \(N\) and under the condition of uniqueness of the absolute maximum \(f_N(0)=1\), the functional of Yu. M. Sukharevskii \((^4)\)
\[ K_N=2\int_0^\pi |f_N(\alpha)|^2\sin\alpha\,d\alpha \tag{7} \]
takes, for the function (6), the greatest possible value (in comparison with the values of this functional for functions of the form (6) with other real or complex coefficients)
\[ K_N=(N+1)^2. \tag{8} \]
The condition that the absolute maximum of \(f_N(\alpha)\) be attained at the single point \(\alpha=0\) is essential here. If two points are allowed, for example \(f_N(0)=f_N(\pi)=1\), then instead of (6), (8) we have
\[ f_N(\alpha)=(N+1)^{-1}(2N+1)^{-1}\sum_{n=0}^{N}(4n+1)P_{2n}(\cos\alpha), \tag{6*} \]
\[ K_N=(N+1)(2N+1). \tag{8*} \]
Here we shall confine ourselves to an example of the exact solution of such an optimal inverse problem for the simplest case \(N=1\), when, according to (1),
\[ g(\theta,\rho)=g_0(\rho). \tag{9} \]
The solution of the direct diffraction problem (2), (3), (4), (9) has the form
\[ \begin{aligned} p(r,\theta,\varphi,\alpha,\beta,1,k) &=\sum_{n=0}^{\infty} i^n(2n+1) \left[ j_n(kr)-\frac{j_n'(\rho)+ig_0j_n(\rho)} {h_n'(\rho)+ig_0h_n(\rho)}h_n(kr) \right]\times\\ &\quad\times \left[ P_n(\cos\theta)P_n(\cos\alpha) +2\sum_{m=1}^{n}\frac{(n-m)!}{(n+m)!} P_n^m(\cos\theta)P_n^m(\cos\alpha)\cos m(\varphi-\beta) \right], \end{aligned} \]
where \(j_n(\rho)=\sqrt{\pi/2\rho}\,J_{n+1/2}(\rho)\); \(h_n(\rho)=\sqrt{\pi/2\rho}\,H_{n+1/2}^{(1)}(\rho)\) are spherical Bessel functions. In particular, according to (5), for the point \(r=r_0\), \(\theta=0\) on the surface of the sphere we obtain
\[ \begin{aligned} p_1(\rho,\alpha) &=\sum_{n=0}^{\infty} i^n(2n+1)P_n(\cos\alpha) \left[ j_n(\rho)-\frac{j_n'(\rho)+ig_0j_n(\rho)} {h_n'(\rho)+ig_0h_n(\rho)}h_n(\rho) \right]\\ &=\sum_{n=0}^{\infty} \frac{i^{\,n+1}(2n+1)P_n(\cos\alpha)} {\rho^2\left[h_n'(\rho)+ig_0h_n(\rho)\right]} . \end{aligned} \]
Hence, using the representation
\[ h_n(\rho)=\frac{e^{i\rho}}{i^{\,n+1}\rho} \sum_{m=0}^{n} \frac{(-)^m}{m!}\frac{(n+m)!}{(n-m)!}\frac{1}{(2i\rho)^m}, \]
we find the relation
\[ f(\alpha,\rho)=\frac{p_1(\rho,\alpha)}{p_1(\rho,0)} = \frac{ \displaystyle\sum_{n=0}^{\infty}(2n+1)c_n(\rho)(2i\rho)^nP_n(\cos\alpha) }{ \displaystyle\sum_{n=0}^{\infty}(2n+1)c_n(\rho)(2i\rho)^n }, \tag{10} \]
where it is denoted
\[ \frac{1}{c_n(\rho)}=\sum_{m=0}^{n}\frac{(-)^m}{m!}\frac{(2n-m)!}{(n-m)!}(n-m+1-i\rho g_0-i\rho)(2i\rho)^m. \tag{11} \]
If the admittance \(g_0\) were a fixed complex number, then from (10), (11) it would follow that
\[ f(\alpha,\rho)=1+O(\rho)\quad \text{as } \rho\to 0 \quad (P_0(\cos\alpha)\equiv 1). \]
However, if one chooses the coefficient \(g_0\) in the form of a function of the parameter \(\rho\), which in the present case is the solution of the inverse problem posed,
\[ g_0(\rho)=1-i\cdot 2/\rho, \tag{12} \]
then from (10), (11) we obtain:
\[ f(\alpha,\rho)=f_1(\alpha)+O(\rho)\quad \text{as } \rho\to 0, \tag{13} \]
which in the limit \(\rho\to 0\) coincides with the optimal expression
\[ f_1(\alpha)=\frac14\sum_{n=0}^{1}(2n+1)P_n(\cos\alpha)=\frac{1+3\cos\alpha}{4}, \]
given by formula (6) for \(N=1\), so that, according to (8), the maximum possible value (under an absolute maximum, at the single point \(\alpha=0\)) of Sukharevsky’s functional (7) is attained:
\[ K_1=4. \tag{14} \]
If, however, an absolute maximum is allowed at two points \(f_1(0)=f_1(\pi)=1\), the solution of the inverse problem
\[ g_0=-i(3/\rho-\rho) \tag{15} \]
leads, according to (10), (11), (6), (8), to the function optimal for \(N=1\),
\[ f(\alpha,0)=f_1(\alpha)=(5\cos^2\alpha-1)/4, \tag{16} \]
which gives the maximum value of the functional (7)
\[ K_1=6. \tag{17} \]
Acoustical
Institute
Received
19 VI 1968
CITED LITERATURE
- G. D. Malyuzhinets, DAN, 60, No. 3 (1948).
- G. D. Malyuzhinets, Some generalizations of the reflection method in diffraction theory, Author’s abstract of doctoral dissertation, “Nauka,” 1950.
- G. D. Malyuzhinets, DAN, 78, No. 3 (1951).
- Yu. M. Sukharevsky, Elektrosvyaz’, No. 4 (1939).