IEEE TRhTSACTIONS ON AXTEXXAS AND PROPAGATION VOL. AP-14, so. 3 MAY, 1Ybb
V. CONCLUSION REFERENCES The characteristics of the waves guided along a plane [I] P. S. Epstein,“On the possibility of electromagneticsurface waves,” Proc. Nat’l dcad. Sciences, vol. 40, pp. 1158-1165, De- interface which separates a semi-infinite region of free cember 1954. [2] T. Tamir and A. A. Oliner, “The spectrum of electromagnetic space from that of a magnetoionic medium are investi- waves guided by a plasma layer,” Proc. IEEE, vol. 51, pp. 317- gated for the case in which the static magnetic field is 332, February 1963. orientedperpendicular to the plane interface. It is [3] &I. A. Gintsburg, “Surface waves on the boundary of a plasma in magnetica field,” Rasprost. Radwvoln i Ionosf., Trudy found that surface waves exist only when w,
Numerical Solution of Initial Boundary Value Problems Involving Maxwell’s Equations in Isotropic Media
KANE S. YEE
Abstracf-Maxwell’s equations are replacedby a set of finite obstacle is moderately large compared to that of an in- difEerence equations. It is shown that if one chooses the field points coming n-ave. appropriately, the set of finite difference equations is applicable for A set of finite difference equations for the system of aboundary condition involving perfectly conducting surfaces. An example is given of the scattering of an electromagnetic pulse by a partial differential equations will be introduced in the perfectly conducting cylinder. early partof this paper. We shall then show that with an appropriate choice of the points at which the various INTRODUCTION field components are to be evaluated, the set of finite OLUTIONS to thetime-dependent Maxwell’s equa- difference equations can be solved and the solution will tionsin general form are unknown except for satisfy the boundary condition. The latter part of this sa few special cases.The difficulty is due mainly to paper will specialize in two-dimensional problems, and the imposition of theboundary conditions. LT,7e shall an example illustrating scattering of an incoming pulse show in this paper how to obtain the solution numer- by a perfectly conducting square will be presented. ically when the boundary condition is that appropriate for a perfect conductor. In theory, this numerical attack AIAXTT-ELL’S EQVATIONAND THE EQUIVALENT SET can be employed for the most general case. However, OF FINITEDIFFERENCE EQUATIONS because of the limited memory capacity of present dal: llaxwell’s equations in an isotropic medium [l]are:’ computers, numerical solutions to a scattering problem aB for which the ratio of the characteristic linear dimen- -+VXE=O, sion of the obstacle tothe m-avelength is largestill at seem to be impractical. We shall show by an example that in the case of two dimensions, numerical solutions are practical even when the characteristic lengthof the
Manuscript receivedAugust 24, 1965; revised January 28, 1966. D = €E, This work was performed under the auspices of the U. S. Atomic Energy Commission. The author is with the Lawrence Radiation Lab., University of California, Livermore, Calif. 1 In MKS system of units. 801 YEE: SOLUTION OF INITIAL BOUNDARY VALUE PROBLEMS 303 where J, p, and E are assumed to be given functions of space and time. In a rectangular coordinate system, (la) and (lb) are equivalent to the following system of scalar equations: dB, aE,aE, --=-.--. (24 dt ay dz dB, dE,dE, ___=__-- (2b) at az ax
dB, dE,aE, (i,j,kl EY - (2c) at ay 8% aD, dH,dH, / _- Jz, (24 at ay az / X- aD, - dHz dH, Ju, (24 Fig. 1. Positions of various field components. TheE-components at dz ax are in the middle of the edges and the H-components are in the center of the faces. dD, dH,dH, Jz, (20 at ax ay BOUNDARY CONDITIONS where we have taken A = (Az,A,, A,).lye denotea grid point of the space as The boundary condition appropriate for a perfectly conducting surface is that the tangential componentsof (i,j, k) = (iAx,jay, KAz) (3) the electric field vanish.This condition also implies and for any function of space and time we put that the normal component of the magnetic field van- ishes on the surface. The conducting surface will there- F(iAx,jay, kAz, %At) = Fn(i,j, k). (4) forebe approximated by a collection of surfaces of A set of finite difference equations for (2a)-(2f) that cubes, the sides of which are parallel to the coordinate will befound convenient for perfectly conducting axes. Plane surfaces perpendicular to the x-axis will be boundary condition is as follows. chosen so asto contain points where E, and E, are For (2a) we have defined. Similarly, plane surfaces perpendicular to the other axes are chosen. BZn+l/'(i,j+ 1.2, K + 4) - Bzn-"2(i,j + 3, K + 3) At GRID SIZEAND STABILITYCRITERION The space grid size must be such that over oneincre- E,"(i,j 3, k 1) - Eyn(i,j 4, K) - + + + mentthe electromagnetic field doesnot change sig- Az nificantly. This means that, to havemeaningful results, the linear dimension of the grid must be only a fraction - E*"(i,j + 1, K + 3) - EZR(i,j,k + 3) * (5) of the wavelength. We shallchoose Ax=Ay =Az. For AY computational stability, itis necessary to satisfy a rela- The finitedifference equationscorresponding to (2b) tion between the space increment and time increment and (k),respectively, can be similarly constructed. At. When E and p are variables, a rigorous stability cri- For (2d) we have terion is difficult to obtain. For constant E and p, com- D,"i + Q,j,k) - D,n-'(i + +,j,K) putational stability requires that At AX)^ + (AY)~+ (AZ)~> cAf = $/;At, (7) - Bz"-l"(i + 4,j + 3, k) - a,n-yi + +,j- 3, K) AY where c is the velocity of light. If cmsI is the maximum ayn-l/S(i+ 1 K + L) - H,n-l/2(i f 1 ' K - 4) light velocity in the region concerned, we must choose - 2,37 2 293, Az d(4~)~+ (AY)~ + (Az)~> cmaxAt. (8) Jzn-1/*(i j, K). (6) + + 3, This requirement putsa restriction on At for the chosen The equations corresponding to (2e) and (2f), respec- Ax, Ay, and Az. tively, can be similarly constructed. The grid points for the E-field and the H-field are MAXWELL'S EQCATIOXSIN Two DIMENSIOXS chosen so as to approximate the condition to be dis- To illustratethe method, me consider a scattering cussed below as accurately aspossible. The various grid problem in two dimensions. We shall assume that the positions are shown in Fig. 1. field components do not depend on thez coordinate of a so4 IEEE TRsNSACTIOh-S ON AhTEhTnTAS AND PROPAGATION BUY
point. Furthermore,we take E and p to be constants and J=O. The only source of our problem is then an "in- cident"wave. This incident wave will be "scattered" after it encounters the obstacle. The obstaclewill be of a few "wavelengths"in its linear dimension. Further simplificationcan be obtained if we observe the fact that in cylindrical coordinates we can decompose any electromagnetic field into"transverse electric" and "transverse magnetic" fields if E and p are constants. The two modes of electromagnetic waves are character- ized by 1) Transverse electric wave (TE)
H, = H, = 0, E, = 0, 1 AT aH, aE, aE, - - - [EZn(i,j+ 1) - ELn(i,j)] (14b) -"a,=x-z' z AY aH, aE, aa, a& E-=-) -E-> (9) 1 AT ay at ax dt - - [aqi+ 1,j) - EP(i,j)]. (14c) + Z Ax and 2) Transverse magnetic wave (TM) KUMERICALCOMPUTATIOMS FOR TXI WAVES For further numerical discussion we shall limit our- E, = & = 0, Hz = 0, selves to the TM waves. In this case we use the finite aE, aH, aH, difference equations (14a)-(14c). The values forEzo(i,j), E-=--- H,1/2(i+$,j), and HZ1l2(i,j-$) are obtained from the dt ax ay incident wave.2 Subsequent values are evaluated from ag, dE* aH, dE, the finite difference equations (14a)-(14c). The bound- p-=--, pdt=d.t-. (10) at aY ary condition is approximated by putting the boundary value of EZn(i,j) equal to zero for any n. Let C be a perfectly conducting boundary curve. ?Ye To be specific, we shall consider the diffraction of an approximate it bya polygon whose sides are parallel to incident T31 wave by a perfectly conducting square. the coordinate axes. If the griddimensions are small The dimensions of the obstacle, aswell as the profile of compared to the wavelength,we expect the approxima- the incident wave, are shown in Fig. 2. tion to yield meaningful results. Letting -
T = ct = $/;t and Ez=O , Hy=O
Ez=O
j =33 E,=O ,ny=o
/ j=l /
j i 1 AT /////////// //////// + - - EZn(i+ $,j + 1) - E,"(i + %,j)] (13a) H,=O AY i= 17 i=49 i=81 Fig. 2. Equivalent problem for scattering of a Thl wave. AT + z- [H~E-~I/~((;+ 4,j + 4) - ~~n+W(i+ $,j - +)] (13b) 2 M'e choose t such that when t=O the incident wave has not en- AY countered the obstacle. LJUV
1.0 A n=5 0.5 1.0 I\ 0.5 n;5 0 I 0 I I I I I I -0'5 If I\ n~15 17.35 - 1.0 - 0.5 0 I I I I I I I\ -0.5 \r n=25 -1.0 0 I I I I I - I I 1.0 ' O7-0.5 n.35 \/- I\ -1.0 n-65 0 I I I I lyrh I 0.5 I\ -0.5 - \/ n.45 -1.0 . of! - 0.5 I.o n=55 \ -1.0 4 n=95 0 I I I I I 0.5 -0.5 \/ I n.65 -1.0 0.5 n=7q I , I I I 0
I I I I I I I 1.0 IO 20 30 40 50 €43 70 80 0.5 A n=85 0 1' Fig. 3. Results of the calculation of E, by means of (14a)-(14c) in I I I I I I I the absence of the obstacle. The ordinate is in volts/meter and 1020394050607080 the abscissa is the number of horizontalincrements. n is the number of time cycles. Fig. 5. E, of the TiLI wave for various time cycles. j=SO.
I .o n 0.5 n'5 0 I I I /I \ I I I I I I I I 0 I -0.5 - I. 0 J O I I I I I n=& - 0.5 \f
I 0 I I I I I -0.5 n=35 v
0 I I I I I n.45 -0.5 \r
- 0.5O pF----%J I I I I 0 n=& - - 0.5 *w 0 0.5 - -0.5 .0.5 n=75 0 At I - 0.5 n.a I I - 0.05 0 -I I I > 0.05 ,,=e5 0.5 I 0 --I -n=95 ' I L I -0.05 0 0.05 n=95 I I I I I 0-- ' I I I I IO 20 30 40 50 60 70 80 -0.05 - I I IO 20 30 40 50 60 70 80 Fig. 6. E, of the Tbl wave for various time cycles. j= 65. Fig. 4. E, of the T&I wave in the presence of the obstacle for various time cycles when j=30. 306 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION MAY
Let the incident wave be plane, with itsprofile being culation was not carried far enough to show this effect. a half sinewave. The width of theincident wave is Figure 5 shows the value of E, for the TAI wave as a taken to be CY units and the square has sides of length function of the horizontal grid coordinate i for j=50. CY units. Since the equations are linear, we can take This line (j= 50) meets the obstacle, and hence we ex- E,= 1 unit. The incident wave will haveonly an E, pect a reflected wave going to the right. These expecta- component and an H, component. We choose tions are borne out in Fig. 5. After the reflected wave from the object meets the right boundary (see Fig. 21, AX = Ay = CY/% (15a) the wave is reflected again. This effect is shown for the and time cycles 75, 85, and 95. AT = cAt = +AX = a/16. (15b) Figure6 is for j=65. This lineforms part of the boundary of the obstacle. Becauseof the required bound-. ,4 finite difference scheme over the whole x-y plane ary condition, E, is zero on the boundary point. To the is impractical; we therefore have to limit the extent of right of the obstacle there isa “partially” reflected wave our calculation region. We assume thatat time t = 0, the of about half the amplitude of a fully reflected wave. To left traveling plane wave is “near” the obstacle. For a the left of the obstacle there is a “transmitted” wave restricted period of time, we can therefore replace the after 85 time cycles. original problems by the equivalent problem shoxn in All these graphs were obtained by means of linear Fig. 2. interpolation between the grid points. They have been The input data are takenfrom the incident xavewith redrawn for reproduction. [(x - 5: + Gt)T &(x, y, t) = sin COhiPARISOW OF THE COMPUTED RESULTSFf’ITH THE 1 KI‘JOWNRESULTS ON DIFFRACTIONOF 0 5 x - 50a + ct 5 Sa (16a) PULSESBY A WEDGE 1 There exist no exact results for the particular exam- lux,y, 0 = -z -%(X, y, 0. (16b) ple we considered here. However, in the case when the obstacleis a wedge,Keller and Blank [2] and Fried- From the differential equation satisfied by Ez n-e con- lander [3] have solved the diffraction problem in closed clude that the results for the equivalent problem (see forms. In addition, Keller [4] has also proposed a meth- Fig. 2) should approximate those of the original prob- od to treat diffraction by polygonal cylinders. To carry lems, provided out the methodproposed by Keller [4], one would have to use some sort of finite difference scheme. The present 0 5 nAr 5 64Ar, difference scheme seems to be simpler to apply in prac- because the artificial boundary conditionswill not affect tice. For a restricted period of time and on a restricted our solution for this period of time. region,our results should be identical with those ob- For n>64, however, only on certain points n-ill the tained from diffraction by a wedge. We present such a results of the equivalent problems approximate those cotnparison along the points on the straight line coin- of the original problems. cident with the upper edge (i.e.,j = 65). Numerical results are presented for the ThI waves Let the sides of a wedge coincide with the rays B=O discussed above. To gain some idea of the accuracy of and e=& Let the physical space be O and a k=- P sin k(a + +) 50 60 Q(’’ ‘)= - cosh RE - cos R(a + +) IO 20 30 40 70 80 Fig. 7. Calculation of E, for various sycles. These results are based - sin .k(~- 4) on (18). The origin of the coordinate and of the time have been adjusted to agree with that used in the numerical calculation. cosh kt - COS k(a - 4) At t = 0, the incident wave hits the corner. The discontinuities of the first two terms across the I lines f3=O0+T and f3= -Oofa are compensated for by [O otherwise. the contributions from the last integral.It can be shown that for Results of the computations based on (18) are shown in Fig. 7. The agreement with the graphson Fig. 6 appears a 3a e=oo=--; p=- to be good, even for this coarse grid spacing. 2 2 ACKNOWLEDGMEST The author wishes to thank Dr. C. E. Leith for help- ful discussions and to express his gratitude to H. Bar- nettand W. P. Crowleyfor their assistance in the course of making the numerical calculations. REFERENCES [l] J. Stratton, Elecfromug~zeticTheory. New York:McGraw-Hill, For our incident wave we have4 1941, p. 23. [2] J. B. Keller and A. Blank, “Diffraction and reflection of pulses by wedges and corners,” Contmm. Pure Appl. Mutlz., vol. 4, pp. 75-94, June 1951. 4 The origin of the wedge is taken to be the upper right-hand [3] F. G. Friedlander, Sound Pulses. New York: Cambridge, 1958. corner of the square. The zerotime here differs from that of the [4] J. B.. Keller, Electuomgnetic TI.luxs. Madison, Wis.: Univ. of 11%- numerical integration. consm Press, 1961.