NAS Technial Memorandu 102024 

Parmelic Study of Elecromagnetic 

Waves Propagtin in Absoring 

Curved S Ducts 

Kenth J. Baumeistr 

Lewis Resarch Center 

Cleveland, Ohio 

July 1989 

I. 

(NASA-TM-102024) PABBHmRIC STUDY OF N89-279 23 

ELECTROHAGPETXC WAVES PROPAGATING IN 

ABSORBIHG CDBVED S DUCTS (NASA. Lewis Research Center) 25 p CSCL 203 Unclas 

I G3/32 0224999 
PARAME 

Na t 

RIC 

ona 

STUDY OF ELECTROMAGNETIC WAVES PROPAGATING 

IN ABSORBING CURVED S-DUCTS 

Kenneth J. Baumeister 

LevJis Research Center 

Cleveland, Ohio 44135 

SUMMARY 

Aeronautics and Space Administration 

A finite-element Galerkin formulation has been developed to study attenuation of transverse magnetic (TM) waves propagating in two-dimensional S-curved 

ducts with absorbing walls. In the frequency range where the duct diameter 

and electromagnetic wave length are nearly equal, the effect of duct length, 

curvature (duct offset), and absorber wall thickness was examined. For a given 

- offset in the curved duct, the length of the S-duct was found to significantly 

,” affect both the absorptive and reflective characteristics of the duct. 

7 straight and a curved duct with perfect electric conductor terminations, power 

attenuation contours were examined to determine electromagnetic wall properties associated with maximum input signal absorption. Offset of the S-duct 

was found to significantly affect the value of the wall permittivity associated 

with the optimal attenuation of the incident electromagnetic wave. 

For a 

INTRODUCTION 

Electromagnetic propagation in curved ducts (wave guides) plays an important role in many practical physical systems. In microwave power generation 

systems, for example (Kennedy, 1985, pp. 315-317>, bends or corners are 

required to alter the direction of the wave. Since curves or corners represent discontinuities, reflection from bends can be significant. Recent work 

(Burkholder, 1987 and Chou, 1987) has been concerned with the absorption of 

electromagnetic waves in curved ducts. 

In the present paper, a finite-element Galerkin formulation has been 

developed to study wave propagation in curved S-shaped ducts with absorbing 

walls. Figure 1 illustrates the various processes of reflection and absorption which can occur when electromagnetic energy flows into a curved duct. 

This paper will focus on the interaction of a propagating duct mode traveling 

down the uniform entrance duct with the curved walls as shown in figure 1. The 

reflection and transmission at the entrance and exit of the curved duct are 

determined by coupling the finite-element solutions in the curved duct to the 

eigenfunctions of the infinite, uniform, perfectly conducting entrance and 

exit ducts. This permits a mu1 timodal representation accounting for 

reflection and mode conversion by the nonuniform curved section. 

In a number of parametric studies, the effect of duct length, curvature 

(duct offset), and wall thickness are examined. In addition, for a PEC (perfect electrically conducting) termination, a computational search is used to 

determine the wall properties which lead to a local maximum attenuation of the 

incident electromagnetic wave for both a straight and curved S-duct. 
NOMENCLATURE 

property term, equation (20) 

property term, equation (21) 

mode amplitude of positive going entrance waves, equation (8) 

mode amplitude of reflected negative going entrance wave, equation (8) 

mode amplitude of positive going exit waves, equation (11) 

character 

d i mens ion 

d i men s i on 

stic duct height 

II 

ess entrance height ba/b 

I1 

ess exit height bb/b 

speed of light in vacuum 

dimensionless frequency, equation (7) 

I1 

x component of magnetic intensity, Hx/Ho 

normalizing magnitude of magnetic intensity 

finite-element approximation to Hx 

wave number 

axial modal wave number, equations (9) and (10) 

dimensionless length, L'/b, 

mode number, equation (13) 

number of modes in expansion, equation (13) 

power of positive going entrance wave 

power in axial direction 

dimensionless thickness of absorber wall 

dimensionless time, 5 t' I 

ba 

2 
I 

X dimensional transverse distance, x'/ba 

I 

Y dimensionless transverse distance, y'lb, 

Z dimensionless axial distance, Z'/ba 

E total complex permittivity, equation (4) 

El permittivity 

I1 

dielectric constant in entrance duct, &a/&O 

I 

EO permittivity in vacuum 

I 

Er complex dielectric constant, 

I 

Er imaginary part of dielectric constant 

R 

Er real part of dielectric constant 

I 

ET total imaginary part of permittivity, equation (4) 

intrinsic impedance 

wavelength 

n 

x 

total permeability, equation (5) 

permeabi 1 i ty 

permeability in vacuum 

I 

Pr relative permeability, 1-1 /po 

I 

Pr imaginary part of relative permeability 

R 

Pr real part of relative permeability 

a' ba 

dimensionless conductance, - 

co "0 

w' b; 

dimensionless angular velocity, - 

cO 

3 

w 
W' angular velocity 

Subscripts 

a entrance region 

b exit region 

Superscripts 

I dimensional quantity 

u approximate quantity 

GEOMETRIC MODEL 

In the finite-element modeling, an S-shaped profile has been chosen to 

approximate the two-dimensional cross-sectional profile that might be found in 

a typical bend, as shown in figure 2. The S-shaped profile can be prescribed 

by a simple third-degree polynomial of the form 

2 

y = h[3@ - 2 ($1 

where the dimensionless duct coordinates are defined as 

I I I 

(1) 

I 

and ba 

h is the offset height of the duct. The S-curve defined by equation (1) has 

zero slope at z/L of 0 and 1; providing a smooth transition from a straight 

entrance to the curved test section. In the foregoing equations, the prime, I, 

is used to denote a dimensional quantity and the unprimed symbols define a 

dimensionless quantity. This convention will be used throughout this report. 

These and all other symbols used in the report are defined in the nomenclature. 

is the height of the straight duct leading into the curved duct and 

GOVERNING EQUATION 

For time harmonic (e+jwt> variations in the electromagnetic field, the 

governing differential equations are the standard Maxwell's equations which 

can be combined to form a single variable property wave equation for the transverse magnetic wave propagation (Silvester, 1983, pg. 48, eq. 4.07). For 

a two-dimensional duct, the scalar form of the wave equation can be written as 

(Baumeister, 1988, eq. 13) 

4 
- 6 (a) + 6- (&) + w 2 pHx = 0 

6Y E6Y 62 E6Z 

(3) 

where the total permittivity including conduction (Cheng, pg. 300 or Harrington, pp. 24 and 25, eq. (1-74) and (1-76)) and the total permeability (Harrington, pg. 25, eq. (1-77)) are defined as 

R Io R I 

E=& -j(er+-)=& r -jET r 

w 

the various parameters are real positive quantities with 

II 

ba 

g=- 

00 

I1 

CE 

and the dimensionless frequency f is defined as 

f'bg bg 

f=--- 1-1 w = 2nf 

(4) 

(7) 

Because of computer storage limitations, the present finite element formulation 

will be restricted to structures where the duct height is the same order of 

magnitude as the wavelength. 

UNIFORM DUCT ANALYTICAL SOLUTIONS 

The analytical solutions of equation (3) for wave propagation in the uniform perfectly conducting duct having an anechoic entrance and exit will be 

employed to give the proper termination boundary condition for the finiteelement region. The analytical solution of equation (3) for TM waves traveling between perfectly conducting parallel plates is given as (Cheng, 1983, 

pg. 458): 

For the e+Jwt time dependence used 

propagating in the positive z direct 

moving in the negative z direction 

here, 

on wh 

+ 

the An represents a wave 

le A, term represents a wave 

5 
The axial wave number kzn in equation (8) is 

=+k 1- ((n - ~)w)~ ((nbakl)w) - < , 

kzn bak 

((n - 1)w)' - ((n - 1)w) 

bak 

kzn = -jk 

bak 

(9) 

(10) 

A similar solution exists at the exit, except only positive going waves 

are considered 

BOUNDARY CONDITIONS 

A variety of boundary conditions will be used in the finite-element solution of equation (3) for the model problems which are displayed in schematic 

form in figures 3 and 4. 

discussed. 

Each of the required conditions will now be briefly 

Input Conditions 

+ 

The analysis assumes a given number Nm of propagating An modes 

(eq. 8). These modes effectively set the level of the magnetic field in the 

finite element region and can be viewed as the equivalent Dirichlet boundary 

condition required for the elliptic boundary value problem as defined by 

equation (31. 

The modal expression represented by equation (8) has been truncated to a 

total of Nm modes of the infinite number possible. Thus, a total of Nm 

unknown modal amplitudes AT, Ai ,. . . ANm have been introduced. Nm 

constraint equations will be required to determine each of these unknown 

reflection coefficients. The equations used to determine these coefficients 

will now be introduced. 

Continuity at Inlet and Exit 

The tangential component of an H field is continuous across an 

interface between two physical real media which are not perfect conductors 

(Kraichman, 1970, eq. (1.61)). Thus, the boundary condition becomes 

., 

6 
where Hxa is the modal representation of the magnetic field in the 

analytical inlet region given by equation (8) and Hx represents the 

finite-element numerical solution to be discussed in detail in a following 

section of this report. 

u 

At the inlet to the curved section, shown in figures 3 or 4, the Hxa in 

the analytical region given by equation (8) must match the magnetic field 

defined by the finite-element nodal points along the boundary interface. Many 

possible mode matching schemes can be employed for this boundary condition, 

such as point collocation, least squares, or weighted residuals. A weighted 

residual approach was used herein with the weighting function equal to the 

eigenfunctions for the uniform infinitely long duct with perfectly conducting 

walls; 

=o at z=O (13) 

(Nm equations, m = 1, 2, 3,. . . Nm) I 

Equation (13) represents Nm separate equations; one for each coefficient 

defined in equation (8). The symbol m has been introduced as a dummy variable to make it distinct from the multiple n mode numbers that make up the 

Hxa analytical function. A similar situation exits at the exit. 

In addition to the tangential component of the magnetic field, the tangential component of the electric field must also be continuous across the interface (Cheng, 1983, eq. (7-52(a))>. Using Maxwell's equations to express the I 

tangential electric field in terms of the magnetic field (Jordan, 1968, 

eq. (7-4)) yields 

I 

(14) 

Perfectly Conducting Wall Conditions 

At a perfectly conducting wall surrounding the duct or at the exit plane 

in figure 4, the tangential component of the electric field vector is zero 

(Cheng, 1983, eq. (7-52(a)) or Kraichman, 1970, eq. (1.69)). Again, using Maxwell's equations to relate the electric field to the magnetic field (Jordan, 

1968, eq. (7-4>>, the component of the gradient of the magnetic filed normal 

to a perfectly conducting wall becomes 

(15) 

7 
FINITE ELEMENT THEORY 

The finite element weak formulation of the wave equation (3) is now 

generated by using the weighted residual approach with the Galerkin approximation. The continuous domain D is first divided into a number of discrete 

triangular areas as shown in figure 5. The finite element aspects of converting equation (3) and the boundary conditions into an appropriate set of global 

difference equations can be found in text books (Burnett, 1987, pp. 561-5641 

and for conciseness will not be presented herein. An exact adaption of this 

standard finite element theory to the duct propagation problem can be found in 

an earlier work (Baumeister, 1988). 

RESULTS AND COMPARISONS 

A number of example calculations are now presented to illustrate the use 

of the code as applied to curved ducts with absorbing walls. 

duct geometries and element arrangements used in the numerical examples are 

discussed. Next, the effect of offset and duct length on the transmitted and 

reflected energy are examined for a fixed wall absorption layer. Then, the 

effect of absorber thickness is considered. Finally, for a PEC exit termination, the power attenuation contours for a curved and straight duct which 

maximize input signal absorption are examined. 

First, typical 

Duct Geometry 

The grid generation package generates the geometries shown in figure 6, 

where a typical straight duct and curved duct with maximum offset (h = 1) are 

shown. The absorber has been placed above and below the duct. All the duct 

geometries are of this form but with different lengths, offsets and absorber 

thickness. The linear triangular finite element grid associated with this 

geometry is shown in figure 7. 

Example 1: Transmitted Power 

The effect of duct curvature and length on transmitted power in an infinite duct are examined in figure 8 for various values of duct offset. 

metrical configuration is shown by the inserted sketch in figure 8. In this 

case, the entrance and exit ducts stretch from minus infinity to plus infinity 

which signifies the absence of reflected energy at the numerical terminations. 

The wall properties were taken to be 7 1.00-2.83j and pw = 4. These 

properties are associated with nearly maximum absorption of a plane TM wave in 

a straight duct at the frequency of unity. For a fixed length of duct, as 

seen in figures 8(a>, (b), and (c), an increase in the duct offset parameter 

h increases the attenuation of the transmitted electromagnetic power (Poynting vector, cross product of the electric and magnetic fields times duct area, 

Baumeister 1988 pg. 23 or Cheng 1983, pg. 330) at the exit of the curved lined 

portion. This effect is most pronounced for the smaller duct lengths as shown 

in figures 8(a) and (b). The magnetic intensity fields inside the duct are 

illustrated in figure 9 for the duct with 0.75 length. As seen in figure 9(a) 

the magnetic intensity beams directly through the duct with grazing contact 

along the absorbing wall till it reaches the exit with very little 

The geo8 
attenuation. In contrast, in figure 9(e> the magnetic field comes in nearly 

normal contact with the wall and quickly dies out. Returning to figure 8(a>, 

note that the dimensionless power drops below the input value of 1 for large 

values of the offset parameter h. This signifies increased reflection of the 

curved portion of the duct. This reflection will now be examined. 

Example 2: Reflected Power 

The effect of duct curvature and length on the reflected power from the 

curved portion of a semi-infinite duct are now examined in figure 10 for various values of duct offset. Again, the geometrical configuration is shown by 

the inserted sketch in figure 10. To highlight the reflected energy in example 

in figure 10, the termination at the exit is a PEC (perfect electric conductor) plate which reflects all the energy which reaches the exit back 

towards the infinitely long entrance duct. 

For a fixed duct length, as seen in figure 10, increasing the liner offset 

h generally decreases the reflection coefficient. However, for ducts with 

lengths of 0.75 and 1.0, an optimum is reached. In these cases, the discontinuity of the curvature appears to reflect significant energy back down the duct 

before it can be absorbed. Figure 11 shows the magnetic fields inside with the 

length of 0.75. For the straight duct the interaction of the field is clearly 

seen in figure ll(a>. In contrast, the field in the duct with large offset 

(h = 1) are comparable to those in figure 9(e>, because the level has been 

severely reduced by absorption into the duct walls. 

Example 3: Absorber Thickness I 

In this section the effect of absorber thickness is briefly examined. 

The configuration considered is again shown by the sketch in the upper portion 

of figure 12. In this case the duct length and offset will be held fixed at 

unity and the wall absorber thickness will be varied. As seen in figure 12 

for thickness of 0.1 or greater, the power variation along the axial length of 

the duct remains unchanged. However, for thickness of 0.05 and 0.01 significant decreases in the power absorbed are seen. I 

, 

Example 4: Axial Discretization in Wall Absorber I 

In performing finite element solutions, the number of axial nodal points 

should be held to a minimum to reduce computer storage as well as solution 

time. This is especially true in the next example where determining the wall 

properties associated with optimum signal absorption requires hundreds of separate calculations starting with large values of wall permittivity and 

permeabi 1 i ty . 

re 

mi 

For linear elements, roughly twelve grid points per wavelength are 

uired to accurately resolve the complex electromagnetic field and the transted or reflected electromagnetic power. The wave length for plane wave 

propagation in lossy material will now be determined from a solution of the 

wave equation. 

equation (3) reduces to 

For plane waves propagating in a homogeneous lossy material, 

9 
6z (%) 6z + o2&pHX = 0 

The solution of which can be written as 

H=He -UZ e -jpz 

X 0 

where 

2 

p=- [ 6- + B]1’2 

2 

RR I1 

T r - ‘rpr B=E~ 

(16) 

(17) 

(18) 

(19) 

(21) 

and the wavelength can be expressed in terms of the propagation phase constant 

13 as follows: 

(22) 2n A=- P 

The number of grid points I in the axial direction to accurately resolve 

the electromagnetic field is 

L 6PL I=12-=-- 

xn (23) 

ax i i1 d i recti on. 

would be required. 

to wavelength rat 

For large va 

For the special case where the duct length is just equal to the axial wavelenath of the electromagnetic wave, twelve nodal points are required in the 

If the duct length is twice the wavelength then 24 nodes 

Thus the number of nodes is just 12 times the duct length 

0. 

ues of the permittivity and permeability associated with 

wall absorbers, the phase constant 

which leads to smaller values of the wavelength according to equation (22) and 

a considerably larger number of axial nodes according to equation (23). 

ever, a plane wave incident obliquely at an interface with a denser medium 

will be bent toward the normal (Cheng, 1983, pg. 353, eq. 8-124(c>> making the 

increases according to equation (19) 

How10 
energy transfer in the wall absorber normal to the direction of the energy 

transfer in the duct itself and thereby partially reducing the axial energy 

transfer in the wall absorber. Since the axial attenuation in the duct wall 

will be much larger than in the air duct, consideration is now given to basing 

the axial grid spacing on the wavelength of the air duct rather than the wall 

absorber wave length. In this case the axial grid point spacing would be 

1=12-- - 12 Lf L 

'air 

(24) 

If such an approximation is valid, larger savings in computer storage are 

pos s i b 1 e. 

The validity of equation (24) in predicting energy attenuations and magnetic fields is now examined for two extremes of wall properties. First the 

moderate value listed in figure 13(a) and the much larger value listed in 

figure 13(b). As seen in the upper portion of figure 13(a), axial spacing 

based on the wall properties using equation (23) results in a much denser grid 

than the lower figure based on the wavelength in air using equation (24). For 

the large wall property value, the grid density is even greater as seen in the 

upper portion of figure 13(b). For the duct configuration shown in the upper 

portion of figure 14, exact calculations for duct attenuation displayed 

in figure 14 indicate that the axial spacing based on air properties, equation 

(24) gives nearly the same results as the axial spacing based on the wall material, equation (23). In addition, the contour plots of the magnetic field 

shown in figure 15(a> and (b) show that the magnetic field in the duct is for 

all practical purposes identical when either equation (23) or (24) is employed 

to set the axial grid point density. Therefore, in the next optimization 

example, equation (24) will be used to set the axial grid spacing. 

Finally, the sensitivity to the number of transverse nodes in the wall 

absorber is determined in figure 16. As seen in figure 16 at least 5 nodes 

are required to accurately estimate the attenuation in a 0.1 thick absorber 

coating with the same range of property variations previously considered. 

Example 5: Attenuation Contours 

Optimizing the wall absorber for maximum attenuation as well as minimizing duct reflection can be an important part of the design of an electromagnetic duct suppressor. In duct acoustics, for example, the maximum possible 

attenuation occurs at the so-called optimum impedance. 

tic mode (analogous to an electromagnetic mode) or more generally for modes 

with common cut-off ratios, the optimum impedance can be determined analytically from semi-infinite duct theory using a single soft-wall mode (Rice, 

1979). Generally, the optimum impedance for a finite length liner will be 

lower than the single mode liner due to generation of higher order modes at 

the leading edge of the finite length liner (Unruh, 1976 and Baumeister, 1984). 

For a particular acousSince electromagnetic propagation and attenuation is analogous to acoustic propagation, an electromagnetic wall impedance should exist which can be 

expected to maximize absorption by the duct wall absorbers. However, in the 

present computer code, the wall properties are specified rather than wall 

11 
impedance. Consequently, the local attenuation optimum will be determined for 

specific values of wall properties rather than the more general global optimum 

analogous to Rice, 1979. 

Consider a plane wave propagating down a duct and incident upon the 

curved section with a PEC exit plane, as shown in the schematic of figure 17. 

The object of the present example is to estimate the wall properties wh ch 

will minimize the reflected signal transmitted back down the duct or 

conversely to maximize the energy absorbed by the wall absorber. Start 

with a single plane wave traveling down the duct and initially assuming 

Dermeabilitv of 4.1 in the curved absorber section of the duct, the ref 

ng 

a wall 

ected 

energy was determined throughout the permittivity plane shown in figure 17. 

With increments of 0.5 taken in the real and imaginary parts of the permittivity, 400 separate calculations were performed at the equally spaced nodes. 

From these calculations, the attenuation contour plots were established using 

i near i n terpol at ion be tween the cal tu 1 ated poi n t 5'. 

The local optimum wall value of the permittivity is seen in figure 17 

occur at a total permittivity of 5.0 - j 5.0. The coordinates of figure 1 

are the real positive values of both and defined in equation (4) 

In this case, the optimum wall permittivity associated with maximum signal 

absorption is represented by the peak contours enclosed in the smallest ci 

of figure 17(a>. For this first iteration, the intrinsic (or characterist 

R I 

impedance of the material squar 

portion of figure 17(a>. 

The maximum attenuation of 

the local optimum point as disp 

dB is expressed in terms of the 

eq. 8) as 

to 

(a) 

cle 

C> 

d is 0.41 + j 0.41 as shown in the upper 

the incoming wave is 23.903 dB associated with 

ayed in figure 17(a> where the attenuation in 

transmitted power Px (Baumeister, 1986, 

dB = 10 loglo % ['I (25) 

The maximum and minimum attenuation in absolute values of dB are listed in figure 17(a> as well as each figure which follows. The dB contours have been normalized in figure 17 as well as the following figures by the simple 

equation: 

(26) 

The second step in the iteration process to determine the optimum wall 

properties is to hold the permittivity fixed and vary the permeability. In 

this case, the optimum wall value of the permeability is seen in figure 17(b> 

12 
to occur at a total permeability of 6.0 - j 8.5. For this second iteration, 

the intrinsic impedance squared has shifted to 1.45 j 0.25 and the maximum 

attenuation has increased to 28.73 dB. 

The third and final iteration is shown in figure 17(c) in which the opt 

mum value of wall permeability was held fixed at the value of 6.0 - j 8.5 

determined during the second iteration. In the third iteration, the optimum 

properties remained fixed at the values determined during the second iterati 

indicating a local convergence over the range considered. Figure 17(c> differs from figure 17(a> in that attenuation can now occur for zero values of 

the complex permittivity since the permeability is now a complex quantity 

which can contribute to the attenuation of the incoming wave. 

For the same geometry as considered in figure 17, the sensitivity of the 

initial starting point on the electromagnetic reflective energy contours is 

examined in figure 18. In this case the initially fixed value of permeability 

is 2.0 rather that 4.1 as used in figure 17(a). A comparison of figure 17(a) 

to figure 18(a> indicates that the first estimate for the optimum has been 

shifted to the right in figure 18(a>. The optimum in the second iteration 

falls slightly off the range of parameters considered. However, the maximum 

attenuation of 27.945 at a permeability of 10 - j 10 and permittivity of 9 - j 

5.5 is nearly equal to the optimum attenuation of 28.73 dB displayed in 

figure 17(c>. Apparently, the optimum contours have slowly varying slopes. 

However, in both cases considered, the intrinsic impedance of the medium are 

nearly equal. Consequently, it may be inferred that any combination of material properties with a similar intrinsic impedance should yield similar values 

of attenuation. 

the same 

attenuat 

than the 

opt i mum 

C1 early, 

the wall 

on has a maximum value of 22.113 dB which is approximately 6 dB 

curved duct considered in figures 17 and 18. More significantly 

ntrinsic impedance has shifted to a new value of 0.1 + j 0.7. 

the curvature of the duct will play a significant role in determ 

materials to obtain the maximum attenuation. 

Figure 19(a> and (b> displays similar results for the straight duct of 

length as considered in figure 17. In this case the local optimum 

ower 

the 

ni ng 

CONCLUDING REMARKS 

A finite-element Galerkin formulation was developed to study transverse 

magnetic (TM) wave propagation in two dimensional curved S-shaped ducts with 

absorbing walls. The derivation from Maxwell's equations assumed that the 

material properties could vary with position resulting in a nonhomogeneous variable property two-dimensional wave equation. This eliminated the necessity 

of finding the boundary conditions between the air duct and the absorbing 

walls. The reflection and transmission at the entrance and exit of the curved 

duct are determined exactly by coupling the finite-element solutions in the 

curved duct to the eigen-functions of an infinite, uniform, perfectly 

conducting duct. 

Example solutions illustrated the relationship of absorption on the 

length, thickness and offset of curved duct absorbing walls. Also, attenuation 

contour plots were presented for PEC ended curved ducts. Local optimum wall 

13 
values of permittivity and permeability as well as the intrinsic impedance 

associated with the optimum were found to be significantly different for 

straight and curved ducts with PEC exits. 

REFERENCES 

1. Baumeister, K.J.: Electromagnetic Propagation in PEC and Absorbing Curved 

S-Ducts. NASA TM-100833, 1988. 

2. Baumeister, K.J.: Finite Element Analysis of Electromagnetic Propagation 

in an Absorbing Wave Guide. NASA TM-88866, 1986. 

3. Baumeister, K.J.; and Horowitz, S.J.: Finite Element-Integral Acoustic 

Simulation of JTl5D Turbofan Engine. J. Vibration, Acoustics Stress, 

Rel. Design, vol. 106, no. 3, July 1984, pp, 405-413. 

4. Burnett, D.S.: Finite Element Analysis. Addison-Wesley, Reading, MA, 

1987. 

5. Burkholder, R.J.; Chuang, C.W.; and Pathak, P.H.: Electromagnetic Fields 

Backscattered from an S-Shaped Inlet Cavity with an Absorber Coating on 

its Inner Walls. (FR-715723-2, Ohio State University; NASA Grant 

NAG3-476) NASA CR-182401, 1987. 

6. Cheng, D.K.: Field and Wave Electromagnetics. Addison-Wesley, 1983. 

7. Chou, R.; Ling, H.; and Lee; S.W.: Reduction of the Radar Cross Section 

of Arbitrarily Shaped Cavity Structures. (UILU-ENG-87-2560, UILU-TR-87-6, 

University of I l l i n o i s a t Urbana-Champaign; NASA Grant NAG3-475) NASA 

CR-180307, 1987. 

8. Harrington, R.F.: Time-Harmonic Electromagnetic Fields. McGraw-Hill, 

9 

10 

11 

1961. 

Jordan, E.C.; and Balmain, K.G.: Electromagnetic Waves and Radiating 

Systems, 2nd ed., Prentice-Hall, 1968. 

Kraichman, M.B.: Handbook of Electromagnetic Propagation in Conducting 

Media. NAVMAT P-2302, U.S. Government Printing Office, Washington, D.C., 

1970. 

Kennedy, G.: Electronic Communication Systems, 3rd ed., McGraw-Hill, 1985. 

12. Rice, E.J.: Optimum Wall Impedance for Spinning Modes - A Correlation 

with Mode Cutoff Ratio. J. Aircraft, vol. 16, No. 5, May 1979, 

13. Silvester, P.P.; and Ferrari, R.L.: Finite Elements For Electrical 

14. Unruh, J.F.: Finite Length tuning for Low Frequency Lining Design. 

pp. 336-343. 

Engineers. Cambridge University Press, Cambridge, England, 1983. 

3. Sound Vibration, vol. 45, no. 1, Mar. 8, 1976, pp. 5-14. 

14 
I NC 1 DENT 

POWER 

TTED 

LINED WALL 

FIGURE 1. - REFLECTIVE AND ABSORPTION CHARACTERISTICS OF LINED 

S-DUCT. 

z 

FIGURE 2. - S-DUCT GEOMETRY AND COORDINATE SYSTEM. 

rPERFECT CONDUCTING 

INTERFACE 

I 

LANALYTICAL I 1z=o REGION 

REG I ON I 

COUPLE ANALYTICAL 

AND NUMERICAL 

THEORIES AT 

INTERFACES 

FIGURE 3. - TWO DIMENSIONAL S-DUCT FINITE ELEMENT MODEL. 

,-PEG EXIT CONDITION 

r PERFECT CONDUCTING 

=L 

INTERFACE 

I I REGION 

Iz=O 

LANALYTl CAL 

REG I ON 

COUPLE ANALYTICAL 

AND NUMERICAL 

THEORIES AT 

INTERFACES 

FIGURE 4. - TWO DIMENSIONAL S-DUCT FINITE ELEMENT MODEL WITH PEC EXIT PLANE. 

15 
r ENTRANCE 

1 DUCT 

FIGURE 5. - ELECTROMAGNETIC BOUNDARY VALUE FORMULATION. 

(A) STRAIGHT DUCT 

(B) CURVED DUCT. 

FIGURE 6. - ABSORBING WALL DUCT CONFIGURATIONS. 

16 
(A) STRAIGHT DUCT. 

(B) CURVED DUCl 

FIGURE 7. - DISCRETIZATION OF AIR FILLED WAVE GUIDE WITH 

ABSORBERS MOUNTED ALONG BOTH UPPER AND LOWER WALLS. 

17 

2- h/ba 

0 

0.25 

0.50 

(A) L = 0.75. 

(B) L = 1.0. 

2 

AXIAL POSITION, x 

(C) L = 2. 

FIGURE 8. - EFFECT OF DUCT OFFSET ON THE 

MAGNITUDE OF THE AXIAL POWER AS A FUNCTION Of POSITION (WALLS EU 1. - j2.83: 

vu= 4.1 AND f = 1). 
RELATIVE 

WAGN I TUDE 

0 1.0 

A .8 

U .6 

D .4 

0 .2 

6 0 

- 

- 2 

I 1 

(A) h/b, = 0. 

(B) h/b, = 0.25. 

(D) h/b, = 0.75. 

(C) h/b, = 0.5. 

(E) h/b, = 1.0. 

FiGURE 9. - EFFECT OF OFFSET OF ABSORBING UALL ON THE CONTOURS OF THE MAGNETIC FIELD WIIH NONHEFLECTiNG EXIT FOR L = 0.75 AND f = 1. 

18 

1.0 

.1 

.01 

t 

h/ba 

FIGURE 10. - EFFECT OF DUCT OFFSET h ON THE AXIAL POWER 

REFLECTION COEFFICIENT AS A FUNCTION OF DUCT LENGTH 

(E" = 1. - 12.83; = 4.1 AND f = 1). 
(A) h/b, = 0. 

(B) h/b, = 0.25. 

(D) h/b, = 0.75. 

RELATIVE 

MAGNITUDE 

0 1.0 

A .8 

D .6 

D .4 

0 0 

.2 

(C) h/b, = 0.5. 

(E) h/b, = 1.0 

FIGURE 11. - FFfICT OF OtFSET OF ABSORBING WALL ON THE CONTOURS OF THE MAGNFlIC FIELD WITH PEC EXIT CONDITION 

(L = 0.75. f = 1). 

-_--___ --- --__-- 

-.5 0 .5 1.0 1.5 2.0 

AXIAL POSITION, x 

FIGURE 12. - EFFECT OF ABSORBER THICKNESS ON THE MAGNITUDE 

OF THE AXIAL POWER AS A FUNCTION OF POSITION (Ew = 

1. - j2.83; Pw= 4.1). 

19 
AX BASED ON WALL 

PROPERTIES 

Eg= 1.5-3.5) 

CI, = 4.1 

AX = 0.02564 

AX BASED ON AIR 

PROPERTIES 

E=l CI=l 

AX = 0.0909 

(A) MODERATE CASE = 1.5-3.5j. 

AX BASED ON VALL 

PROPERTIES - AX = 0.0119 

AX BASED ON AIR 

PROPERTIES 

E=l u=1 1 AX = 0:0909 

(6) LAHbE CASL EW= 10.0-10.0). 

FIGURE 13. - DISCRETIZATION OF AIR FILLED WAVE GUIDE WITH 

ABSORBERS MOUNTED ALONG BOTH UPPER AND LOWER WALLS. 

20 

,-ABSORBERS 

- ,-Ew = 10.0-lO.Oj 

,' pw = 4.1 I 

/ 

h, v 0 

t 

AIR SPACING 

I 

I 

0 .02 .04 .06 .08 .10 

AXIAL ELEMENT WIDTH. AX 

FIGURE 14. - ELECTROMAGNETIC ENERGY TRANSMISSION ATTENUATION AS A FUNCTION OF AXIAL NODE SPACING IN ABSORBER 

AND DUCT FOR CURVED DUCT (f = 1. L = 1. h = 1 WITH 5 

VERTICAL NODES IN ABSORBER REGION). 
AX = 0.02564 

BASEO ON WALL 

PROPERTIES 

E, = 1.5-3.51 

P" = 4.1 

AX = 0.0909 

BASED ON AIR 

PROPERTIES 

(A) MODERATE CASE E, = 1.5-3.5) P, = 4.1 

AX = 0.0119 

BASEO ON WALL 

PROPERTIES 

E, = 10.0-lO.0j 

P, = 4.1 

40 

m 30 u 

V + 

3 

L AX = 0.0909 

BASED ON AIR L 20 

PROPERTIES 0 

s 

F 

L 

+ 

10 

(B) LARGE CASE E, = 10.0-10.0 j Pw= 4.1. 

FIGURE 15. - CONTOUR PLOTS OF MAGNETIC FIELD AMPLITUDE. 

ABSORBER T< 

h= 1 I 

/ - LEw = 1.5-3.5j 

LI, = 4.1 

AX = 0.02564 

- ,-E, = 10.0-lO.0j 

/' p, = 4.1 / 

AX = 0.0119 

0 2 4 6 a 10 

NUMBER OF VERTICAL GRID POINTS IN 0.1 THICK ABSORBER 

FIGURE 16. - ELECTROMAGNETIC ENERGY TRANSMISSION ATTENUATION AS A FUNCTION OF VERTICAL NODES IN ABSORBER FOR I 

CURVED DUCT (f = 1. L = 1. h = 1). I 

21 
0.1 

1 

I I I I I 

310$6T 1 , I , I 5 

-1 

REAL PART PERREABILITY 

(B) SECOND ITERATION - p SEARCH (E= 5.0-5.Oj). 

I I 

= 1.45-0.253 

I 1 I 

PEC 7 

I 1 I I 1 

PEC 7 = p/E = 0.162 + 0.098j 

/ ABSORBER 7. '\ 1 3 

1 

1 

dBMAX = 21.158 

--? dB,,, = 0.0 .L 

1 

(A) INITIAL ITERATION - E SEARCH (I&= 2.0) 

l1 

p/E = 1.30 - .3146j 

\ 

5 

3 :I dB,,,, = 27.945 

dB,,, = 10.253 

'tin 0.1 /------ 

-1 

0 2 4 6 8 10 

(B) SECOND ITERATION - p SEARCH(€= 9.0-5.5j). 

FIGURE 18. - SENSITIVITY OF INITIAL STARTING POINT ON ELECTRGWGNETIC REFLECTIVE ENERGY CONTOURS FOR CURVED DUCT 

WITH PEC EXIT (f = 1, L = 1. h = 1). 

22 
11 QOpt = p/& = 0.082 + ,5741 

\ 

I I 

I > 

t 

I I 1 

.2 

= lp------ 0.1 

E 20.405 

0.0 

5 :F 

BMAx = 22.113 

B,,,,,, = 7.090 

REAL PART PERMEABILITY 

(8) SECOND ITERATION - p SEARCH (E= 1.0-7.Oj). 

STRAIGHT DUCT WITH PEC EXIT ( f = 1. L = 1, h = 0.0). 

FIGURE 19. - ELECTROMAGNETIC REFLECTIVE ENERGY CONTOURS FOR 

23 
National Aeronautics and 

Space Administration 

1. Report No. 

NAS TM-102024 

Report Documentation Page 

2. Government Accession No. 3. Recipient's Catalog No. 

4. Title and Subtitle 

Parmetric Study of Electromagnetic Waves Propagtin in 

Absoring Curved S Ducts 

5. Report Date 

7. Author@) I 8. Performing Organization Report No 

9. Performing Organization Name and Address 

Nationl Aeronautics and Space Adminstraion 

Lewis Resarch Center 

Clevland, Ohio 4135-3191 

2. Sponsoring Agency Name and Address 

Kenth J. Baumeistr 

10. Work Unit No. 

505-62-21 

11. Contract or Grant No. 

13. Type of Report and Period Covered 

Technial Memorandu 

1 E-4761 

Nationl Aeronautics and Space Adminstraion 

Washington, D.C. 20546-0001 

14. Sponsoring Agency Code 

~ 

5. Supplementary Notes 

5. Abstract 

A finte-elmnt Galerkin formulation has ben devlopd to study atenuio of transverse magnetic (TM) waves 

proagtin in two-dimensoal S-curved ducts with absorbing wals. In the frequncy range wher the duct 

diametr and elctromagnetic wave length are nearly equal, the efct of duct length, curvature (duct ofset), and 

absorber wal thicknes was examined. For a given ofset in the curved duct, the length of the S-duct was found 

to signfcatly afect both the absorptive and reflctive charcteristics of the duct. For a straight and a curved 

duct with perfct elctric conductor terminations, power atenuio contours were examined to detrmine 

elctromagnetic wal properties asocited with maxiu input signal absorption. Ofset of the S-duct was found 

to signfcatly afect the value of the wal permitvy asocited with the optimal atenuio of the incdet 

elctromagnetic wave. 

Unclasifed - Unlimted 

Subject Category 32 

7. Key Words (Suggested by Author@)) 

Finte elmnts 

Electromagnetic waves 

Coatings 

9. Security Classif. (of this report) 

Unclasifed 

20. Security Classif. (of this page) 21. No of pages 22. Price' 

Unclasifed I 24 I A03 

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