ISSN 0965-5425, Computational Mathematics and Mathematical Physics, 2007, Vol. 47, No. 9, pp. 1514–1527. © Pleiades Publishing, Ltd., 2007. Original Russian Text © S.A. Zapunidi, A.V. Popov, 2007, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2007, Vol. 47, No. 9, pp. 1576– 1590.
Physical Pattern of Wave Emission in a Wedge-Shaped Region: Generalization of the Transverse Diffusion Method S. A. Zapunidi and A. V. Popov Institute of Terrestrial Magnetism, Ionosphere, and Radio Wave Propagation, Russian Academy of Sciences, Troitsk, Moscow oblast, 142092 Russia e-mail:
[email protected] Received February 2, 2007
Abstract—An analysis of the rigorous solution to the problem of waves emitted by sources arbitrarily distributed along a wedge face is used to propose a generalization of the Malyuzhinets heuristic transverse diffusion method. Mathematically, the problem is reduced to the numerical solution of a parabolic equation in ray coordinates with prescribed discontinuities on the boundaries of the shadow zone of partial plane waves or with a distributed right-hand side. The physical concept of the phase synchronism of emitted and diffracted waves is stated. DOI: 10.1134/S0965542507090126 Keywords: diffraction theory, Sommerfeld integral, parabolic equation, ray coordinates, phase synchronism, Malyuzhinets transverse diffusion method, numerical method for parabolic equations
INTRODUCTION The Malyuzhinets transverse diffusion method is a generalization of the Leontovich–Fock parabolic equation (see [1]). Specifically, the diffracted field is represented as a superposition of waves of various types, each of which is described by its own parabolic equation written in ray coordinates. The diffraction phenomenon is manifested in the diffusion of the wave amplitude along curvilinear wavefronts through the common boundaries of partial waves. This method agrees well with exact solutions to a number of model problems (see [2, 3]) and gives the asymptotics of the wave field not only in a narrow neighborhood of the shadow-zone boundary but also in deep shadow. However, the method has not been widely applied because of certain fundamental and computational difficulties, namely, an ambiguous partition of the total wave field into partial components and the lack of boundary conditions on their matching lines for determining a unique solution. Moreover, the coefficients of the parabolic equation written in curvilinear ray coordinates may have singularities. As a result, a special analysis of the behavior of its solutions is required and special computational schemes have to be developed. These issues are considered below in the context of the classical problem of inhomogeneous waves emitted by a vibrating wedge face. The exact solution to this problem was given in the form of a Sommerfeld integral (see [4]). Because of its computational complexity, this solution is unsuitable for direct computations in practice, but its analysis allows us to answer the above questions, which still remain open in the transverse diffusion method. Since the transverse diffusion method is inapplicable in its primary formulation, we refuse to describe the emitted wave by a parabolic equation. Instead, the emitted wave is represented as a superposition of inhomogeneous plane waves generated by individual components of the complex Fourier transform of the wedge face’s vibration amplitude, which is given. For each component, we introduce its own shadow-zone boundary determined by its phase synchronism with the diffracted cylindrical wave propagating away from the wedge edge. In the case of a discrete set of such waves, the emitted wave has an amplitude jump on each shadow-zone boundary, which can be compensated for by the solution to the parabolic equation describing the diffracted cylindrical wave. Multiple jumps can merge to form a smeared shadow-zone boundary. In this case, the emitted wave constructed no longer satisfies the Helmholtz equation, but the discrepancy it generates in the entire domain has a cylindrical wave phase. That is why an asymptotic solution to the problem can be constructed by adding a diffracted wave satisfying an inhomogeneous parabolic equation with a right-hand side specified by the discrepancy of the emitted wave. Apparently, this procedure is a well-defined mathematical expression of the idea of transverse diffusion offered in Malyuzhinets’ early works. Compared with the exact solution to the wedge face emission problem for particular examples, this approach demonstrates good accuracy and computational efficiency. 1514
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1. EXACT SOLUTION TO THE EMISSION PROBLEM AND THE TRANSVERSE DIFFUSION METHOD In the wedge-shaped region r > 0, 0 < ϕ < Φ, we consider a monochromatic wave field satisfying the Helmholtz equation 2
∆E + k E = 0
(1)
with the boundary conditions E ( ρ, 0 ) = F ( ρ ),
E ( ρ, Φ ) = 0
(2)
and the suppression condition E ( ρ, ϕ ) < ∞
for Im k > 0
(3)
(see [1, 5]). Here, ρ = kr is the dimensionless distance from the edge of the wedge. For simplicity, the consideration is restricted to the case Φ > π, when the inactive wedge face ϕ = Φ is not directly exposed to the sources specified in the half-plane ϕ = 0. The exact solution to this problem was given in [4] in the form of a Sommerfeld integral (see [5]): 1 – iρ cos α [ s ( ϕ + α ) – s ( ϕ – α ) ] dα, E ( ρ, ϕ ) = -------- e 2πi
∫
Γ
(4)
+
where the contour Γ+ is shown in Fig. 1. In the shaded areas, the Sommerfeld kernel exp(–iρ cos α) increases as Im α ∞. The analytic function s(α) is regular in the strip 0 < Re α < Φ + π and is determined by solving a system of linear functional equations that follow from boundary conditions (2): s ( α ) – s ( – α ) = 2 f ( α ), s ( α + Φ ) – s ( –α + Φ ) = 0,
(5)
Im α S
Γ+
Re α –π
0
Fig. 1. COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
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where f (α) is determined by the Fourier transform of the given boundary value F(ρ): ∞
i sin α iρ cos α f ( α ) = ------------- e F ( ρ ) dρ. 2
∫
(6)
0
By construction, f (α) is an odd 2π-periodic function. Its singularity in the complex plane of α lies in the semi-infinite strips given by 2Nπ < Re α < (2N + 1)π, Im α > 0 and (2N – 1)π < Re α < 2Nπ, Im α < 0. By virtue of Eqs. (5), the function s(α) has the same singularities in the semi-infinite strip –π < Reα < 0, Imα < 0 and, by assumption, does not have any other singularities in the strip –π < Reα < Φ + π (physically, this means that there are no other sources but those described by boundary condition (2)). Equations (5) are solved using the modified Fourier transform: i ⎛ s ( α ) = -------------- ⎜ 2 2π ⎝
ε + i∞
–ε + i∞
∫
+
–ε – i∞ iωΦ
⎞ – iωα dω, ⎟ S ( ω )e ⎠ ε – i∞
∫
e S ( ω ) = -------------------------------2π sin ( ωΦ )
(7)
i∞
∫
f ( α )e
iωα
dα
– i∞
(see [6]). This solution describes the wave field produced by sources arbitrarily distributed along the face ϕ = 0. In the general case, it is too difficult to compute (7), since it requires a fourfold integral transform of the boundary function F(ρ): F ( ρ ) ⇒ f ( α ) ⇒ S ( ω ) ⇒ s ( α ) ⇒ E ( ρ, ϕ ).
(8)
In the simplest case of uniformly distributed sources (i.e., for F(ρ) ≡ 1), the solution is found by the method of [6]: i⎛ s ( α ) = --- ⎜ 8⎝
ε + i∞
–ε + i∞
∫
+
–ε – i∞
⎞ cos [ ( Φ – α )ω ] ⎟ ------------------------------------------- dω. ⎠ ⎛π ⎞ ε – i∞ sin --- ω sin ( Φω ) ⎝2 ⎠
∫
(9)
Since the integrand in (4) is odd and in view of the easy-to-check relation π sin ⎛ ---- α⎞ ⎝ ⎠ Φ π -, s ( α – π ) – s ( α + π ) = ------- --------------------------------------------2 2Φ π π cos ------- – cos ⎛ ---- α⎞ ⎝Φ ⎠ 2Φ
(10)
the integration path in the Sommerfeld integral can be reduced to a saddle contour S: α = iγ – π – arcsin ( tanh γ ) (see Fig. 1). The contour passes through the pole α = –π/2 – ϕ of s(ϕ + α), the residue at which corresponds to an emitted plane wave propagating normally to the emitting face in the sector 0 < ϕ < π/2 (see Fig. 2, which shows the diffraction pattern for a plane wave emitted by sources uniformly distributed along the face ϕ = 0). The contribution of the saddle point α = –π is determined by the standard method and has the form of a cylindrical diffracted wave as if it propagated away from the wedge edge. As a result, we obtain the asymptotic representation E ( ρ, ϕ ) ~ { e
iρ sin ϕ
ρ→∞
exp [ i ( ρ + π/4 ) ] } 0 < ϕ < π/2 + --------------------------------------- A ( ϕ ), 2πρ
(11)
where the scattering pattern A(ϕ) = s(ϕ – π) – s(ϕ + π) is explicitly determined by formula (10). Since s(α) is singular, this representation does not make sense on the shadow-zone boundary ϕ = π/2, which is actually the source of a “boundary” wave (see [1]). A uniform asymptotic representation of the solution has the form E ( ρ, ϕ ) ~ { e ρ→∞
iρ sin ϕ
iρ
} 0 < ϕ < π/2 + e U ( ρ, ϕ ),
(12)
(see [2]). Here, the slowly varying wave amplitude U(ρ, ϕ) satisfies the following parabolic equation in ray COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
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Diffusion region ϕ = π/2 Emitted plane wave
E(ρ, 0) = 1 Diffracted wave
E(ρ, Φ) = 0 Fig. 2.
coordinates (ρ, ϕ): 2
∂U U 1 ∂ U 2i ------- + i ---- + -----2 ---------2- = 0. ∂ρ ρ ρ ∂ϕ
(13)
Formula (12) is derived from exact solution (4) by proceeding to a steepest descent contour S (see Fig. 1) 2
(α + π) and approximating the Sommerfeld kernel exp(iρcosα) by exp iρ ⎛ 1 – --------------------⎞ , which holds in the ⎝ 2 ⎠ neighborhood of the saddle point α = π as ρ ∞: 2
1 (α + π) U ( ρ, ϕ ) = -------- exp – iρ -------------------- [ s ( ϕ + α ) – s ( ϕ – α ) ] dα. 2πi 2
∫
(14)
S
Thus, the emitted plane wave Eg = {eiρsinϕ}0 < ϕ < π/2 appears as a residue at the integrand’s pole crossed by the integration contour when it is deformed, and the diffracted wave Ed = U(ρ, ϕ)exp(iρ) propagates from the shadow-zone boundary due to diffusion along the cylindrical fronts r = const. This result is a special case of the heuristic transverse diffusion principle proposed by Malyuzhinets for solving a wide class of diffraction problems (see [1]). In accordance with this concept, the high-frequency (k ∞) asymptotic representation of the wave field is a superposition of waves of various natures (incident, reflected, and diffracted) propagating along rays of generalized geometric optics. These waves are governed by a parabolic equation written in the corresponding system of ray coordinates. As applied to the given problem in the general case of nonuniformly distributed sources, the Malyuzhinets concept can be described as follows [1]. The rays and fronts of the emitted wave are constructed, and the parabolic equation of transverse diffusion ∂U ∂ ( ln h η ) 1 ∂ 1 ∂U - U + ----- ------ ⎛ ----- -------⎞ = 0 ik 2 ------- + -----------------∂ξ ∂ξ h η ∂η ⎝ h η ∂η ⎠
(15)
written in ray coordinates (ξ, η) is solved together with equations of form (13) for two cylindrical “boundary” waves existing on both sides of the shadow-zone boundary (see Fig. 3, which shows the diffraction pattern in the case of nonuniformly distributed sources). However, the numerical implementation of this program encounters a number of difficulties. First, this approach is suitable only for a sufficiently smooth function F(ρ). Next, the given source distribution can generate several or even an infinite number of radial congruences with their own shadow-zone boundaries. Finally, the obvious requirement that the total field COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
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ZAPUNIDI, POPOV Emitted plane wave
Diffusion region
E(ρ, 0)= F(ρ) Diffracted wave
Fig. 3.
be continuous and smooth on the shadow-zone boundaries is insufficient for determining a unique solution to the system of coupled parabolic equations because the boundary conditions are fewer than required. An attempt to introduce additional boundary conditions leads to a rapidly oscillating component appearing in the partial wave amplitudes Uj (ξ, η), which violates the original assumption of the method, while conditions for its elimination are difficult to formulate in a form suitable for numerical implementation. In this context, it seems reasonable to reconsider the foundations of the transverse diffusion method and somewhat modify the concept of emitted and diffracted waves. 2. HEURISTIC ARGUMENT: THE CONCEPT OF PHASE SYNCHRONISM The rigorous solution to the problem makes it possible to give a physically correct definition of the wave emitted by the face ϕ = 0 and to clarify the mechanism of originating a diffracted wave in the corner domain 0 < ϕ < Φ. Explicit formulas (7) are of little use for the analysis, since, even in the simplest example discussed above, when F(ρ) ≡ 1, the transform s(α) cannot be expressed in terms of elementary functions. However, in view of the above argument, the solution can be partitioned into an emitted and a diffracted wave by tracing the singularities of f(α) in the semi-infinite strip –π < Reα < 0, Imα < 0, since they coincide with the singularities of the desired function s(α). Consider an elementary harmonic component of the boundary condition: F(ρ) = exp(–ipρ). A solution to Helmholtz equation (1) that satisfies the boundary condition E0(ρ, 0) = F(ρ) is easy to construct: 2
E 0 ( ρ, ϕ ) = exp { – iρ [ p cos ϕ + sin (ϕ 1 – p ) ] }.
(16)
For |p| < 1, setting p = cosϕ0 , we obtain the plane wave E0(ρ, ϕ) = exp[–iρcos(ϕ – ϕ0)] coming from the direction ϕ = ϕ0 in the sector –π < ϕ < 0. Calculating the angular spectrum sin α f ( α ) = ----------------------------------------2 ( cos ϕ 0 – cos α )
(17)
and taking into account the first equation in system (5), we see that s(α) has a simple pole at the point α = ϕ0. A wave emitted by the half-plane ϕ = 0 into the corner domain 0 < ϕ < Φ corresponds to ϕ0 ranging in the interval (–π, 0). Since the integrand in (4) is odd, the original integration contour Γ+ is equivalent to C. According to Fig. 4, if ϕ0 < ϕ < π + ϕ0 , then the transition to the saddle path S is accompanied by the intersection of the pole of s(α + ϕ) lying at the point α = ϕ0 – ϕ on the real line. As a result, the asymptotic repCOMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
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Im α
C S
–π
ϕ0 –ϕ
Re α 0
Fig. 4.
resentation of the solution is supplemented by the term ⎧ exp [ – ip cos ( ϕ – ϕ 0 ) ], E 0 ( ρ, ϕ ) = ⎨ ⎩ 0, π + ϕ 0 < ϕ < Φ,
0 < ϕ < ϕ 0 + π,
(18)
which corresponds to a plane wave propagating in the corner sector 0 < ϕ < ϕ0 + π (see Fig. 5, where the fronts of the wave emitted by the wedge face and the fronts of the diffracted cylindrical wave are shown by solid lines, while the shadow-zone boundary of the emitted wave is shown by the dashed line). Note that, in contrast to E0(ρ, ϕ), the emitted wave thus constructed satisfies both boundary conditions (2), but it is not a solution to the Helmholtz equation in the whole corner domain 0 < ϕ < Φ, since it has a discontinuity –exp(iρ) on the line ϕ = ϕ0 + π. According to the concept of [1], this jump is the source of a diffracted wave that compensates for the discontinuity in the geometric-optics part of field (18) and propagates on both sides of the shadow-zone boundary due to transverse diffusion. In this example, the position of the shadow-zone boundary completely agrees with the intuitive representation of rays and fronts of the emitted wave. The result of analyzing the exact solution in the case of inhomogeneous emitted waves is less obvious. Consider two characteristic examples. 1. The boundary condition F(ρ) = exp(–qρ) corresponds to exponentially decaying in-phase sources continuously distributed along the face. This boundary value is associated with the inhomogeneous plane wave 2
E 0 ( ρ, ϕ ) = exp { ρ [ –q cos ϕ + i sin ( ϕ 1 + q ) ] },
(19)
which propagates normally to the emitting face ϕ = 0. At first glance, the shadow-zone boundary for this wave is the ray leaving the edge at an angle of ϕ = π/2. However, in the asymptotics of the rigorous solution, plane wave (19) occurs only in the range of observation angles q π 0 < ϕ < ϕ = --- – arcsin ------------------, 2 2 1+q COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
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Fig. 5.
which can easily be seen if we write the condition for intersecting the saddle contour S: α ≡ β + iγ = iγ – π – arcsin ( tanh γ ) with the complex pole of s(α + ϕ): α = ϕ0 – ϕ = –ϕ – π/2 – iArshq . The shadow-zone boundary deviates from the normal to the wavefront because the phase velocity of inhomogeneous wave (19) (i.e., the rate of change in the phase along the ray ϕ = π/2) differs from the velocity of propagation of locally plane waves forming the diffracted wave Ed ~ U(ρ, ϕ)exp(iρ). Condition (20) determines the line ϕ = ϕ on which the emitted wave is in phase with Ed(ρ, ϕ): 2
q E 0 ( ρ, ϕ ) = exp iρ – ρ ------------------ . 2 1+q
(21)
2. The boundary condition F(ρ) = exp(iqρ) with q > 1 corresponds to a slow harmonic wave that travels along the face ϕ = 0 and excites in its neighborhood the surface wave 2
E 0 ( ρ, ϕ ) = exp { ρ [ iq cos ϕ – sin ( ϕ q – 1 ) ] },
(22)
which propagates along the axis x = rcosϕ and exponentially decays in the normal direction y = rsinϕ. The rigorous solution to the problem yields the correct shadow-zone boundary for surface wave (22). Setting ϕ0 = –π – iArshq , we again obtain formula (17) for the angular spectrum f(α). This formula shows that s(α) has a pole ϕ0 in the semi-infinite strip –π < Reα < 0, Imα < 0. The condition of its intersection with S gives the domain where the emitted surface wave exists 2
q –1 0 < ϕ < ϕ = arcsin -----------------q
(23)
and its value at the shadow-zone boundary ϕ = ϕ 2
q –1 E 0 ( ρ, ϕ ) = exp iρ – ρ ------------------ . q
(24)
As in the previous example, the jump in the geometric-optics part of the field has the phase ρ and can be compensated for by a cylindrical boundary wave of the form U(ρ, ϕ)exp(iρ). Figure 6 displays the shadowzone boundary and the phase fronts of the emitted and diffracted waves for both examples. The fronts of the emitted and diffracted waves are depicted by solid lines, and the shadow-zone boundary (the line of phase COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
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synchronism) is shown by the dotted line. Figure 6a presents the case of exponentially decaying in-phase sources distributed along the wedge face. The case of a traveling surface wave is shown in Fig. 6b. Based on the analysis of these examples combined with the superposition principle, the following general result can be formulated for emitted waves in a wedge-shaped region. Concept. Each harmonic component of the boundary condition generates an emitted plane wave E 0 (ρ, ϕ) (homogeneous or inhomogeneous) that propagates in a sector contiguous to the emitting face. The boundary of the sector is uniquely determined by the condition of phase synchronism with a diffracted cylindrical wave that compensates for the jump in the emitted field originating on the shadow-zone boundary. The amplitude of the jump is constant or decays exponentially away from the wedge edge (when a surface wave or inhomogeneous plane wave is excited). According to the transverse diffusion principle, the asymptotics of the diffracted wave can be found by solving the Malyuzhinets parabolic equation (13) in polar coordinates (ρ, ϕ). If the boundary condition E(ρ, 0) = F(ρ) contains a finite number of harmonics, multiple shadow-zone boundaries arise with each partial wave having its own jump. For arbitrary F(ρ), these jumps can merge to form a smeared shadow-zone boundary. Mathematically, this situation is manifested in the nonzero right-hand side appearing in Eq. (13). 3. IMPROVED TRANSVERSE DIFFUSION PRINCIPLE For sources arbitrarily distributed along the wedge face, the solution to the emission problem is partitioned into emitted and diffracted waves in terms of Sommerfeld integrals. Consider a superposition of plane waves 1 –iρ cos ( α – ϕ ) E 0 ( ρ, ϕ ) = ----- e f ( α ) dα, πi
∫
(25)
C
where f(α) is the angular spectrum (6) of the boundary value F(ρ) and the contour C is the polygonal line (–i∞, 0, –π, −π + i∞) in the complex plane of α (if the singularities of f(α) approach the boundary of the regularity strips, the contour C bypasses them on the right and from above). It is easy to see that, in the sector 0 < ϕ < π, when integral (25) is absolutely convergent, E0(ρ, ϕ) satisfies Helmholtz equation (1), the boundary condition E0(ρ, 0) = F(ρ), and suppression condition (3). For ϕ > π, integral (25) diverges and its analytic continuation satisfies neither the suppression condition nor the boundary condition on the inactive face ϕ = Φ. To construct a physically correct representation of the solution, we use the heuristic argument proposed in the previous section. Specifically, each harmonic component of the emitted wave propagates in a sector bounded by the phase synchronism line with an originating diffracted wave. In Sommerfeld representation (25), this line is the curvilinear contour Sϕ α = iγ + ϕ – π – arcsin ( tanh γ )
(26)
in the complex plane of α on which the phase of the exponential is ρ (obviously, this is the above contour S shifted by ϕ). According to this concept, the emitted wave is defined as an integral of form (23) taken along part of the contour C lying to the right of Sϕ: 1 – iρ cos ( α – ϕ ) E 0 ( ρ, ϕ ) = ----- e f ( α ) dα. πi
∫
(27)
Cϕ
For simplicity, we restrict ourselves to the case when F(ρ) decreases sufficiently rapidly as ρ ∞ and the function f(α) has no singularities on C. According to (26), the truncated contour Cϕ is the union of the poly–
+
gonal line C ϕ : (–i∞, 0, ϕ – π) and the interval C ϕ : (–π + Arth ( sin ϕ ) , −π + i∞) when ϕ < π/2 (see Fig. 7a, 0
which shows Cϕ for 0 < ϕ < π/2) and the polygonal line C ϕ : ( – iArth ( sin ϕ ) , 0, ϕ – π) when π/2 < ϕ < π (see Fig. 7b, which shows Cϕ for π/2 < ϕ < π). For ϕ > π, the contour Cϕ degenerates and E 0 (ρ, ϕ) vanishes identically. The emitted wave (25) thus defined has the following properties: (a) E 0 (ρ, ϕ) satisfies the boundary condition on the emitting face ϕ = 0 and is found from the Fourier transform of F(ρ) by formula (6). (b) E 0 (ρ, ϕ) satisfies the suppression condition. COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
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ZAPUNIDI, POPOV (a)
(b)
Fig. 6.
(c) E 0 (ρ, ϕ) vanishes for ϕ > π and, hence, satisfies the boundary condition on the inactive face ϕ = Φ. Since the integration limits in (27) depend on ϕ, the function E 0 (ρ, ϕ) does not satisfy the wave equation. Substituting it into Eq. (1) gives a discrepancy that is the source of the diffracted wave. Calculating the righthand side of the equation 2
1 ∂ ∂E 1∂ E --- ------ ⎛ ρ ---------0⎞ + -----2 ----------2-0 + E 0 = R ( ρ, ϕ ), ρ ∂ρ ⎝ ∂ρ ⎠ ρ ∂ϕ
(28)
we easily obtain R ( ρ, ϕ ) = e
iρ
2
sin ϕ A ( ρ, ϕ ) + B ( ρ, ϕ ) exp ⎛ – ρ -------------⎞ , ⎝ cos ϕ ⎠
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where A(ρ, ϕ) and B(ρ, ϕ) are slowly varying functions. As expected, the phase of the right-hand side of Eq. (28) is ρ and the arising discrepancy can be compensated for by a cylindrical diffracted wave of the form E d ( ρ, ϕ ) = u ( ρ, ϕ ) exp ( iρ ).
(30)
Representing the desired wave field as the sum of the emitted and diffracted waves, i.e., E ( ρ, ϕ ) = E 0 ( ρ, ϕ ) + E d ( ρ, ϕ ),
(31)
we obtain an equation of form (28) for Ed(ρ, ϕ) with the right-hand side R(ρ, ϕ). The second term on the right-hand side of (28) decays exponentially as ρ ∞ and makes no contribution to the scattering pattern. Therefore, following the transverse diffusion principle (see [1]), we have an inhomogeneous parabolic equation for the slowly varying wave amplitude u(ρ, ϕ): 2
∂u u 1 ∂ u 2i ------ + i --- + -----2 ---------2 = – A ( ρ, ϕ ). ∂ρ ρ ρ ∂ϕ
(32)
This heuristic argument is confirmed by an analysis of rigorous solution (4). As was noted above, since the integrand is odd, integral (4) can be equivalently switched to the Sommerfeld contour C. Next, taking into account that the singularities of f(α) and s(α) partially coincide and deforming the integration contours in domains where the integrand is regular, we infer from (4) and (27) that the diffraction field Ed = E – E 0 satisfies the integral representation ⎧⎛ ⎧ ⎞ 1 ⎪⎜ ⎪ ------⎟ e –iρ cos ( α – ϕ ) [ s ( α ) – 2 f ( α ) – s ( 2ϕ – α ) ] dα + ⎪ 2πi ⎨ ⎜ ⎟ ⎪⎝ – + ⎪ Sϕ Sϕ ⎠ ⎩ ⎪ ⎪ ⎪ ⎫ ⎪ ⎪ – iρ cos ( α – ϕ ) ⎪+ e [ s ( α ) – s ( 2ϕ – α ) ] dα ⎬, 0 < ϕ < π/2, ⎪ 0 ⎪ ⎪ Sϕ ⎭ ⎪ ⎪ ⎧⎛ ⎪ ⎞ E d ( ρ, ϕ ) = ⎨ 1 ⎪ ⎜ ⎟ e –iρ cos ( α – ϕ ) [ s ( α ) – s ( 2ϕ – α ) ] dα ------+ ⎨ ⎪ 2πi ⎜ ⎟ ⎪⎝ – +⎠ ⎪ S S ϕ ⎩ ϕ ⎪ ⎪ ⎪ ⎫ ⎪ ⎪ – iρ cos ( α – ϕ ) ⎪+ e [ s ( α ) – 2 f ( α ) – s ( 2ϕ – α ) ] dα ⎬, π/2 < ϕ < π, ⎪ 0 ⎪ ⎪ Sϕ ⎭ ⎪ ⎪ ⎪ 1 – iρ cos ( α – ϕ ) - e [ s ( α ) – s ( 2ϕ – α ) ] dα, π < ϕ < Φ, ⎪ ------⎩ 2πi S
∫ ∫
∫
∫ ∫
(33)
∫
∫
ϕ
–
0
+
where S ϕ , S ϕ , and S ϕ are the saddle path segments cut off by C (see Fig. 7). The asymptotics of integrals (33) as ρ ∞ are determined by the neighborhood of the saddle point α = – + ϕ – π and by the endpoints of the contours S ϕ : α = –ib(ϕ) and S ϕ : α = –π + ib(ϕ), where b(ϕ) = Arth ( sin ϕ ) . For ϕ ≠ 0, it follows from (26) that the π-contribution of the endpoints is exponentially small: 2
sin ϕ –1 O ρ exp ⎛ – ρ -------------⎞ . In the neighborhood of the saddle point, where α = ϕ – π + (i – 1)γ with |γ| 1, the ⎝ cos ϕ ⎠ COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
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ZAPUNIDI, POPOV Im α
(a) Sϕ
Im α
(b)
+
Cϕ
+
Sϕ
0
–π
Sϕ
0
Re α
Re α
0
–π
0 Cϕ
ϕ–π
ϕ–π
– Cϕ –
Sϕ
Fig. 7.
exponential factor has the form 2
e
– iρ cos ( α – ϕ )
≈e
(α + π – ϕ) iρ 1 – -----------------------------2
≈e
2
ρ(i – γ )
(34)
and integrals (33) are asymptotically reduced to Fresnel integrals. The transverse diffusion method calculates the diffracted wave Ed(ρ, ϕ) without explicitly determining ρ 2 s(α). A prompting argument is that the exponential exp –i --- ( α + π – ϕ ) is an exact solution to Malyuzhi2 nets equation (11). We replace the Sommerfeld kernel in integrals (33) by approximation (34) and substitute u = Ed exp(–iρ) into the left-hand side of Eq. (13). The differentiation of the variable integration limit α = ϕ – π yields the contribution of the emitted wave f(α) and, up to exponentially small terms, we obtain inhomogeneous parabolic equation (32) with the right-hand side ⎧ f '(ϕ – π) 0 < ϕ < π, ⎪ ----------------------, 2 A ( ρ, ϕ ) = ⎨ iπρ ⎪ ⎩ 0, π < ϕ < Φ. Equation (32) with the boundary conditions u(ρ, 0) = u(ρ, Φ) = 0 and the boundedness condition as ρ has a unique solution and is easily solved by finite-difference methods (see [2]).
0
This approach considerably reduces the amount of computations as compared with the numerical implementation of solution (4)–(7). Specifically, instead of four-fold integration in (8) for each “observation point” (ρ1, ϕ1), we only need to perform the double integral transform F(ρ) ⇒ f(α) ⇒ E 0 (ρ1, ϕ1) and solve Eq. (34) numerically, which gives an amplitude distribution of the diffracted wave in the entire sector 0 < ϕ < Φ, 0 < ρ < ρ1. In many model problems, f(α) can be found analytically, which further reduces the amount of computations. When the distribution of sources along the wedge face is given numerically, the emitted wave E 0 (ρ, ϕ) can be expressed using formulas (6) and (27) directly in terms of the boundary condition F(ρ) as the integral transform ∞
E 0 ( ρ, ϕ ) =
∫ F ( ξ )K ( ξ, ρ, ϕ ) dξ. 0
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PHYSICAL PATTERN OF WAVE EMISSION
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1.8 10 1.6 1.4
5
1.2 1.0
0
0.8 –5
0.6 0.4
–10 0.2 –10
–5
0
5
10
Fig. 8.
10
5
0
–5
–10
–10
–5
0
5
10
Fig. 9.
Here, the kernel 1 i [ ξ cos α – ρ cos ( α – ϕ ) ] sin α dα K ( ξ, ρ, ϕ ) = ------ e 2π
∫
Cϕ
is expressed in terms of an inexact Hankel function [7], i.e., a Sommerfeld integral with the truncated contour Cϕ. The constructed asymptotic solution to the emission problem is not yet completely consistent with the heuristic concept developed in Section 2, since the asymptotic representation of emitted wave (27) involves a contribution of the form A0(ϕ)exp(iρ)/ ρ from the saddle point α = ϕ – π, i.e., the addition of the diffracted wave. When the spectrum f(α) is given analytically, an absolutely correct partition of the solution into emitted and diffracted waves can easily be indicated. For this purpose, the integration path Cϕ in (27) COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
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1526
ZAPUNIDI, POPOV
10
5
0
–5
–10
–10
–5
0
5
10
Fig. 10.
is deformed until it coincides with the steepest descent contour Sϕ without intersecting the singularities of the integrand, and the emitted wave is defined as the arising integral over a closed contour encircling (in the positive direction) all the singularities of f(α) lying to the right of Sϕ. If the singularities of f(α) comprise a finite set of poles, the emitted wave is the sum of plane waves of form (18) with their own shadow-zone boundaries, i.e., the lines of phase synchronism. In the case of branch points, there appears an integral around the cut portion cut off by Sϕ. In the former case, the emitted wave satisfies the Helmholtz equation everywhere except for the partial shadow-zone boundaries specifying the discontinuities of the diffracted wave. In the latter case, we obtain a distributed right-hand side of form (29) with A(ρ, ϕ) ≡ 0, since the contribution of the saddle point is included in the diffracted wave. This situation can be illustrated by a numerical example. Suppose that a wave field is excited by in-phase sources distributed along the active wedge face and having the amplitude F(ρ) = ρ . The angular spectrum is π sin α iπ/4 -e f ( α ) = – ------- --------------------4 ( cos α ) 3/2 and has a branch point at α = –π/2. To obtain a single-valued branch of the solution, we perform the cut (−π/2, –π/2 –i∞). According to the above technique, the “true” or “pure” emitted wave is determined as a contour integral around of the cut (–π/2 – iArth ( cos ϕ ) , –π/2): 1 E˜ 0 ( ρ, ϕ ) = ----πi
°∫
e
– iρ cos ( α – ϕ )
f ( α ) dα.
(35)
π – --- – iArth ( cos ϕ ) 2
By a change of variables and integration by parts, formula (35) is reduced to the converging integral ρ E˜ 0 ( ρ, ϕ ) ∼ ------π
cot ϕ
∫ 0
2
e
ρ ( i sin ( ϕ 1 + x ) – x cos ϕ )
dx x cos ϕ – i sin ⎛ ϕ ------------------⎞ ------- , ⎝ 2⎠ x 1+x
(36)
which is easy to evaluate by numerical methods. In this case, the emitted wave does not contain any cylindrical boundary wave arising due to transverse diffusion through the smeared shadow-zone boundary ϕ = π/2. The latter is determined by the numerical integration of Eq. (28) with the right-hand side given by (29). Its absolute value is displayed in Fig. 8 for the width of the wedge-shaped region Φ = 230°. The real part of COMPUTATIONAL MATHEMATICS AND MATHEMATICAL PHYSICS
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PHYSICAL PATTERN OF WAVE EMISSION
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Re E (ρ, ϕ) 2 ρ = 7.8
1 0 –1 –2 –3 0
50
100
150
200
250
300
350 ϕ°
Fig. 11.
the pure emitted wave (36) is shown by shades of grey in Fig. 9. The real part of the total wave field calculated by the modified transverse diffusion method is presented in Fig. 10, which illustrates the spread of the emitted wave into the shadow zone. A quantitative estimate of the transverse diffusion effect is given in Fig. 11, which compares the total wave field (solid curve) with the emitted wave (dashed curve) on a circle of dimensionless radius ρ ≡ kr = 7.8. CONCLUSIONS An analysis of the rigorous solution to the problem of waves emitted by a vibrating wedge face has shown that the wave field created in the exterior of the wedge by sources arbitrarily distributed along its face can be described by an improved transverse diffusion method. The physical concept of phase synchronism makes it possible to give an adequate definition of an emitted wave and to indicate the region where diffracted waves originate (multiple or smeared shadow-zone boundaries of plane partial waves). The generalization proposed for the parabolic equation method gives an effective technique for computing wave diffraction in a wedge-shaped region. It can also be used to construct asymptotics of wave fields in more complicated problems. REFERENCES 1. G. D. Malyuzhinets, “Development in Our Concepts of Diffraction Phenomena,” Usp. Fiz. Nauk 69, 321–334 (1959). 2. A. V. Popov, “The Solution to Diffraction Theory Parabolic Equation Using the Finite Difference Method,” Zh. Vychisl. Mat. Mat. Fiz. 8, 1140–1144 (1968). 3. G. D. Malyuzhinets and L. A. Vainshtein, “Transverse Diffusion in Diffraction by Impedance Cylinder of Large Radius, Part 1: Parabolic Equation in Ray Coordinates,” Radiotekh. Elektron. 6 (8), 142–153 (1961). 4. G. D. Malyuzhinets, “Inversion Formula for Sommerfeld Integral,” Dokl. Akad. Nauk SSSR 118, 1099–1102 (1958). 5. V. M. Babich, M. A. Lyalinov, and V. E. Grikurov, Sommerfeld–Malyuzhinets Method in Diffraction Problems (VVM, St. Petersburg, 2004) [in Russian]. 6. G. D. Malyuzhinets, “The Radiation of Sound by the Vibrating Boundaries of an Arbitrary Wedge 1,” Akust. Zh. 1 (2), 144–164 (1955). 7. M. M. Agrest and M. Z. Maksimov, Theory of Incomplete Cylindrical Functions and Their Applications (Atomizdat, Moscow, 1965; Springer-Verlag, Berlin, 1971).
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