Proceedings of the International Conference DAYS on DIFFRACTION 2016, pp. 431–434
c 2016 IEEE 978-1-5090-5800-6/16/$31.00 ⃝
Spacetime triangle diagram technique for sectoral horn waveguides Andrei B. Utkin INOV – Inesc Inova¸c˜ ao and CeFEMA, IST, Technical University of Lisbon, Portugal; e-mail:
[email protected] Extension of the STTD (spacetime triangle diagram) technique — a powerful method of solving problems related to generation and propagation of acoustic and electromagnetic signals (finite pulses) — to the case of wave motion within a sectorial waveguide is discussed. The paper shows how a general modal solution, which describes the wave-pulse generation by a source arbitrarily distributed within a sectoral horn, can be constructed for the Dirichlet or Neumann boundary conditions imposed on the horn walls. Providing an exact solution, readily implementable for particular sources using general software packages for scientific computing, the obtained integral formulas represent a powerful tool for analysis of pulsed wave propagation in the sectorial horn waveguides.
• the model geometry does not possess angular symmetry, so the coordinate φ must be explicitly separated for any source configuration. The modal expansions must match boundary conditions imposed on the lateral and top/bottom walls of the horn. Instantiation of these conditions depends on the physical meaning of the wavefunction, which may be an acoustic wave or an electromagnetic parameter resulted from some scalarization procedure. 2
Statement of the problem
Geometry of the problems of wave motion in question is shown in Fig. 1. The most general treatment of wave motion imProviding inherently causal solutions directly in plies consideration of sources and, thus, leads to the the spacetime domain, the STTD (spacetime triinhomogeneous wave equation angle diagram) technique [1] has been successfully applied to solve a wide range of electromagnetic □ψ(τ, r) = j(τ, r) (1) and scalar problems of wave motion. At the moment, the STTD methodology is developed for nu- Here □ is the d’Alembert (wave) operator [8] for the merous models resulting in the wave and wave- positive spacetime metric signature [+ − − −], ψ like hyperbolic partial differential equations writ- the wavefunction, j the source, r the coordinate of ten in the Cartesian, cylindrical, and spherical co- the observation point, and τ the observation time ordinate systems [2–5]. In 2012, specific STTDs expressed in the units of length using the wavefront were constructed for the elliptic cylinder coordinate system [6], enabling the general solution to the causal inhomogeneous problem of wave propagation in an elliptical waveguide to be found as an expansion in terms of transient Mathieu modes. Based on preliminary theoretical considerations made by Borisov [2], the present study describes extension of the STTD technique to another practically important case (see, e.g., [7]) of the sectoral horn waveguide constrained top and bottom by two parallel planes. The adopted methodology has the following specificity: • the consideration is carried out in the cylindrical (rather than spherical) coordinates (ρ, φ, z), Figure 1: Schematic representation of the sec• the unseparated spatial variable is untraditional torial horn waveguide. (coordinate ρ rather than z), 1
Introduction
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velocity. The wave operator can be written in the [6]. Remarkably, all of them obey the relations cylindrical coordinate system ρ, z, τ as follows: ( mπ )2 ∂φ2 Φm (φ) = − Φm (φ) , 2θ □ = ∂τ2 − ∇2 = ∂τ2 − ρ−1 ∂ρ (ρ∂ρ ) − ρ−2 ∂φ2 − ∂z2 . (2) (7) ( kπ )2 Zk (z) . ∂z2 Zk (z) = − 2l Thus, in the general case of wave generation in the In this way, the solution of the inhomogeneous inisectorial horn waveguide, one has the following initial boundary problem reduces to an expansion in tial (with respect to τ ) and boundary (with respect terms of transient modes whose amplitudes can be to φ, z) value problem found by substitution of (5) into (3) [ 2 ] [ ] ∂τ − ρ−1 ∂ρ (ρ∂ρ ) − ρ−2 ∂φ2 − ∂z2 ψ(τ, ρ, φ, z) 1 1 ( mπ )2 ( kπ )2 2 + Ψmk (τ, ρ) ∂τ − ∂ρ (ρ∂ρ ) + 2 = j(τ, ρ, φ, z), (3) ρ ρ 2θ 2l ψ(τ, ρ, φ, z) ≡ 0 for τ < 0, = J (τ, ρ) , (8) mk
Ψmk (τ, ρ) ≡ 0 for τ < 0. in which the boundary conditions posed at the planes φ = ±θ and z = ±l may be of the Dirichlet To make the problem suitable for application of the (ψ = 0) or Neumann (∂n ψ = 0) type. STTD technique, we need to represent the hyperNote 1. Concrete physical meaning of the wave- bolic PDE in the canonical form, which can be done function ψ depends on the physical problem under by the substitution Ψmk = ρ1/2 ψmk . This leads to ( ) ( )] consideration. In the case of acoustics applications, [ 1 ( mπ )2 1 kπ 2 ψ directly represents a scalar acoustic wave, while ∂τ2 − ∂ρ2 + 2 − + ψmk (τ, ρ) ρ 2θ 4 2l in the electromagnetics, the scalar function ψ represents an electromagnetic field component (like Ez ), = jmk (τ, ρ) := ρ−1/2 Jmk (τ, ρ), (9) an electromagnetic field component times indepenψmk (τ, ρ) ≡ 0 for τ < 0. dent variable (like ρEφ , ρBρ ), or a result of more complicated procedure of scalarization of Maxwell’s Here, finally, we have a problem for unknown amequations (scalar potential), see [2] for more details. plitudes ψmk (τ, ρ) of the general modal solution to Note 2. Particular choice of the boundary con- the problem of wave propagation along a sectorial ditions depends on both the physical nature of the horn waveguide ∞ ∑ ∞ wavefunction and the material properties of the cor∑ ρ1/2 ψmk (τ, ρ)Φm (φ)Zk (z). responding waveguide planes. A classic example ψ(τ, ρ, φ, z) = m=1 k=1 is the homogeneous Dirichlet conditions on all the (10) planes, ψ(τ, ρ, ±θ, z) = 0,
ψ(τ, ρ, φ, ±l) = 0,
(4) 3
which admits separation of φ and z via the expansion ( ) ) ∞ ∑ ∞ ( ∑ j Jmk (τ, ρ, φ, z) = (τ, ρ) Φm (φ) Zk (z) ψ Ψmk m=1 k=1
(5) where ) ) ( ( φ+θ z+l Φm (φ) = sin mπ , Zk (z) = sin kπ . 2θ 2l (6) For other boundary conditions one has similar representations for Φm and Zk in terms of trigonometric functions. They can be found using the source extrapolation technique akin to that described in
Construction and analysis of the general solution
The next stage of application of the STTD technique implies constructing the unique solution to the reduced problem (9) using the Riemann–Volterra integral formula. Remarkably, Riemann function for the PDE in question is not known. However, we may notice that ( mπ )2 1 ( mπ 1 ) ( mπ 1 ) − = − + = ν(ν + 1) , 2θ 4 2θ 2 2θ 2 (11) 1 where ν := mπ − . Thus, the PDE of problem (9) 2θ 2 can be rewritten in the form ] [ ν(ν + 1) 2 2 2 ∂τ − ∂ρ + + s ψmk (τ, ρ) = jmk (τ, ρ) , ρ2 (12)
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Figure 2: Examples of STTDs for two basic cases: (a) no explicit source limitation imposed and (b) the source jmk is constrained in time (pulse of duration T ) and space (0 < ρ < ρ0 ) — here the actual domain of integration is defined by Ω∗ = Ω ∩ Rect(0, 0, ρ0 , T ) = Ω ∩ ([0, ρ0 ] × [0, T ]). where s := kπ 2l . In the absence of the third term in the square brackets, (12) turns into the 1D Klein– Gordon equation with known Riemann function (see, e.g., [9–11]) ( √ ) 2 2 R1 (τ, ρ; τ ′ , ρ′ ) = J0 s (τ − τ ′ ) − (ρ − ρ′ )
The corresponding 2D domain of integration Ω and spacetime triangle diagrams for some practical particular cases are shown in Fig. 2. 4
Conclusive remarks
Despite certain degree of complexity, the obtained general modal solution represents a powerful tool for analysis of pulsed wave propagation in the sectorial horn waveguides. It is an exact solution, which can be readily instantiated for particular sources using general software packages for scientific computing. As is the case for all solutions obtained with ( 2 ) 2 2 ρ + ρ′ − (τ − τ ′ ) ′ ′ R2 (τ, ρ; τ , ρ ) = Pν (14) the help of the STTD technique, for finite sources, 2ρρ′ the integration is carried out within a limited region, assuring the boundedness of the wavefunction (Pν being the Legendre function). Thus, we can support (i.e., correct representation of the signal or construct the desired Riemann function using Olev- pulsed wave in question). skii’s theorem [13] as follows (13) (J0 being the Bessel function of the first kind of order zero) and in the absence of the forth term — into the Euler–Poisson–Darboux equation, whose Riemann function is [12]
References 2 2) ρ2 + ρ′ − (τ − τ ′ ) R(τ, ρ; τ , ρ ) = Pν 2ρρ′ [1] Wikipedia contributors, 2016, Spacetime tri( 2 ) ∫ ρ−ρ′ angle diagram technique, Wikipedia, The Free 2 ′2 ρ +ρ −ζ + dζPν Encyclopedia, https://en.wikipedia.org/ ′ 2ρρ τ −τ ′ w/index.php?title=Spacetime_triangle_ ( √ ) 2 diagram_technique. × ∂ζ J0 s (τ − τ ′ ) − ζ 2 . (15) [2] Borisov, V. V., 1991, Nonsteady-State Fields in Waveguides, Leningrad State University Press, Now one can apply the Riemann–Volterra method, Leningrad. which — in view of zero initial conditions — readily yields [5] [3] Borisov, V. V., 1996, Electromagnetic Fields of ∫∫ Transient Currents, Leningrad State Univer1 dτ ′ dρ′ R(τ, ρ; τ ′ , ρ′ )jmk (τ ′ , ρ′ ) ψmk (τ, ρ) = sity Press, Leningrad. 2 Ω ′ ∫ ∫ [4] Utkin, A. B., 2013, Localized waves emanated ) ( ) 1 ρ+τ ′ τ −|ρ−ρ | ′ ( by pulsed sources: the Riemann–Volterra ap= dρ dτ R τ, ρ; τ ′ , ρ′ jmk τ ′ , ρ′ . 2 ρ−τ 0 proach, In: Non-Diffracting Waves, Wiley(16) VCH, Weinheim, pp. 287–306. ′
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[5] Utkin, A. B., 2011, Electromagnetic waves [10] Utkin, A. B., 2012, Droplet-shaped waves: casual finite-support analogs of X-shaped waves, generated by line current pulses, In: Wave J. Opt. Soc. Am. A, Vol. 29(4), pp. 457–462. Propagation, InTech: Vienna, pp. 483–508. [6] Utkin, A. B., 2012, The Riemann–Volterra [11] Utkin, A. B., 2013, Droplet-shape wave protime-domain technique for waveguides: a case duced by line macroscopic current pulse of fistudy for elliptic geometry, Wave Motion, nite length, Proc. Int. Conf. Days on DiffracVol. 49(2), pp. 347–363. tion 2013, pp. 145–150. [7] Pandey, S., Deshpande, R. A., Ranade, S., [12] Borisov, V. V., Manankova, A. V., Utkin, 2012, Designing of CSIW horn antenna, IOSR A. B., 1996, Spherical harmonic representaJournal of Electronics and Communication tion of the electromagnetic field produced by Engineering, Vol. 7(6), pp. 12–16. a moving pulse of current density, J. Phys. A: Math. Gen., Vol. 29(15), pp. 4493–4514. [8] Wikipedia contributors, 2016, D’Alembert operator, Wikipedia, The Free Encyclopedia, [13] Olevskii, M. N., 1952, On the Riemann funchttps://en.wikipedia.org/w/index.php? tion for the differential equation ∂ 2 u/∂x2 − title=D%27Alembert_operator. ∂ 2 u/∂τ 2 + (p1 (x) + p2 (τ )) u = 0, Dokl. Akad. Nauk SSSR, Vol. 87(3), pp. 337–340. [9] Borisov, V. V., Utkin, A. B., 1995, The transient electromagnetic field produced by a moving pulse of line current, J. Phys. D: Appl. Phys., Vol. 28(4), pp. 614–622.
