Radio Science

Diffraction by a wedge with a face of variable impedance

Authors


Abstract

[1] The two-dimensional problem of diffraction by a wedge with a face of variable impedance is explicitly solved by a probabilistic random walk method. The solution admits numerical simulation based on simple scalable algorithms with unlimited capability for parallel processing. The diffracted field is represented as a mathematical expectation of a specified functional on trajectories of random motions determined by the configuration of the problem. The solution is not significantly more difficult than a similar probabilistic solution of the problem with constant impedance faces.

1. Introduction

[2] For more than a century problems of wave diffraction by an infinite wedge have been at the center of the mathematical theory of diffraction starting from the landmark papers of Poincaré [1892, 1897] and Sommerfeld [1896], who presented explicit descriptions of wavefields generated by a plane incident wave in an infinite wedge with ideally reflecting faces. However, they used very different methods. The solution of Poincaré was based on the method of separation of variables, and it led to a solution represented by an infinite series of Bessel functions multiplied by associated trigonometric polynomials. The solution of Sommerfeld was in terms of the so-called Sommerfeld integral representing wavefields as superpositions of plane waves. Although these two solutions look very different, they are equivalent and can be easily converted from one to the other. It took about fifty years before a problem of diffraction by a wedge with impedance boundary conditions was solved. In the 1950s this problem was independently solved by Maliuzhinets [1951, 1955], Senior [1959] and Williams [1959]. They used techniques that are similar in general but different in detail that employed Sommerfeld's representation of wavefields to reduce the problem to certain functional equations in the complex plane. In the following decades the Maliuzhinets method was applied to a number of two- and three-dimensional problems of diffraction by infinite wedges. However, all of these problems either had boundary conditions with constant coefficients or even simpler boundary conditions with the coefficients proportional to the distance from the vertex, as considered by Felsen and Marcuvitz [1972].

[3] Here we consider a two-dimensional problem of diffraction by a wedge with a variable impedance at its face. We employ a novel probabilistic approach to wave propagation that is not restricted to problems with simple canonical geometries or to problems with special (constant) boundary conditions. The probabilistic approach has already been successfully applied by the authors to a standard two-dimensional problem of diffraction by a wedge with constant impedances, as well as to more challenging three-dimensional problems of diffraction by a plane angular sector [Budaev and Bogy, 2004] and by an infinite wedge with anisotropic face impedances [Budaev and Bogy, 2006b]. Most recently, the probabilistic method was applied to diffraction by an arbitrary convex polygon with side-wise constant impedances [Budaev and Bogy, 2006a]. This problem with nontrivial geometry does not have known conventional closed-form solutions, but its probabilistic solution is not considerably more complex than the solutions of the other problems mentioned above. Here we take the next step and extend the probabilistic method in a way which makes it possible to handle boundary conditions with variable coefficients. This is the centerpiece of the paper, and it is based on further development of the idea of analytical continuation introduced by Budaev and Bogy [2005c] and Budaev and Bogy [2006a], who applied it to other problems of interest.

[4] The paper is organized as follows. In section 2 we formulate the problem and convert it to a form suitable for the application of the random walk method. Section 3 deals with the problem with constant coefficients. Most of this material is published by Budaev and Bogy [2005b], but it is included here also to make the presentation of section 4 transparent and self-consistent. The main material of the paper is concentrated in section 4, and section 5 presents numerical experiments based on the obtained solution.

2. Formulation of the Problem

[5] Let (r, equation image) be standard polar coordinates and let equation image = equation image(0, α) be an angular domain

equation image

Then the problem of diffraction of the plane incident wave

equation image

in the wedge equation image consists of the computation of the wavefield U(r, θ), which obeys the Helmholtz equation ∇2U + k2U = 0, accompanied by the standard conditions at infinity and by the boundary conditions

equation image

where B1(r) and B2(r) are impedances assumed to satisfy the inequalities

equation image

which guarantee that the problem has a unique bounded solution.

[6] To reduce insignificant detail, we limit ourselves to the analysis of the configuration characterized by the inequalities α > π and θ* < απ, which guarantee that the face θ = 0 is illuminated by the incident wave, while the face θ = α is located in the shadow zone, as shown in Figure 1. Additionally, to focus only on diffraction we assume that the impedance of the illuminated face is constant, so that B1(r) = const. Then elementary analysis suggests that we seek the total field U(r, θ) in the form of the superposition

equation image

of the yet unknown scattered fields U1(r, θ) and U2(r, θ) together with the two predefined components

equation image

where

equation image

and

equation image

is the reflection coefficient of the surface with a constant impedance B1.

Figure 1.

Diffraction by a wedge with faces of nonconstant impedance.

[7] From (5) and (6) it follows that the scattered fields Un(r, θ), where n = 1,2, must be solutions of the Helmholtz equation that satisfy the conditions at infinity

equation image

the interface conditions

equation image
equation image

and the boundary conditions

equation image

where

equation image

is a function of two variables, but it is actually defined only on the faces θ = 0 and θ = α of the wedge.

[8] It should be emphasized that the boundary value problem (9)–(13) is formulated and will be solved without the assumption B1(r) = const which has been mentioned above. This assumption makes it possible to define the geometrically reflected wave by an elementary formula Ureflected = KUr, where Ur is a plane from (6) and K is the constant reflection coefficients from (8). As a result, the reduction of the original problem of diffraction to the boundary value problem is not complicated by insignificant detail, so that the paper may be focused on its principal innovation which is the procedure for the solution of the boundary value problem (9)–(13), which remains valid in most general case when both impedances B1(r) and B2(r) are arbitrary nonconstant functions.

3. Solution for the Case of Constant Impedances

[9] Formulas (9)–(12) determine two similar boundary value problems for the unknown wavefields U1 (r, θ) and U2 (r, θ), and these problems differ only by the position of the auxiliary interface, which is located either at the half line θ = ϕ1 or at the half line θ = ϕ2. Therefore we need only to find a method of computation of the wavefield U(r, θ; ϕ) ≡ U(r, θ) that is smooth everywhere inside the wedge equation image(0, α) except at the ray θ = ϕ where it has a discontinuity on the half line θ = ϕ described by the interface conditions

equation image

The field U(r, θ) must also satisfy the condition at infinity

equation image

and the boundary conditions (12) and (13).

[10] In the particular case of constant impedances B1 and B2 this problem was explicitly solved by Budaev and Bogy [2005b] by a probabilistic random walk method. Even though the solution obtained by Budaev and Bogy [2005b] was restricted to constant impedances, it is important for our current purposes and therefore is worth reproducing here.

[11] Assume first that the values of U(r, θ) are known on the rays θ = α1 and θ = α2 which are selected in such a way that either 0 < α1 < α2 < ϕ or ϕ < α1 < α2 < α. Then, seeking U(r, θ; ϕ) in the product form

equation image

we come to the problem of finding the amplitude u(r, θ) that vanishes at infinity and satisfies the transport equation

equation image

accompanied by the boundary conditions

equation image

where f1(r) and f2(r) are predefined functions. In the case when f1(r) and f2(r) are analytic and bounded in the first quarter 0 < arg(r) < π/2 of the complex plane, the auxiliary field u(r, θ) can be represented by the Feynman-Kac formula

equation image

where E denotes the mathematical expectation computed over the trajectories of the radial and angular motions ξt and ηt that are controlled by the stochastic equations

equation image

driven by standard one-dimensional Brownian motions wt1 and wt2. The process Pt = (ξt, ηt) starts from the initial position P0 = (r, θ) and stops at the exit time t = τ, defined as the first time when Pt eventually hits one of the faces ηt = 0 or ηt = α.

[12] It is easy to see that for any t > 0 the angular motion ηt is contained in the segment [0, α] and the radial motion ξt runs in the first quarter 0 < arg(ξ) < π/2 of the complex plane, drifting to an unreachable point ξ = i/2k. Such localization of ξt ensures that S(t) has a positive imaginary component which improves the convergence in (19). On the other hand, the fact that ξt runs in the complex plane implies that the boundary function f(ξ, η) must be analytic in the domain 0 < arg(ξ) < π/2 and that it should not grow too fast there along the random walk.

[13] Next we employ the formula (19) to derive the expression for the field U(r, equation image) defined in the entire wedge equation image(0, α) with the interface condition (14) and with the boundary conditions (12) and (13) where B1(r) and B2(r) are constants.

[14] Let U(r, equation image) be already known on the boundaries equation image = 0 and θ = α of the wedge equation image(0, α) as well as on both sides of the interface θ = ϕ ± 0. Then, the value of U(r, θ) with θ ≠ ϕ can be evaluated by the formula (19) applied to that wedge ϕ < θ < α or 0 < θ < ϕ which contains the observation point (r, θ). It is easy to see that this formula leads to the the expression

equation image

where S(t) has the same meaning as in (19). The mathematical expectation in (19) is computed over the trajectories of the random motion (ξt, ηt) which is similar to the motion from (19) and which stops at the earliest of the times t = τ1 or t = t1, where τ1 is the exit time through the side of the original wedge equation image(0, α) and t1 is the first time when ηt hits the interface η = ϕ.

[15] There are two unknown quantities U(ξequation image, ηequation image) and U(ξequation image, ηequation image) in the right-hand side of (19), but both of them can be evaluated using the boundary or interface conditions (12) or (14).

[16] To compute U(ξt1, ηt1), we observe that the auxiliary exit point ηt1 takes one of two values η = ϕ ± 0 which is determined by the location of the observation point (r, θ). Therefore the value of U(ξt1, ηt1) can be represented by the expression

equation image

which is an immediate consequence of the interface condition (14), and which can be rewritten in the probabilistic form

equation image

where dη = ±dt is a random number with two equally possible values and

equation image

The last formula implies that δ = 1 in the case when the angular motion ηt crosses the interface η = ϕ from the right to the left, δ = −1 if the interface is crossed from the left to the right and δ = 0 in the case when the interface is touched but is not intersected.

[17] To compute the value U(ξτ1, ηequation image) of the field U(r, equation image) on the boundary of the wedge equation image(0, α) where the impedance boundary conditions (12) are imposed, we rewrite these conditions in the form

equation image

where two possible values of

equation image

correspond to the two different faces of the wedge equation image(0, α). Then, taking into account the identity epdt = 1 − pdt + o(dt), we convert (25) to the expression

equation image

which represents U(ξequation image, ηequation image) through the value of the field U(r, θ) at the point (ξequation image, ηequation image + dη) located in the interior of the wedge equation image(0, α).

[18] Since both of the points (ξequation image, equation image + dη) and (ξequation image, ηequation image + dη) that appear in (23) and (27) are located inside equation image(0, α) but not on the interface θ = ϕ, the values of U(ξequation image, ϕ + dη) and U(ξequation image, ηequation image + dη) can be evaluated by formulas like (21), and repeating obvious iterations we eventually arrive at the final expressions

equation image
equation image

where the mathematical expectation is computed over the trajectories of the stochastic processes ξt, ηt, δ(tν, ϕ) and λt described below.

[19] The radial motion ξt retains its meaning from (20), while the angular motion ηt is controlled by the stochastic equations

equation image

which imply that ηt is the so-called Brownian motion with reflections at the times t = τ1, t = τ2, …, when it touches the boundary of the interval (0, α). Inside this interval, ηt runs as a standard Brownian motion but every time when it hits the boundary it is deterministically reflected back into the interval.

[20] The angular motion ηt completely determines the other stochastic processes λt and δ(tν, ϕ), which are controlled by the equations

equation image

and

equation image

where {tν} is the sequence of times when the angular motion ηt touches the fixed point η = ϕ. The process λt may be considered as a measure of the time that the angular motion ηt spends on the boundary of the interval (0, α), and correspondingly, this process is known as the ‘local time’ of the reflected Brownian motion ηt on the boundary.

4. Solution for the General Case of Nonconstant Impedances

[21] By analyzing the above described approach to the problem with constant impedances it is easy to see that in order to extend the approach to problems with arbitrary impedances it is necessary to find analytical continuation of the boundary impedances to the complex space (r, θ) with complex radial and angular coordinates.

[22] Consider first the simplest nontrivial case when B2 is still constant but B1(r) is a piecewise constant function

equation image

where b0, b1 and L > 0 are given numbers.

[23] It is clear that the pieces I0(L) = {equation image = 0, 0 < rL} and I1(L) = {θ = 0, r > L} of the face θ = 0 can be characterized by the analytic equations

equation image
equation image

where

equation image

Indeed, applying the cosine theorem to the triangle ΔOPQ shown in Figure 2, we see that rp = equation image(r, θ, p) and that this triangle collapses to the segment (OP] in the case when rp = pr. We conclude that if the vertex Q belongs to the segment (0, L], then the equality rp = pr must remain valid for all p ≥ L.

Figure 2.

Analytic description of the segment (0, L].

[24] The last result implies that if the point Pt = (ξt, ηt) moves in the complex space, then at any time t = t* when ηt* = 0, the point Pt belongs to the interval Iν(t*) where the value ν(t*) = 0 or ν(t*) = 1 of the index νt is determined by the entire preceding trajectory of the point Pt rather than by its position at the time t = t*. In other words, equations (34) and (35) provide analytic continuation of the real one-dimensional intervals I0 and I1 to two-dimensional surfaces in the complex space formed by the pairs (r, θ), where both of the components are considered as complex variables. Correspondingly, these formulas define the analytical continuation of the boundary piecewise function B1(r) to the complex space (r, θ).

[25] Since B1(r) is analytically continued to the complex space, the field U(r, θ) can be computed by the same variation of the random walk method as employed in the previous section and the resulting solution has the form (28) with the phase S(t) computed by a more complex rule than that in (29). More precisely, S(t) is a stochastic process that starts from the initial position S(0) = 0 and evolves thereafter as

equation image

where ν(t) is an index determining which part of the face θ = 0 is touched.

[26] The value of ν(t) is explicitly defined by the formula

equation image

which requires tracing the value of the radical

equation image

along the trajectory of the motion Pt. This procedure provides a clear and unambiguous definition of v(τ), but it hides the relationship between ν(τ) and the global structure of the trajectory of the motion Pt, which becomes transparent from a closer look at (39).

[27] It is obvious that for any p > 0 the argument of the first radical in (39) has the value

equation image

where the branches of the multivalued functions equation image and arg(ζp) are fixed by the cut along the ray [p, ∞). Indeed, since ξt and ηt start from the real positive values ξ0 = r and η0 = θ, and since these motions are contained in the domains 0 ≤ arg(ξt) < π/2 and 0 ≤ ηt < α ≤ 3π/2, the complex number ζt = equation image is contained in the domain 0 ≤ arg(ζt) < 2π. This guarantees that equation image never crosses the cut [p, ∞) where p ≥ 0 and therefore that (40) holds for any possible trajectory of Pt.

[28] To compute the second radical in (39), we need to trace the trajectory of the point ζt = equation image, which starts from the position ζ0 = re−iθ in the domain arg(ζ) < 0 and stops at the point ζτ = ξτ located in the quarter 0 < arg(ζτ) < π/2. Taking into account the restraints 0 ≤ arg(ξt) < π/2 and 0 ≤ ηt < 3π/2 we conclude that ζt is contained in the domain −3π/2 < arg(ζ) < π/2. Such information about the continuous motion ζt guarantees that it crosses the ray arg(ζ) = 0 an odd number of times, as illustrated in Figure 3.

Figure 3.

Trajectory of ζt = equation image and determination of ρτ = inf{pn}.

[29] Let equation imageτ = {pn} be the set of all positive points pn > 0 where the trajectory of ζt intersects the ray arg(ζ) = 0, and let N(p, equation imageτ) be the number of points of the set equation imageτ located to the left of p. Then, the argument of equation image takes the values

equation image

which is completely determined by the disposition of the parameter p with respect to the set equation imageτ and does not depend on other details of the trajectory of ζt. Indeed, if N(p, equation imageτ) is even, then ζt intersects the branch cut [p, ∞) an odd number of times and this leads to the first line of (41). Otherwise, if N(p, equation imageτ) is odd, then the trajectory of ζt has an even number of intersections with the cut [p, ∞) and this leads to the second option of (41).

[30] Finally, combining (39) with (40) and (41), we get

equation image

and combining (42) with (34) and (35), we arrive at the remarkable formula

equation image

which makes it possible to evaluate ν(t) by tracing the intersections of the trajectory of ζt = equation image with the positive semiaxis, but without tracing the radical (39).

[31] The previous result can be straightforwardly generalized to the representation of the field U(r, equation image) in the special case when the impedance B1(r) has a piecewise constant structure

equation image

where {Ln} is a monotonically increasing sequence with L0 = 0, and bn are some constants. Indeed, observing that the intervals In can be represented as In = I(Ln + 1)\I(Ln), where I(L) is the domain in the space (r, θ) defined as

equation image

we readily come to the representation of the field U(r, θ) in the form (28) with the phase S(t) defined by the stochastic equation

equation image

with the index ν(t) defined as

equation image

where equation imageτ is the full intersection of the trajectory of ζt = equation image with the semiaxis arg(ζ) = 0. Comparison of (47) with (44) shows that if the point

equation image

belongs to the interval In, then ντ = n and bν(t) = B1(ρt, 0). As a result, the solution (46) can be written in the form

equation image

which does not rely on the piecewise structure of B1(r), and which therefore can be extended to the case when B1(r) is a virtually arbitrary function of the real variable r > 0.

[32] The probabilistic representation (28), (46) of the wavefield U(r, θ) in the wedge equation image(0, α) with the variable impedance B1(r) and constant impedance B2(r) can be easily generalized to describe this field in the case when both of the impedances are arbitrary functions. Indeed, straightforward reiteration of the above reasonings leads to the representation of the field U(r, θ) by the mathematical expectation

equation image
equation image

where ξt, ηt, δ(tv, ϕ) and λt retain their meaning from (28)–(32) and ρt is the random process controlled by the stochastic equations

equation image

where equation imaget0(r; ηt) and equation imagetα(r; ηt) are the intersection of the trajectory of the motions

equation image

with the semiaxis arg(ζ) = 0.

5. Numerical Examples

[33] To illustrate the suitability of the obtained solution of the problem of diffraction by a wedge with a face of variable impedance, we conducted numerical simulations for the configuration considered by Osipov [2004] and Budaev and Bogy [2005b]. In this configuration the wave number is fixed as k = 1, and a plane incident wave U*(r, θ) = eequation image with θ* = 43° propagates in the wedge 0 < θ < 266° with impedance boundary conditions on its faces.

[34] In Figure 4 we consider four cases with the face θ = 0 having a fixed constant impedance B1 = 5 and with the face θ = 266° having four different impedances. In the first and the second cases the second impedance has constant values B2 = 5 and B2 = 1/5, respectively. The total wavefield corresponding to B2 = 5 is shown by a thick solid line, and it obviously agrees with the results of Osipov [2004] and Budaev and Bogy [2005b]. The field corresponding to B2 = 1/5 is shown by a thin line additionally marked by dots. In the third and the forth cases the impedance of the face θ = 266° is variable and defined as B2(r) = −i/(r2 + 1) and B2(r) = 5(1 + cos(/4)), respectively. The total fields in these cases are shown by × and ° marks. As expected the total fields corresponding to different impedances of the face θ = 266° are practically undistinguishable in the domain 0° < θ < 150° but they differ from each other considerably in the domain 200° < θ < 266°.

Figure 4.

Total fields in a wedge with B1 = 5.

[35] Results of four other numerical simulations are shown in Figure 5. In all cases presented in Figure 5 the face θ = 0 has a constant impedance B1 = 1/5 while the impedance of the face θ = 266° are either constant or piecewise constant functions. The bold solid line shows the total field in the case with B2 = 1/5, which has also been considered by Osipov [2004] and Budaev and Bogy [2005b]. The field corresponding to B2 = 5 is shown by a thin line additionally marked by dots. In the two other cases the impedance B2 is defined by

equation image

The total wavefields corresponding to these cases are marked by crosses and circles, respectively.

Figure 5.

Total fields in a wedge with B1 = 1/5.

[36] All of the computations presented here were obtained by the averaging of 1500 discrete random walks with the time increment dt = 0.01. The computations were carried out on a 900 Mhz notebook PC using MATLAB code, which was only a few lines longer than the code published by Budaev and Bogy [2005b] for the similar problem with constant impedances.

6. Conclusion

[37] The version of the probabilistic random walk method surveyed by Budaev and Bogy [2005a] made it possible to obtain a solution of the problem of diffraction by a wedge with a nonconstant impedance faces. The obtained solution is not considerably more complex than the similar solution of the standard problems of diffraction by a wedge with constant impedances, and it admits numerical simulation based on a simple very short algorithm. These features confirm that the probabilistic approach is well suited for further applications to problems of diffraction with complex shapes and boundary conditions.

Acknowledgments

[38] This research was supported by NSF grant CMS-0408381.

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