Computational Electromagnetics
Chen Aixin School of Electronic and Information Engineering
Chapter 2 Analytical Methods
2.1 Introduction
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The most commonly used analytical methods in solving EM-related problems include:
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(1) separation of variables (2) series expansion (3) conformal mapping (4) integral methods
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Perhaps the most powerful analytical method is the separation of variables.
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2.2 Separation of Variables
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The method of separation of variables (sometimes called the method of Fourier)
It entails seeking a solution which breaks up into a product of functions, each of which involves only one of the variables. For example,
A solution of the form in Eq. is said to be separable in x, y, z, and t .
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To determine whether the method of independent separation of variables can be applied to a given physical problem, we must consider
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the PDE describing the problem the shape of the solution region the boundary conditions
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The three elements that uniquely define a problem.
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We shall always take these three major steps:
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(1) separate the (independent) variables (2) find particular solutions of the separated equations, which satisfy some of the boundary conditions (3) combine these solutions to satisfy the remaining boundary conditions
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The homogeneous scalar wave equation
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If we let an arbitrary constant ?k2 be the common value of the two sides
solution or
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Notice that if k = 0, the time dependence disappears and Eq. becomes Laplace’s equation.
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2.3 Separation of Variables in Rectangular Coordinates
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Helmholtz equation In rectangular coordinates, becomes
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We let
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We obtain Each term must be equal to a constant since each term depends only on the corresponding variable. We conclude that so that
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Suppose we choose Then Or Introducing the time dependence gives
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where ω = kc is the angular frequency of the wave. The solution in Eq. represents a plane wave of amplitude A propagating in the direction of the wave vector k = kxax + kyay + kzaz with velocity c.
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2.4 Separation of Variables in Cylindrical Coordinates
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Helmholtz’s equation Let Substituting into Eq. and dividing by RFZ/ρ2 yields where n = 0, 1, 2, . . . and n2 is the separation constant.
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Thus And Dividing both sides of Eq. by ρ2 leads to
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where μ2 is another separation constant.
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Hence And
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If we let
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The three separated equations become
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The solutions or or
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Let x = λρ and replace R by y; the equation becomes Bessel’s equation It has a general solution of the form
where Jn(x) and Yn(x) are Bessel functions of the first and second kinds of order n and real argument x. Yn is also called the Neumann function.
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If x in Eq. is imaginary so that we may replace x by jx, the equation becomes modified Bessel’s equation This equation has a solution of the form
where In(x) and Kn(x) are modified Bessel functions of the first and second kind of order n.
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For small values of x, some typical Bessel functions (or cylindrical functions)
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For the Bessel function of the first kind
where Γ(k + 1) = k! is the Gamma function.
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This is the most useful of all Bessel functions.
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Some of its important properties and identities (also hold for Yn(x))
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For the Neumann function
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when n > 0
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If n = 0
where γ = 1.781 is Euler’s constant and
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For the modified Bessel function of the first kind
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For the modified Bessel function of the second kind
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If n > 0
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if n =0
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Other functions closely related to Bessel functions are Hankel functions of the first and second kinds, defined respectively by
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Hankel functions are analogous to functions exp(±jx) just as Jn and Yn are analogous to cosine and sine functions.
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This is evident from asymptotic expressions
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With the time factor ejωt, and represent inward and outward traveling waves, respectively, while Jn(x) or Yn(x) represents a standing wave. With the time factor e-jωt , the roles of and are reversed.
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Any of the Bessel functions or related functions can be a solution to Eq. depending on the problem. If we choose R(ρ) = Jn(x) = Jn(λρ) with the solutions of and and apply the superposition theorem
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The solution is
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Introducing the time dependence, we get
where ω = kc.
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2.5 Separation of Variables in Spherical Coordinates
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The wave equation becomes
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We substitute into the equation. Multiplying the result by r2 sin2θ/RHF gives
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Since the left-hand side of this equation is independent of φ, we let
where m, the first separation constant, is chosen to be nonnegative integer such that U is periodic in φ.
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This requirement is necessary for physical reasons that will be evident later.
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Thus Eq. reduces to
where λ is the second separation constant.
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Let λ = n(n+1), the separated equations are
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The solution to Eq. Is or
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If we let
,Eq. Becomes
which has the solution
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Functions zn(x) are spherical Bessel functions and are related to ordinary Bessel functions Zn+1/2 according to
Zn+1/2(x) may be any of the ordinary Bessel functions of half-integer order, Jn+1/2(x), Yn+1/2(x), In+1/2(x), Kn+1/2(x), ,and while zn(x) may be any of the corresponding spherical Bessel functions jn(x), yn(x),in(x),kn(x), ,and .
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Bessel functions of fractional order are, in general, given by
where J-ν and I-ν are, respectively, obtained from Eqs. by replacing ν with-ν.
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Since Gamma function of half-integer order is needed in Eq., it is necessary to add that
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Thus the lower order spherical Bessel functions are as follows:
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Other zn(x) can be obtained from the series expansion in Eqs. of Bessel functions of fractional order or the recurrence relations and properties of zn(x) presented as below
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By replacing H in Eq. with y, cosθ by x, and making other substitutions, we obtain Legendre’s associated differential equation
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Its general solution is of the form
where and are called associated Legendre functions of the first and second kind, respectively.
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and can be obtained from ordinary Legendre functions Pn(x) and Qn(x) using
where -1 < x < 1. We note that
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Pn(x) and Qn(x) are Legendre functions of the first and second kind, respectively.
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Typical associated Legendre functions are:
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Higher-order associated Legendre functions can be obtained using Eqs. of and is unbounded at x = ±1, and hence it is only used when x = ±1 is excluded.
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Substituting and applying superposition theorem,we obtain
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Note that the products H(θ)F(φ) are known as spherical harmonics.
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2.6 Concluding Remarks
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Analytic solutions are of major interest as test models for comparison with numerical techniques. The emphasis has been on the method of separation of variables, the most powerful analytic method. In the course of applying the method of separation of variables, we have encountered some mathematical functions such as Bessel functions and Legendre polynomials.
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We conclude this chapter by remarking that the most satisfactory solution of a field problem is an exact analytical one. In many practical situations, no solution can be obtained by the analytical methods now available, and one must therefore resort to numerical approximation, graphical or experimental solutions. (Experimental solutions are usually very expensive, while graphical solutions are not so accurate).
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