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On the eigenstructure of a Sturm-Liouville problem with an impedance boundary condition.


Structural properties of the eigenvalues eigenvalues

statistical term meaning latent root.
 and eigenfunctions of a Sturm-Liouville problem with an impedance impedance, in electricity, measure in ohms of the degree to which an electric circuit resists the flow of electric current when a voltage is impressed across its terminals.  boundary condition boundary condition
n. Mathematics
The set of conditions specified for behavior of the solution to a set of differential equations at the boundary of its domain.
 are studied herein. This entails a Robin boundary condition In mathematics, a Robin boundary condition imposed on an ordinary differential equation or a partial differential equation is a specification of a linear combination of the values of a function and the values of its derivative on the boundary of the domain.  with a complex boundary parameter. The thrust of the present investigation is provided by the mapping of Neumann to Dirichlet eigen-problems that obtains as the complex boundary parameter varies from zero to the point at infinity The point at infinity, also called ideal point, is a point which when added to the real number line yields a closed curve called the real projective line, .  along a fixed direction in the complex plane. Various properties of the spectrum and modal Mode-oriented. A modal operation switches from one mode to another. Contrast with non-modal.

1. modal - (Of an interface) Having modes. Modeless interfaces are generally considered to be superior because the user does not have to remember which mode he is in.
 functions are explored.

AMS AMS - Andrew Message System  Subject Classification: 34B24, 34L05, 34M25. Keywords: Sturm-Liouville problem, impedance boundary condition, non-self-adjoint boundary value problem In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions. .

1. Introduction

Herein, we explore the eigenstructure of the Sturm-Liouville boundary value problem (SL-BVP) with an impedance boundary condition (IBC IBC International Building Code
IBC Iraq Body Count
IBC Institutional Biosafety Committee
IBC Inflammatory Breast Cancer
IBC International Business Company
IBC Independence Blue Cross
IBC Insurance Bureau of Canada
IBC International Broadcasting Convention

u"(x) + [lambda] x u(x) = 0, 0 < x < L, u'(0) - [sigma] x u(0) = 0, u'(L) + [sigma] x u(L) = 0. (1.1)

The complex boundary parameter [sigma] = [[sigma].sub.R] + [??] x [[sigma].sub.I] is assumed to have the same value at both ends of the interval J = [0,L]. Attention will be restricted to this case as it is the one most relevant to the extension of the results for the eigenstructure of the equilateral triangle equilateral triangle

perfect geometrical representation of triune God. [Christian Symbolism: Appleton, 102]

See : Trinity
 [4, 5] from real to complex values of [sigma] which furnishes the motivation for the present one-dimensional study.


Figure 1 illustrates the parallel plate waveguide waveguide, device that controls the propagation of an electromagnetic wave so that the wave is forced to follow a path defined by the physical structure of the guide.  problem which gives rise to the SL-BVP, Equation (1.1). Either an acoustic [8, pp. 485-496] or electromagnetic electromagnetic /elec·tro·mag·net·ic/ (-mag-net´ik) involving both electricity and magnetism.


pertaining to or emanating from electromagnetism.
 [3, pp. 81-84] time-harmonic wave of angular frequency In physics (specifically mechanics and electrical engineering), angular frequency ω (also referred to by the terms angular speed, radial frequency, and radian frequency) is a scalar measure of rotation rate.  [omega] is propagating in the z-direction with no field variation in the y-direction. The complex material parameter [sigma] is related to the wall impedance which accounts for field penetration into the walls of the waveguide. After the SL-BVP has been solved for the eigenvalue eigenvalue

In mathematical analysis, one of a set of discrete values of a parameter, k, in an equation of the form Lx = kx. Such characteristic equations are particularly useful in solving differential equations, integral equations, and systems of
 [lambda] and the corresponding eigenfunction Eigenfunction

One of the solutions of an eigenvalue equation. A parameter-dependent equation that possesses nonvanishing solutions only for particular values (eigenvalues) of the parameter is an eigenvalue equation, the associated solutions being the
 (mode) u(x), the real physical field may be reconstructed re·con·struct  
tr.v. re·con·struct·ed, re·con·struct·ing, re·con·structs
1. To construct again; rebuild.


U(x, z, t) = Re{u(x) x [e.sup.[??]{([omega]t - [gamma]z)}, [gamma] := [square root of - [lambda]]. (1.2)

This same SL-BVP arises in the study of the vibrating string A vibration in a string is a wave. Usually a vibrating string produces a sound whose frequency in most cases is constant. Therefore, since frequency characterizes the pitch, the sound produced is a constant note.  with sidewise side·wise  
adv. & adj.

Adv. 1. sidewise - toward one side; "the car slipped sideways into the ditch"; "leaning sideways"; "a figure moving sidewise in the shadows"
sideway, sideways

 displacement displacement, in psychology: see defense mechanism.

Same as offset. See base/displacement.
 of the end supports [6, pp. 133-134].

Much of the current mathematical literature devoted to SL-BVPs is concerned primarily with the self-adjoint case [1, 10]. However, the SL-BVP described by Equation (1.1) is non-self-adjoint for complex values of [sigma]. Fortunately, a classic reference on the non-self-adjoint case is provided by [2, pp. 298-305] where it is shown that this problem reduces to the study of the solutions of a single transcendental equation (Math.) an equation into which a transcendental function of one of the unknown or variable quantities enters.

See also: Transcendental

In what follows, we will derive and exhaustively study this transcendental equation. The principal focus will be on what happens to the eigenstructure of the Neumann problem ([sigma] = 0) as [sigma] proceeds along rays emanating from the origin toward the point at infinity in the complex plane. We will find that, for the most part, there is a natural homotopy connecting the Neumann modes to those of a corresponding Dirichlet problem In mathematics, a Dirichlet problem is the problem of finding a function which solves a specified partial differential equation (PDE) in the interior of a given region that takes prescribed values on the boundary of the region.  (IBC-Dirichlet modes). However, we will show that, under appropriate conditions, there will be two Neumann eigenvalues that have no Dirichlet counterpart. The determination of the precise nature of these latter eigenvalues together with that of their corresponding eigenfunctions ("missing modes") is the primary concern of the present investigation.

2. Solution of the Sturm-Liouville Problem

Ignoring the boundary conditions, the general solution to Equation (1.1) is

u(x) = cos (v[pi]x/L - [delta]), [lambda] = [(v[pi]/L).sup.2]. (2.1)

Application of the boundary condition at x = 0 yields

tan ([delta]) = [sigma]L/v[pi], (2.2)

while imposition of the boundary condition at x = L yields

tan (v[pi] - [delta]) = [sigma]L/v[pi]. (2.3)

Equations (2.2) and (2.3) may be combined to produce the transcendental equation

(2[delta] + n[pi]) tan ([delta]) = [sigma]L, v = 2[delta]/[pi] + n; (2.4)

with n an integer integer: see number; number theory .

Observe that in Equation (1.1) when [sigma] [right arrow] 0 we recover the Neumann problem, u'(0) = 0 = u'(L), whose solution, u(x) = [N.sub.n](x) := cos (n[pi]x/L), is obtained from Equation (2.4) with [delta] = 0 [??] v = n. Thus, we may profitably view [sigma] as a continuation parameter which provides a homotopy extending from this well understood problem to that of the impedance boundary condition. Throughout the ensuing en·sue  
intr.v. en·sued, en·su·ing, en·sues
1. To follow as a consequence or result. See Synonyms at follow.

2. To take place subsequently.
 development we will avail ourselves of this important observation.

Furthermore, note that if the normal derivative remains bounded then [sigma] [right arrow] [infinity infinity, in mathematics, that which is not finite. A sequence of numbers, a1, a2, a3, … , is said to "approach infinity" if the numbers eventually become arbitrarily large, i.e. ] yields the Dirichlet problem, u(0) = 0 = u(L), whose solution, u(x) = [+ or -][D.sub.[n[+ or -]1](x) := [+ or -] sin ((n [+ or -] 1)[pi]x/L), is obtained from Equation (2.4) with [delta] = [+ or -][pi] 2) v = n [+ or -] 1. In that case, the homotopy may be further extended to lead from a Neumann mode to a corresponding Dirichlet mode. As will subsequently be shown, for the impedance boundary condition this is usually, although by no means always, the case.

Introduction of z : = [delta] + n[pi]/2 into Equation (2.4) reduces it to

z tan (z) = [sigma]L/2, (2.5)

if n is even and

z cot (z) = -[sigma]L/2, (2.6)

if n is odd. These two cases may then be separately studied from the graphical representation of their respective complex transformations [8, p. 909].

However, in the case of the equilateral triangle [4, 5] which is our ultimate goal, such a reduction is not available. Thus, Equation (2.4) itself will be numerically approximated using MATLAB (MATrix LABoratory) A programming language for technical computing from The MathWorks, Natick, MA ( Used for a wide variety of scientific and engineering calculations, especially for automatic control and signal processing, MATLAB runs on Windows, Mac and . This will permit us to trace out trajectories in the complex [delta]-plane or, equivalently, in the complex v-plane as the complex boundary parameter [sigma] is varied.


As shown in Figure 2, beginning with [sigma] = 0 (the Neumann problem), we will allow [sigma] to move along a ray inclined at an angle [theta Theta

A measure of the rate of decline in the value of an option due to the passage of time. Theta can also be referred to as the time decay on the value of an option. If everything is held constant, then the option will lose value as time moves closer to the maturity of the option.
] to the horizontal toward the point at infinity. We may then track the trajectory Trajectory

The curve described by a body moving through space, as of a meteor through the atmosphere, a planet around the Sun, a projectile fired from a gun, or a rocket in flight.
 of each Neumann eigenvalue in order to determine whether it eventually approaches a Dirichlet eigenvalue (in which case it will be called an IBC-Dirichlet mode) or migrates to infinity (in which case it will be called a "missing mode"). We will eventually find that these are the only two possible types of asymptotic behavior.

3. SL-BVP/IBC Solution Properties

Before constructing a taxonomy taxonomy: see classification.

In biology, the classification of organisms into a hierarchy of groupings, from the general to the particular, that reflect evolutionary and usually morphological relationships: kingdom, phylum, class, order,
 of the asymptotic nature of the eigenstructure of our SL-BVP with an IBC as [sigma] [right arrow] [infinity] (at which time, the mysterious icons on Figure 2 will become intelligible), it behooves us to first catalog catalog, descriptive list, on cards or in a book, of the contents of a library. Assurbanipal's library at Nineveh was cataloged on shelves of slate. The first known subject catalog was compiled by Callimachus at the Alexandrian Library in the 3d cent. B.C.  some important properties of the eigenvalues and eigenfunctions of Equation (1.1).

We begin with the observation that, without loss of generality Without loss of generality (abbreviated to WLOG or WOLOG and less commonly stated as without any loss of generality) is a frequently used expression in mathematics. , we may restrict our attention to the case Im([sigma]) [greater than or equal to] 0 since, by taking complex conjugates complex conjugate
Either one of a pair of complex numbers whose real parts are identical and whose imaginary parts differ only in sign; for example, 6 + 4i and 6 - 4i are complex conjugates.

Noun 1.
 in Equation (2.4), we find that [sigma] [??] [bar.[sigma]] [??] [delta] [??] [bar.[delta]] [??] v [??] [bar.v]. Thus, in the event that Im([sigma]) < 0, we can obtain trajectories in either the [delta]-plane or v-plane by reflection about the real axis of the the diameter of the sphere which is perpendicular to the plane of the circle.

See also: Axis
 corresponding trajectories for [bar.[sigma]].

Furthermore, taking complex conjugates in Equation (1.1) itself reveals that [sigma] [??] [bar.[sigma]] [??] [lambda] [??] [bar.[lambda]] =: [mu], u(x) [??] [bar.u](x) =: v(x) where [mu] and v(x) form the solution to the adjoint Ad´joint

n. 1. An adjunct; a helper.
 boundary value problem

v'(x) + [mu] x v(x) = 0, 0 < x < L,

v'(0) - [bar.[sigma]] x v(0) = 0,

v'(L) + [bar.[sigma]] x v(L) = 0. (3.1)

Since the SL-BVP is non-self-adjoint for complex [sigma], the eigenvalues and eigenfunctions can be complex and, most importantly Adv. 1. most importantly - above and beyond all other consideration; "above all, you must be independent"
above all, most especially
, eigenfunctions corresponding to distinct eigenvalues are not necessarily orthogonal At right angles. The term is used to describe electronic signals that appear at 90 degree angles to each other. It is also widely used to describe conditions that are contradictory, or opposite, rather than in parallel or in sync with each other.  with respect to the complex inner product < f(x); g(x) >:= [[integral.sup.L.sub.0] f(x)[bar.g](x) dx. However, we do have the biorthogonality relationship involving the eigenfunctions of the boundary value problem, Equation (1.1), and those of of the adjoint boundary value problem, Equation (3.1),

< [u.sub.p](x), [v.sub.q](x) >:= [[integral].sup.L.sub.0] [u.sub.p](x)[[bar.v].sub.q](x) dx = 0. (3.2)

In turn, this provides us with the eigenfunction expansion

f(x) = [[infinity].summation summation n. the final argument of an attorney at the close of a trial in which he/she attempts to convince the judge and/or jury of the virtues of the client's case. (See: closing argument)  over (k=1)] < f(x), [v.sub.k](x) > [u.sub.k](x), (3.3)

where the series converges in the mean for f(x) [member of] [L.sup.2](0, L) [2, pp. 310-312].

We have already seen via Equations (2.5) and (2.6) that the modes naturally partition A reserved part of disk or memory that is set aside for some purpose. On a PC, new hard disks must be partitioned before they can be formatted for the operating system, and the Fdisk utility is used for this task.  according to according to
1. As stated or indicated by; on the authority of: according to historians.

2. In keeping with: according to instructions.

 the parity of n. This fact is underscored by

Theorem theorem, in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiom in that a proof is required for its acceptance.  1 The even/odd numbered modes are symmetric/antisymmetric, respectively, on the interval [0, L] for all values of [sigma].

Proof. Equation (3) may be recast re·cast  
tr.v. re·cast, re·cast·ing, re·casts
1. To mold again: recast a bell.

 as [u.sub.n](x) = cos (v[pi]/L (x - L/2) + n[pi/ 2). Thus, if n is even/odd then [u.sub.n](x) is a cosine/sine, respectively, centered at x = L/2.

4. The Case of Real [sigma]

In the case of real [sigma], the eigenstructure of the SL-BVP with an IBC defined by Equation (1.1) has been exhaustively treated in [9, pp. 90-98]. As a springboard for the study of the case of complex [sigma] in the next section, we review here the highlights of those results.

When [sigma] is real, the problem is self-adjoint so that the eigenvalues are real and the eigenfunctions can also be chosen to be real. Moreover, eigenfunctions corresponding to distinct eigenvalues are orthogonal with respect to the real inner product < f(x); g(x) >:= [[integral.sup.L.sub.0] f(x)g(x) dx. Because of their very different behaviors, we treat the cases [sigma] [greater than or equal to] 0 and [sigma] < 0 separately. We will eventually see that, taken together, they display the characteristic behaviors exhibited in the general case of complex [sigma].

In what follows, we denote de·note  
tr.v. de·not·ed, de·not·ing, de·notes
1. To mark; indicate: a frown that denoted increasing impatience.

 the dependence of the eigenvalues and eigenfunctions upon [sigma] by [[lambda].sub.n]([sigma]) = [(v([sigma])[pi]/L).sup.2] and un(x; [sigma]), respectively, for n = 0, 1, .... The subscript (1) In word processing and scientific notation, a digit or symbol that appears below the line; for example, H2O, the symbol for water. Contrast with superscript.

(2) In programming, a method for referencing data in a table.
 n is chosen so that when [sigma] = 0 they reduce to the corresponding values for the Neumann problem, [d.sub.n](0) = 0 [??] [v.sub.n](0) = n and [u.sub.n](x; 0) = [N.sub.n](x) := cos(n[pi]x/L). Also, the Dirichlet problem has the same eigenvalues [[lambda].sub.n](0) but with the restriction n = 1, 2, ... and its corresponding eigenfunctions are denoted by un(x; 0) = [D.sub.n](x) := sin(n[pi]x/L).

4.1. The Case [sigma] [greater than or equal to] 0


The case of the radiation boundary condition ([sigma] > 0) is by far the simplest in that the eigenvalues are not only real but are in fact all positive. Moreover, n < [[upsilon up·si·lon or yp·si·lon
Symbol The 20th letter of the Greek alphabet.
].sub.n]([sigma]) < n + 1. It also possesses the simplest asymptotic behavior in that

[MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression.  NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. .]. (4.1)

That is, the Neumann mode [N.sub.n](x) "morphs" analytically into the Dirichlet mode [D.sub.n+1](x) as [sigma] ranges from 0 to [infinity]. This is illustrated in Figure 3 which displays this homotopy between fundamental modes (n = 0) for 0 [less than or equal to] [sigma] [less than or equal to] -[infinity].

4.2. The Case [sigma] < 0

The case of the absorbing boundary condition ([sigma] < 0) is more complicated in that the eigenvalues while still real are no longer all positive. However, for n = 2, 3, ... we have n - 1 < [v.sub.n]([sigma]) < n. These so-called IBC-Dirichlet modes possess the simple asymptotic behavior described by



That is, the Neumann mode [N.sub.n](x) "morphs" analytically into the Dirichlet mode - [D.sub.n-1](x) as [sigma] ranges from 0 to -[infinity]. This is illustrated in Figure 4 which displays this homotopy between [N.sub.2](x) and -[D.sub.1](x) for 0 [greater than or equal to] [sigma] [greater than or equal to] -[infinity].


This leaves open the case of the "missing modes" n = 0, 1. Since there are no Dirichlet modes for n = -1, 0 we clearly do not have the simple asymptotic behavior described by Equation (4.2). Thus we are confronted with the question: "What happens to the missing n = 0, 1 modes as [sigma] [right arrow] -[infinity]?". The solution to the mystery of the missing modes naturally decomposes into two special cases each of which we now explore separately.


For n = 0, [v.sub.0]([sigma]) is pure imaginary and, consequently, [[lambda].sub.0]([sigma]) is negative. Specifically, for [sigma] [right arrow] -[infinity],

[[delta].sub.0]([sigma]) [approximately equal to] - [sigma]L/2 x [??] [v.sub.0]([sigma]) [approximately equal to] - [sigma]L/[pi] x [??] [??] [u.sub.0](x; [sigma]) [approximately equal to] cosh (-[sigma]x + [sigma]L/2). (4.3)

This asymptotic expression for [u.sub.0](x; [sigma]) becomes unbounded as [sigma] [right arrow] -[infinity]. However, if we first scale it by its value at an endpoint, cosh ([sigma]L/2), we find that this normalized mode approaches 1 at the two endpoints and 0 elsewhere. Such singular limiting behavior, necessary since this does not approach a Dirichlet mode, is on prominent display in Figure 5.

For n = 1, [[nu].sub.1]([sigma]) initially decreases along with [sigma] until it vanishes when [sigma] = - 2/L. Specifically,


However, if we first scale [u.sub.1](x; [sigma]) by its value at the endpoint x = 0, cos ([delta]), we find that this normalized mode approaches the straight line [u.sub.1](x; -2/L) = 1 - 2/L x x.

As [sigma] continues to decrease we have, for [sigma] < - 2/L, pure imaginary [[delta].sub.1]([sigma])+ [pi]/2 thereby producing a [v.sub.1]([sigma]) which is also pure imaginary and, consequently, [[lambda].sub.1]([sigma]) becomes negative. Specifically, for [sigma] [right arrow] -[infinity],

[[delta].sub.1]([sigma]) [approximately equal to] - [pi]/2 - [sigma]L/2 [??] [??] x [v.sub.1]([sigma]) [approximately equal to] - [sigma]L/[pi] x [??] [u.sub.1](x) [approximately equal to] [??] x sinh sinh
hyperbolic sine


Abbreviation of hyperbolic sine
 ([sigma]x - [sigma]L/2). (4.5)

This asymptotic expression for [u.sub.1](x; [sigma]) becomes unbounded as [sigma] [right arrow] -[infinity]. However, if we first scale it by its value at the endpoint x = 0, [??] x sinh (- [sigma]L/2), we find that this normalized mode approaches [+ or -]1 at the left-/right-hand endpoint, respectively, and 0 elsewhere. The resulting unbounded derivative, necessary since this does not approach a Dirichlet mode, is evident in Figure 6.

5. The Case of Complex [sigma]


In the case of complex [sigma], as previously noted, the SL-BVP defined by Equation (1.1) is non-self-adjoint and the eigenstructure is consequently complex. Defining the residual function of Equation (2.4) as

[[rho].sub.n]([delta]; [sigma]) = |(2[delta] + n[pi]) tan ([delta]) - [sigma]L|; (5.1)

we note that the sought-after eigenvalues are determined by its local minima. Next, define

[sigma] = r x [e.sup.x[theta]]; with [theta] fixed and 0 [less than or equal to] r [less than or equal to] + [infinity]. (5.2)

Figure 7, which is a contour contour or contour line, line on a topographic map connecting points of equal elevation above or below mean sea level. It is thus a kind of isopleth, or line of equal quantity.  plot of [[rho].sub.n], displays the resultant trajectory in the [delta]-plane for n = 0 as r varies with [theta] = [pi]/4. As occurred in the real case with [sigma] [greater than or equal to] 0, [delta] varies from 0 to [pi]/2 only now it makes an excursion excursion /ex·cur·sion/ (eks-kur´zhun) a range of movement regularly repeated in performance of a function, e.g., excursion of the jaws in mastication.  into the complex plane rather than traveling along the real axis. The attendant complex mode morphing Transforming one image into another; for example, a car into a tiger. The term comes from metamorphosis. Morphing programs work by marking prominent points, such as tips and corners, of the before and after images.  from [N.sub.0](x) to [D.sub.1](x) is made explicit by Figure 8.

Figure 9 displays the corresponding trajectories, this time in the [nu]-plane, for the the first four modes over the full range of values 0 [less than or equal to] [theta] [less than or equal to] [pi]. As is evident from each of these plots, for some values of [theta] mode morphing occurs, i.e [[lambda].sub.n]([sigma]) [right arrow] [[lambda].sub.n[+ or -]1](0), (IBC-Dirichlet modes) while for other values of [theta] we observe |[[lambda].sub.n]([sigma])| [right arrow] [infinity] (missing modes). We are going to devote our remaining efforts to clarifying this asymptotic behavior.


In order to achieve this, we will require detailed knowledge of the properties of trajectories in the v-plane. We commence with

Theorem 2 1. Let [[delta].sub.n](r) = [[delta].sup.R.sub.n] (r) + [??] x [[delta].sup.I.sub.n](r) and [v.sub.n](r) = [v.sup.R.sub.n] (r) + [??] x [v.sup.I.sub.n](r). Then, the trajectories [[delta].sub.n](r) and [v.sub.n](r) have the same slope.

(a) If n [not equal to] 0 then the trajectory [v.sub.n](r) makes an angle [theta] measured counterclockwise from the real axis at [v.sub.n](0) = n.

(b) If n = 0 then the trajectory [v.sub.0](r) makes an angle [theta]/2 measured counterclockwise from the real axis at [v.sub.0](0) = 0.

2. As r [right arrow] [infinity], either [v.sub.n](r) [right arrow] n [+ or -] 1 or |[v.sub.n](r)| [right arrow] [infinity].

(a) if [v.sub.n](r) [right arrow] n [+ or -] 1 then the trajectory [v.sub.n](r) makes an angle [pi] - [theta] measured counterclockwise from the real axis at [v.sub.n](1) = n [+ or -] 1.

(b) if |[v.sub.n](r)| [right arrow] [infinity] then the trajectory [v.sub.n](r) goes off to infinity at an angle [theta] - [pi]/2 measured counterclockwise from the real axis as r [right arrow] [infinity].


1. Since [v'.sub.n](r) = 2/[pi] x [[delta]'.sub.n](r), both trajectories have slope [[delta].sup.I'.sub.n](r)=[[delta].sup.R'.sub.n](r).


(a) r = 0 [??] [sigma] = 0 so that Equation (2.4) produces [[delta].sub.n](0) = 0) [[nu].sub.n](0) = n. Substitution of Equation (5.2) into Equation (5.3) with subsequent differentiation with respect to r yields

[[delta]'.sub.n](r) = Le{[theta] [cos.sup.2] ([[delta].sub.n](r)) sin (2[[delta].sub.n](r)) + 2[[delta].sub.n](r) + n[pi] : (5.3)

Thus, [[delta]'.sub.n](0) = [Le.sup.[??][theta]]/n[pi] [??] tan [empty set] := [[delta].sup.I'.sub.n] (0)/[[delta].sup.R'.sub.n](0) = tan [theta] [??] [empty set] = [theta].

(b) For n = 0, Equation (5.3) produces [[delta]'.sub.0](0) = [infinity] so that the slope tan [empty set] := [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is indeterminate That which is uncertain or not particularly designated.

INDETERMINATE. That which is uncertain or not particularly designated; as, if I sell you one hundred bushels of wheat, without stating what wheat. 1 Bouv. Inst. n. 950.
. However, by Equation (5.3),


Thus, tan [empty set] = tan ([theta] - [empty set]) [??] [empty set] = [theta]/2.

2. As r = |[sigma]| [right arrow] [infinity], Equation (2.4) clearly implies that either [[delta].sub.n](r) [right arrow] [+ or -] [pi]/2, in which case [v.sub.n](r) [right arrow] n [+ or -] [infinity], or |[[delta].sub.n](r)| [right arrow] [infinity], in which case |[v.sub.n](r)| [right arrow] [infinity].

(a) If [[delta].sub.n](r) [right arrow] [+ or -] [pi]/2 then [[nu].sub.n](r) [right arrow] n [+ or -] 1 and [[delta]'.sub.n](r) [right arrow] 0 by Equations (2.4) and (5.3), respectively. Thus, as r [right arrow] [infinity], [[delta]'.sub.n](r) [right arrow] [[epsilon].sub.1] [e.sup.[??][empty set]] and [[delta].sub.n](r) [right arrow] [+ or -] [pi]/2 [+ or -][[epsilon].sub.2][e.sup.[??][empty set]] where [[empty set].sub.1], [[epsilon].sub.2] [right arrow] 0. Comparison of the arguments of both sides of Equation (5.3) reveals that [empty set] = [theta] + 2[empty set] [??] [empty set] = -[theta].

(b) If |[[delta].sub.n](r)| [right arrow] [infinity] then tan ([[delta].sub.n](r)) [right arrow] [??]. Inserting this into Equation (2.4) and equating e·quate  
v. e·quat·ed, e·quat·ing, e·quates
1. To make equal or equivalent.

2. To reduce to a standard or an average; equalize.

 real and imaginary parts Noun 1. imaginary part - the part of a complex number that has the square root of -1 as a factor
imaginary part of a complex number

complex number, complex quantity, imaginary, imaginary number - (mathematics) a number of the form a+bi where a and b are real
 produces 2[[delta].sup.R.sub.n] [right arrow] rL sin ([theta]) and -2[[delta].sup.I.sub.n] [right arrow] rL cos ([theta]). Thus, [[delta].sup.I.sub.n](r)/ [[delta].sup.R.sub.n] (r) [right arrow] -cot ([theta]) = tan (theta] - [phi]/2).

Returning now to Figure 9, note the following important structural features. For modes n = 0 and n = 1, there is a critical angle [[theta].sup.-.sub.n] such that for [theta] < [[theta].sup.-.sub.n] we have the mode morphing [N.sub.n](x) [??] [D.sub.n+1](x) while, for [theta] > [[theta].sup.-.sub.n], mode n is missing in the previously defined sense as r [right arrow] 1. For all other modes n [less than or equal to] 2, there are two critical angles [[theta].sup.-.sub.n] and [[theta].sup.+.sub.n]. For [theta] < [[theta].sup.-.sub.n] we have the mode morphing [N.sub.n](x) [??] [D.sub.n+1](x) and for [theta] > [[theta].sup.+.sub.n] we have the mode morphing [N.sub.n](x) [??] -[D.sub.n-1](x) as r [right arrow] [infinity]. In the intermediate regime [[theta].sup.-.sub.n] < [theta] < [[theta].sup.+.sub.n], mode n is missing as r [right arrow] [infinity].

Our next result concerns the trajectories at these critical angles.

Theorem 3 1. The critical trajectories all possess a corner located in the [delta]-plane at the roots of the function

[[tau].sub.n]([delta]) := sin (2[delta]) + 2[delta] + n[pi]: (5.4)

The smallest root with Re([[delta].sup.-.sub.n]) > 0 corresponds to [[theta].sup.-.sub.n] and the smallest root with Re([[delta].sup.+.sub.n]) < 0 corresponds to [[theta].sup.+.sub.n].

2. All of these corners of the critical trajectories are 90[delta].

3. In the special case, [theta] = -[pi], n = 1, the corner is located at [delta] = [pi]/2 with a corresponding [sigma] = - 2/L.


1. Clearly, there can be no transition from a bounded mode-morphing trajectory to the unbounded trajectory of a missing mode without a singularity (1) See technology singularity.

(2) (Singularity) An experimental operating system from Microsoft for the x86 platform written almost entirely in C#, a .NET managed code language. Released in 2007, Singularity is a non-Windows research project.
 exhibiting a corner. At such a corner, the derivative [[delta]'.sub.n](r) must either vanish or become unbounded. Inspection of Equation (5.3) reveals that the derivative does not vanish since Im([[delta].sub.n](r) > 0) for r > 0. Further perusal of Equation (5.3) shows that the derivative becomes unbounded if and only if [[delta].sub.n](r) is located at a zero of the denominator denominator

the bottom line of a fraction; the base population on which population rates such as birth and death rates are calculated.

, i.e. at a root of Equation (5.4). These critical values [[delta].sup.[+ or -].sub.n] occur at branch points [7, pp. 404-408] of Equation (2.5) if n is even and of Equation (2.6) if n is odd. Observe that once [[delta].sup.[+ or -].sub.n] is known then Equation (2.4) may be used to find the critical angle and modulus See modulo.  from [[sigma].sup.[+ or -].sub.n] = [r.sup.[+ or -].sub.n] [e.sup.[??][[theta].sup.[+ or -].sub.n]].

2. The corner will be 90[delta] if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] is pure imaginary since, in that and only that case, the tangent tangent, in mathematics.

1 In geometry, the tangent to a circle or sphere is a straight line that intersects the circle or sphere in one and only one point.
 vector is multiplied by a pure imaginary as we pass through the corner thereby producing the posited rotation by [+ or -]90[delta]. But,


by l'Hopital's Rule. Thus, [L.sup.2] = -1 [??] L = [+ or -][??].

3. Set [theta] = -[pi] and n = 1. Then, since the eigenvalue is real, the corner on the critical trajectory occurs as it passes through [v.sub.1] = 0 [??] [[delta].sub.1] = -[pi]/2. Thus, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] by l'Hopital's Rule.


Returning once again to Figure 9, note that the corner on the critical trajectory corresponding to [[theta].sup.[+ or -].sub.n] is a bifurcation Bifurcation

A term used in finance that refers to a splitting of something into two separate pieces.

Generally, this term is used to refer to the splitting of a security into two separate pieces for the purpose of complex taxation advantages.
 point where one path leads to n"1 and the other veers off to infinity at an angle of [[theta].sup.[+ or -].sub.n] -[pi]/2 (Theorem 2, Part 2b). Turning attention to Figure 10 which is an amalgam of Figure 9, the top frame for the even pinumbered modes and the bottom frame for the odd numbered modes, observe that the boundaries of adjacent modal regions formed from these bifurcated bi·fur·cate  
v. bi·fur·cat·ed, bi·fur·cat·ing, bi·fur·cates
To divide into two parts or branches.

To separate into two parts or branches; fork.

 trajectories align, the result in both cases being a corresponding partition of the [nu]-plane.

Furthermore, the trajectories corresponding to adjacent modes overlap but do not intersect In a relational database, to match two files and produce a third file with records that are common in both. For example, intersecting an American file and a programmer file would yield American programmers.  in the sense that they never come in contact for the same value of [sigma]. Specifically we have the following


Theorem 4 1. A trajectory emanating from n never intersects a trajectory emanating from n [+ or -] 1 for the same value of [sigma].

2. A trajectory emanating from n intersects a trajectory emanating from n [+ or -] 2 forthe same value of [sigma] only at their common branch (bifurcation) point. In fact, [[delta].sup.+.sub.n] = [[delta].sup.-.sub.n-2] - [pi] and [[theta].sup.+.sub.n] = [[theta].sup.-.sub.n-2]. At the common branch point there is a modal deficiency and along the bifurcated trajectory there is modal ambiguity.


1. According to Equation (2.4), [v.sub.n] = [v.sub.n+1] if and only if [[delta].sub.n] = [[delta].sub.n+1] + [pi]/2 and tan ([[delta].sub.n+1] + [pi]/2) = tan ([[delta].sub.n+1]). But, this would require that - cot ([[delta].sub.n+1]) = tan ([[delta].sub.n+1]) which is impossible.

2. From Equation (5.4), we have

[[tau].sub.n-2]([delta]) = sin (2([delta] - [pi])) + 2([delta] - [pi]) + n[pi] = [[tau].sub.n]([delta] - [pi]), (5.5)

which implies that the roots of [[tau].sub.n]([delta]) are the roots of [[tau].sub.n-2]([delta]) shifted to the left by [pi]. Thus, [[delta].sup.+.sub.n] = [[delta].sup.-.sub.n-2] - [pi] and [[theta].sup.+.sub.n = [[theta].sup.-.sub.n-2]. Now, Equation (5.3) implies that


so that [[delta].sub.n](r) and [[delta].sub.n-2](r) - [pi] satisfy the same differential equation differential equation

Mathematical statement that contains one or more derivatives. It states a relationship involving the rates of change of continuously changing quantities modeled by functions.
 with different initial conditions. By the fundamental Existence and Uniqueness Theorem for ordinary differential equations ordinary differential equation

Equation containing derivatives of a function of a single variable. Its order is the order of the highest derivative it contains (e.g., a first-order differential equation involves only the first derivative of the function).
 [2, pp. 1-11], they can only intersect at a singularity and such an intersection is equivalent to [v.sub.n-2](r) = [v.sub.n](r). At this common branch point, modes n - 2 and n coalesce co·a·lesce  
intr.v. co·a·lesced, co·a·lesc·ing, co·a·lesc·es
1. To grow together; fuse.

2. To come together so as to form one whole; unite:
 thereby producing a modal deficiency. Beyond the branch point, there is modal ambiguity in that it is not clear which mode to associate with which bifurcation branch.

The modal deficiency and ambiguity established in the previous theorem must be taken into account when utilizing the eigenfunction expansion of Equation (3.3). Also, this theorem alleviates the need to calculate [[delta].sup.+.sub.n] and [[theta].sup.+.sub.n] since they are obtainable from [[delta].sup.-.sub.n-2] and [[theta].sup.-.sub.n-2], respectively.


Figure 11 shows the level curves of |[[tau].sub.0]([delta])| (see Equation (5.4)) where the branch point [[delta].sup.-.sub.0] is on prominent display. We next study [[delta].sup.-.sub.n] as n varies.

Theorem 5 1. As n [right arrow] 1, [[sigma].sup.-n] [right arrow] [-ln ((3 + 2n)[pi]) + [??] x (n + 3/2)[pi]] = L.

2. For 0 [less than or equal to] [theta] [less than or equal to] [pi]/2, all modes are IBC-Dirichlet modes (i.e. there are no missing modes).


1. As n [right arrow] 1, Re([[delta].sup.-.sub.n]) [right arrow] [(3[??]/4).sup.-] and [[delta].sup.[I.sup.-.sub.n]] := Im([[delta].sup.-.sub.n]) [right arrow] + [infinity]. Thus, tan ([[delta].sup.[I.sup.-.sub.n]]) [right arrow] [??] and, from [[tau].sub.n]([[delta].sup.-.sub.n]) = 0, [[delta].sup.[I.sup.-.sub.n]] [right arrow] 1/2 ln ((3 + 2n)[pi]). Solving for [[sigma].sup.-.sub.n] in Equation (2.4) yields [[sigma].sup.-.sub.n] [right arrow] [-ln ((3 + 2n)[pi]) + [??] x (n + 3=2)[pi]]/L.

2. Since [[theta].sup.-.sub.n] := arg [[sigma].sup.-.sub.n], we have [[theta].sup.-.sub.n] = [tan.sup.-1] [-(n + 3/2)[pi]= ln ((3 + 2n)[pi])] [right arrow] [tan.sup.-1] (-[infinity]) = [([pi]/2).sup.+]. Therefore, all of the critical angles lie in the range [pi]/2 < [[theta].sup.-.sub.n] [less than or equal to] [pi].

5.1. The Case Re([sigma]) [greater than or equal to] 0

Because of the second part of Theorem 5, the asymptotic behavior for Re([sigma]) [greater than or equal to] 0 is especially simple. With reference to Figure 12, there is a complete modal homotopy from each of the Neumann modes to a corresponding Dirichlet mode. Specifically, [N.sub.n](x) [??] [D.sub.n+1](x) for all n. This asymptotic behavior, illustrated by Figure 8 for n = 0 and [theta] = [pi]=4, is directly analogous analogous /anal·o·gous/ (ah-nal´ah-gus) resembling or similar in some respects, as in function or appearance, but not in origin or development.

 to the case of [sigma] real and positive except that now the homotopy passes through the complex plane.

5.2. The Case Re([sigma]) < 0


The asymptotic behavior for Re([sigma]) < 0, is directly analogous to the case of [sigma] real and negative in that there are always two missing modes with the remainder being IBC-Dirichlet modes. However, which modes are missing is now determined by the value of [theta]. Specifically, if [[theta].sup.-.sub.n] < [theta] < [[theta].sup.-.sub.n-1] then modes n and n + 1 will be missing. This is shown graphically in Figure 13. Table 1 lists many branch points [[delta].sup.-.sub.n] together with their corresponding critical angles [[theta].sup.-.sub.n].

5.2.1 IBC-Dirichlet Modes

If [theta] < [[theta].sup.-.sub.n] then [N.sub.n](x) [??] [D.sub.n+1](x) analogous to the case of the mode morphing displayed in the case of Re([sigma]) [greater than or equal to] 0. However, if [theta] > [[theta].sup.+.sub.n] = [[theta].sup.-.sub.n-2] then [N.sub.n](x) [??] -[D.sub.n-1](x) characteristic of the mode morphing displayed in the case of [sigma] real and negative. This latter mode-morphing behavior is on display in Figure 14 for n = 2 and [theta] = 3[pi]=4.

5.2.2 Missing Modes

If [[theta].sup.-.sub.n] < [theta] < [[theta].sup.+.sub.n] = [[theta].sup.-.sub.n-2] then mode n is missing. The resulting rightward procession pro·ces·sion  
1. The act of moving along or forward; progression.

2. Origination; emanation; rise.

a. A group of persons, vehicles, or objects moving along in an orderly, formal manner.
 of missing modes as [theta] is lowered from [pi] to [pi]/2 is shown in Figure 15. The missing modes for [pi]=2 < [theta] < [pi] exhibit a peculiar singular behavior as r = |[sigma]| [right arrow] [infinity].

Theorem 6 For [pi]/2< [theta] < [pi], all missing modes oscillate To swing back and forth between the minimum and maximum values. An oscillation is one cycle, typically one complete wave in an alternating frequency.  between [+ or -]1 at the endpoints and approach zero elsewhere.

Proof. Simply let |[delta] = [[delta].sub.R] + [??] x [[delta].sub.I]| [right arrow] [infinity] in Equation (2.1). At the endpoints, [u.sub.n] [right arrow] [+ or -]([e.sup.[[delta].sub.I]]/2) x [cos ([[delta].sub.R]) - [??] x sin ([[delta].sub.R])] and normalization In relational database management, a process that breaks down data into record groups for efficient processing. There are six stages. By the third stage (third normal form), data are identified only by the key field in their record.  by [e.sup.[[delta].sub.I]]/2 reveals the oscillatory oscillatory

characterized by oscillation.

oscillatory nystagmus
see pendular nystagmus.
 behavior at the endpoints as well the approach to zero for interior points.

This oscillatory singular behavior is in evidence in Figures 16 (even numbered modes) and 17 (odd numbered modes).

6. Conclusion

The bulk of this paper has been devoted to exploring the asymptotic nature of the eigenstructure of the SL-BVP with an IBC, Equation (1.1), as [sigma] [right arrow] 1. Our main results may be summarized as follows (see Figure 2):

Theorem 7 (Asymptotic Behavior of SL-BVP/IBC Eigenstructure) Consider the SL-BVP with an IBC described by Equation (1.1) with [sigma](r) = [re.sup.[??][theta]] for fixed [theta] and 0 [less than or equal to] r [less than or equal to] [infinity].



1. If 0 [less than or equal to] [theta] [less than or equal to] [pi]/2 then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for all n.

2. If [pi]/2 < [theta] [less than or equal to] [pi] then there exists n([theta]) such that



(c) and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] for k = n([theta]) - 1, n([theta]).

These observations are directly applicable to the rectangular rec·tan·gu·lar  
1. Having the shape of a rectangle.

2. Having one or more right angles.

3. Designating a geometric coordinate system with mutually perpendicular axes.
 waveguide [8, pp. 503-509].

Furthermore, these results lead naturally to the question of the corresponding asymptotic nature of the eigenstructure of the Laplacian on an equilateral triangle with an impedance boundary condition. The special cases of the radiation boundary condition [4] and the absorbing boundary condition [5] have already received exhaustive treatment. The one-dimensional results of the present paper seem to indicate that these two special cases exhibit the full spectrum of possible asymptotic behavior as [sigma] [right arrow] [infinity]. The final installment of this series of papers, Eigenstructure of the Equilateral Triangle, Part V: The Impedance Boundary Condition, will address this problem.




The author thanks Mrs. Barbara A. McCartin for her indispensable aid in constructing the figures. This paper is dedicated to the memory of our beloved Mother, Dorothy Frances (Kelly) McCartin, on this 25th Anniversary of her departure from her family. Gone but not forgotten!


[1] Amrein W. O., Hinz A. M. and Pearson D. B., Sturm-Liouville Theory In mathematics and its applications, a classical Sturm-Liouville equation, named after Jacques Charles François Sturm (1803-1855) and Joseph Liouville (1809-1882), is a real second-order linear differential equation of the form

: Past and Present, Birkhauser, Basel, 2005.

[2] Coddington E. A. and Levinson N., Theory of Ordinary Differential Equations, McGraw-Hill, New York New York, state, United States
New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of
, NY, 1955.

[3] Mahmoud S. F., Electromagnetic Waveguides: Theory and Applications, Peter Peregrinus Ltd., London, UK, 1991.

[4] McCartin B. J., 2004, Eigenstructure of the Equilateral Triangle, Part III: The Robin Problem, International Journal of Mathematics and Mathematical Sciences International Journal of Mathematics and Mathematical Sciences (IJMMS) is a biweekly refereed mathematics journal. It was founded in 1978 by Lokenath Debnath. External links
  • Current website
  • Website prior to 3 March 2001
, Vol. 2004(16), pp. 807-825.

[5] McCartin B. J., 2007, Eigenstructure of the Equilateral Triangle, Part IV: The Absorbing Boundary Condition, International Journal of Pure and Applied Mathematics, Vol.37(3).

[6] Morse P. M., Vibration and Sound, Acoustical Society of America The Acoustical Society of America (ASA) is an international scientific society dedicated to increasing and diffusing the knowledge of acoustics and its practical applications. History
The ASA was instigated by Wallace Waterfall, Floyd Watson, and Vern Oliver Knudsen.
, Melville, NY, 1976.

[7] Morse P. M. and Feshbach H., Methods of Theoretical Physics, Part I, McGraw-Hill, New York, NY, 1953.

[8] Morse P. M. and Ingard K. U., Theoretical Acoustics acoustics (ək`stĭks) [Gr.,=the facts about hearing], the science of sound, including its production, propagation, and effects. , McGraw-Hill, New York, NY, 1968.

[9] Strauss W. A., Partial Differential Equations partial differential equation

In mathematics, an equation that contains partial derivatives, expressing a process of change that depends on more than one independent variable.
: An Introduction, Wiley, New York, NY, 1992.

[10] Zettl A., Sturm-Liouville Theory, American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, which it does with various publications and conferences as well as annual monetary awards to mathematicians. , Providence, RI, 2005.

Brian J. McCartin

Applied Mathematics, Kettering University The university boasts that the majority of its' seniors are employed or accepted to graduate schools before graduation and that one out of 15 alumni either own their own business or are high-level managers in leading companies (see Notable Alumni).  

1700 West Third Avenue, Flint flint, mineral
flint, variety of quartz that commonly occurs in rounded nodules and whose crystal structure is not visible to the naked eye. Flint is dark gray, smoky brown, or black in color; pale gray flint is called chert.
, MI 48504-4898 USA

Table 1: Branch Points and Critical Angles

      n    [[delta].sup.-.sub.n]     [[theta].sup.-.sub.n]

      0    2.106196+1.125364[??]           .7150[pi]
      1    2.178042+1.384339[??]           .6500[pi]
      2    2.214676+1.551574[??]           .6172[pi]
      3    2.237591+1.676105[??]           .5970[pi]
      4    2.253497+1.775544[??]           .5832[pi]
      5    2.265277+1.858384[??]           .5730[pi]
      6    2.274400+1.929404[??]           .5653[pi]
      7    2.281699+1.991571[??]           .5591[pi]
      8    2.287689+2.046852[??]           .5541[pi]
      9    2.292704+2.096626[??]           .5499[pi]
     10    2.296970+2.141891[??]           .5464[pi]
    100    2.346069+3.229037[??]           .5074[pi]
  1,000    2.354804+4.373567[??]           .5010[pi]
 10,000    2.356019+5.524184[??]           .5001[pi]
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Author:McCartin, Brian J.
Publication:Global Journal of Pure and Applied Mathematics
Geographic Code:1USA
Date:Apr 1, 2007
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