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UDC 534.26
MATHEMATICAL PHYSICS
V. M. ASTAPENKO
REFLECTION OF SOUND BY AN IMPEDANCE CORRUGATED SURFACE
(Presented by Academician L. M. Brekhovskikh, 27 IV 1970)
Let, in the \(xy\)-plane, there be located the trace of a corrugated surface periodic with period \(2c\) along the \(y\)-axis (in \(xyz\)-space, parallel to the \(z\)-axis), each period of which consists of a part of a convex curve, having an axis of symmetry parallel to the \(x\)-axis, and a part of the straight line \(x=0\) (Fig. 1).
Denote by \(D\) the region lying to the right \((x>0)\) of the above-mentioned curve, and let a plane wave be incident from the region \(D\) onto its boundary \(\partial D\)
\[ P_0=\exp[-ik(\alpha x-\beta y)] \qquad (\alpha^2+\beta^2=1) \]
with wave number \(k\) and direction cosines \(\alpha\) and \(\beta\).
Statement of the problem. It is required to find a function \(P=P(x,y;k)\), twice continuously differentiable in the region \(D\), continuous on the boundary \(\partial D\), and quadratically integrable in neighborhoods of the corner points of \(\partial D\), regular in \(k\) for \(|k|<\pi/2c\), which satisfies the following conditions: in the region \(D\), the Helmholtz equation
\[ (\Delta+k^2)P=0; \tag{1} \]
on one part of the boundary \(\partial D\) (the set of the above-mentioned convex curves \(\Gamma\)), the Neumann boundary condition
\[ \partial P/\partial n\big|_{\Gamma}=0 \tag{2} \]
(\(n\) is the normal to \(\Gamma\), directed into \(D\)); on the other part of \(\partial D\) (the set of segments of the straight line \(x=0\), \(L\)), the third boundary condition
\[ \partial P/\partial x+ikg(y;k)P\big|_{L}=0 \tag{3} \]
(\(g(y;k)\) is a function periodic with period \(2c\), continuous in \(y\), regular in \(k\) for \(|k|<\pi/2c\), satisfying the condition \(\operatorname{Re} g\geq 0\)); everywhere in the region \(D\), the quasiperiodicity condition
\[ P(x,y+2c)=P(x,y)e^{2ik\beta c}; \tag{4} \]
the Maliuzhinets radiation condition \((^1)\)
\[ \sup_D |(P-P_0)e^{-ik\beta y}|<\infty \qquad \text{for } \operatorname{Im} k>|\operatorname{Re} k|. \tag{5} \]
Fig. 1
We note that the problem of diffraction of a plane wave by a rigid, frequently periodic corrugated surface was first posed and solved by G. D. Maliuzhinets. In the present work the method proposed by G. D. Maliuzhinets in solving the above-mentioned problem is applied.
On the basis of the fact that any solution of equation (1) satisfying conditions (4) and (5), as \(x\to 0\) and \(|k|<\pi/2c\), is representable as—
in the following form (²):
\[ P=P_0+ae^{ik(\alpha x+\beta y)}+O(e^{-\sigma x}),\qquad \sigma\geqslant \sigma_0>0 \tag{6} \]
(\(a\) is a constant), we seek the solution of problem (1)—(5) in the form
\[ P=P_0+\frac{\alpha+iB(\alpha,\beta;k)}{\alpha-iB(\alpha,\beta;k)}e^{ik(\alpha x+\beta y)} +\frac{2\alpha u(x,y;k)}{\alpha-iB(\alpha,\beta;k)}e^{ik\beta y}. \]
Next, substituting \(P\), as defined by the last equality, into conditions (1)—(5), we reformulate the diffraction problem posed above as the following boundary-value problem for the function \(u\):
\[ \left(\Delta+2ik\beta\frac{\partial}{\partial y}+k^2\alpha^2\right)u=0,\qquad (x,y)\in D; \tag{7} \]
\[ \frac{\partial u}{\partial n}+ik\beta\frac{\partial y}{\partial n}u\bigg|_{\Gamma} =-ik\beta[\cos(k\alpha x)-B\sin(k\alpha x)/\alpha]\frac{\partial y}{\partial n} +k[\alpha\sin(k\alpha x)+B\cos(k\alpha x)]\frac{\partial x}{\partial n}\bigg|_{\Gamma}; \tag{8} \]
\[ \frac{\partial u}{\partial x}+ikgu\bigg|_{L}=k(B-ig); \tag{9} \]
\[ u=O(e^{-\sigma x})\quad \text{as } x\to\infty; \tag{10} \]
\[ u(x,y+2c)=u(x,y); \tag{11} \]
\[ \begin{aligned} 4cB={}&-\alpha\int_{\Gamma_0}\sin(2k\alpha x)\frac{\partial x}{\partial n}\,dl -B\int_{\Gamma_0}[\cos(2k\alpha x)-1]\frac{\partial x}{\partial n}\,dl \\ &-2\alpha\int_{\Gamma_0}u\sin(k\alpha x)\frac{\partial x}{\partial n}\,dl -2i\beta\int_{\Gamma_0}u\cos(k\alpha x)\frac{\partial y}{\partial n}\,dl +2i\int_{L_0}[1+u(0,y)]g\,dy. \end{aligned} \tag{12} \]
Here \(\Gamma_0\) and \(L_0\) denote, respectively, the parts of \(\Gamma\) and \(L\) for \(|y|<c\).
The integral relation (12), connecting \(B\) and \(u\), was obtained from Green’s formula for the domain \(D_0\) (\(D_0\) is the part of \(D\) for \(|y|<c\)), applied to the functions \(P\) and \(\cos(k\alpha x)\exp(-ik\beta y)\).
Using the regularity of the functions \(u\), \(B\), and \(g\) with respect to \(k\) in a neighborhood of \(k=0\), we expand them in power series in \(k\):
\[ B=\sum_{p=0}^{\infty}B_pk^p,\qquad u=\sum_{p=0}^{\infty}u_pk^p,\qquad g=\sum_{p=0}^{\infty}g_pk^p. \]
If the resulting expression is substituted into conditions (7)—(12) and all terms having the same power of \(k\) are collected, then, similarly to what was done in (²), we obtain a recurrent sequence of boundary-value problems for the Laplace and Poisson equations:
\[ \Delta u_p+2i\beta\frac{\partial u_{p-1}}{\partial y}+\alpha^2u_{p-2}=0,\qquad (x,y)\in D; \tag{13} \]
\[ \begin{aligned} \frac{\partial u_p}{\partial n} +i\beta\frac{\partial y}{\partial n}u_{p-1}\bigg|_{\Gamma} ={}&-i\beta\left[ \frac{1-(-1)^p}{2}\frac{(i\alpha x)^{p-1}}{(p-1)!} -\sum_{r=1}^{p-1}\frac{1-(-1)^r}{2i}\frac{(i\alpha x)^r}{\alpha r!}B_{p-r-1} \right]\frac{\partial y}{\partial n} \\ &+\alpha\left[ \frac{1+(-1)^p}{2i}\frac{(i\alpha x)^{p-1}}{(p-1)!} +\sum_{r=0}^{p-1}\frac{1+(-1)^r}{2}\frac{(i\alpha x)^r}{\alpha r!}B_{p-r-1} \right]\frac{\partial x}{\partial n}\bigg|_{\Gamma}; \end{aligned} \tag{14} \]
\[ \frac{\partial u_p}{\partial x} +i\sum_{r=0}^{p-1}g_ru_{p-r-1}\bigg|_{L} =B_{p-1}-ig_{p-1}; \tag{15} \]
\[ u_p=O(e^{-(\pi/c-\varepsilon)x})\quad \text{as } x\to\infty\qquad (0<\varepsilon<\pi/c); \tag{16} \]
\[ u_p(x,y+2c)=u_p(x,y); \tag{17} \]
\[ 4cB_p =-\frac{1-(-1)^p}{2i}\frac{(2i\alpha)^p}{p!}\alpha M_p -\sum_{r=2}^{p}\frac{1+(-1)^r}{2}\frac{(2i\alpha)^r}{r!}B_{p-r}M_r+ \]
\[ +2i\alpha\sum_{r=1}^{p-1}\frac{1-(-1)^r}{2}\frac{(i\alpha)^r}{r!}\,\mu_{p-r}^r -2i\beta\sum_{r=0}^{p-1}\frac{1+(-1)^r}{2}\frac{(i\alpha)^r}{r!}\,\lambda_{p-r}^r +i\nu_p+i\sum_{r=0}^{p-1}\sigma_{p-r}^r . \tag{18} \]
Here the following notation has been introduced:
\[ M_p=\int_{\Gamma_0} x^p\frac{\partial x}{\partial n}\,dl,\qquad \mu_p^r=\int_{\Gamma_0} x^r u_p\frac{\partial x}{\partial n}\,dl,\qquad \lambda_p^r=\int_{\Gamma_0} x^r u_p\frac{\partial y}{\partial n}\,dl, \]
\[ \nu_p=\int_{L_0} g_p\,dy,\qquad \sigma_p^r=\int_{L_0} g_r u_p\,dy \tag{19} \]
\[ (u_p=B_p=g_p=0\quad \text{for }p<0). \]
In addition, in deriving equality (18) it was taken into account that the function \(u_0\equiv 0\). The latter assertion follows directly from the uniqueness of the solution of the boundary-value problem
\[ \Delta u_0=0,\quad (x,y)\in D,\qquad \left.\frac{\partial u_0}{\partial n}\right|_{\partial D}=0; \]
\[ u_0=O(e^{-\pi x/c})\quad \text{as }x\to\infty; \]
\[ u_0(x,y+2c)=u_0(x,y). \]
From equality (18), for \(p=0\), we obtain
\[ B_0=\frac{i}{2c}\nu_0=\frac{i}{2c}\int_{L_0} g_0\,dy. \tag{20} \]
Similarly, from the same equality for \(p=1\), the relation follows
\[ B_1=\bigl[i(\nu_1+\sigma_1^0)-2\alpha^2M_1-2i\beta\lambda_1^0\bigr]/4c \tag{21} \]
(\(M_1\) is the area bounded by the curve \(\Gamma_0\) and the straight line \(x=0\)).
The functionals \(\lambda_1^0\) and \(\sigma_1^0\) are determined by the solution of the first boundary-value problem (for the Laplace equation)
\[ \Delta u_1=0,\quad (x,y)\in D,\qquad \left.\frac{\partial u_1}{\partial n}\right|_{\Gamma} =-i\beta\frac{\partial y}{\partial n} +B_0\left.\frac{\partial x}{\partial n}\right|_{\Gamma}, \]
\[ \left.\frac{\partial u_1}{\partial x}\right|_{L}=B_0-ig_0,\qquad u_1=O(e^{-\pi x/c})\quad \text{as }x\to\infty, \tag{22} \]
\[ u_1(x,y+2c)=u_1(x,y). \]
Consider a function \(\varphi\), harmonic in \(D\), periodic with period \(2c\), and tending exponentially to zero as \(x\to\infty\), whose normal derivative on \(\Gamma\) takes the value
\[ \left.\frac{\partial\varphi}{\partial n}\right|_{\Gamma} =\left.\frac{\partial y}{\partial n}\right|_{\Gamma}. \]
If Green’s formula for the domain \(D_0\) is applied to the functions \(u_1\) and \(\varphi\), then we obtain
\[ \lambda_1^0=\frac{1}{2}i\beta\lambda_y,\qquad \lambda_y=-2\int_{\Gamma_0}\varphi\frac{\partial y}{\partial n}\,dl. \]
Here \(\lambda_y\) is the added-mass coefficient of the grating \((^2)\), obtained from the corrugated surface by adding to the latter its mirror reflection with respect to the \(y\)-axis.
Substituting the result obtained above into relation (21),
\[ B_1=\bigl[i(\nu_1+\sigma_1^0)+\beta^2\lambda_y-2\alpha^2M_1\bigr]/4c. \tag{23} \]
Thus, the field far from the grating in the first approximation in \(k\) as \(k\to 0\) is determined by the formula
\[ P=P_0+\frac{\alpha+iB_0+iB_1k+O(k^2)} {\alpha-iB_0-iB_1k+O(k^2)} \,e^{ik(\alpha x+\beta y)}; \]
where \(B_0\) and \(B_1\) have been found in the form of expressions (20) and (23) and are entirely and completely computed from the boundary value at \(L\) of the solution of the first boundary-value problem and from the coefficient of the attached mass of the corresponding grating.
To find the reflection coefficient of a plane wave from the original corrugated surface in any approximation in \(k\) as \(k \to 0\), one must use equality (18), whose right-hand side is expressed through the functionals (19), which depend only on the shape of the surface under consideration (the shape coefficients). The above-mentioned shape coefficients are expressed through solutions of the corresponding terms of the recurrent sequence of boundary-value problems for the Poisson equation. Specifying a particular function \(g(y; k)\) entering the boundary condition on the corrugated surface, and the shape of this surface, the recurrent sequence mentioned above can conveniently be computed on an electronic computer.
Acoustics Institute
Moscow
Received
27 IV 1970
CITED LITERATURE
- G. D. Malyuzhinets, Symposium on Wave Diffraction, Abstracts of Reports, Odessa, 1960, Publishing House of the Academy of Sciences of the USSR, 1960.
- V. M. Astapenko, G. D. Malyuzhinets, Acoustical Journal, vol. 3 (1970).
- L. I. Sedov, Plane Problems of Hydrodynamics and Aerodynamics, Moscow–Leningrad, 1950.