608
Xovder
R,eflection Factor of Gradual-Transition Absorbers for
Electromagnetic and Acoustic Waves*
KLAUS WALTHER?
Summary-Absorbers for electromagnetic or acoustic waves are someoscillations a s l / X increases. A satisfactory perdescribed, for which a good impedance match and low reflection
formance of absorbers can be specified by the requirefactor can be achievedby providing a gradual transition of material
well designedsound
and
constants into the lossy medium. The reflection factor can be calcu- ment r S 10 percent.For
microwave
absorbers
this
condition
can
be
realized
in
lated by means of a Riccati-merential equation. General conclusions from the WKB-perturbationmethodcanbedrawnforabthe range l / X > O . 2 . For low frequencies a largelayer
sorbers, the layer thickness of which is either very small or very
thickness 1 would thus be required. One idea for the
large in comparisonto the wavelength.For “thin” layers,wave
reduction of the IIX-cutoff value is to apply lossy maenergypenetratesthewholethickness
of theabsorber.Suitable
terials with a high index of refraction. Thus, the waveaveragevalues of the material constants are derived to describe
below
only lengthwithinthematerialcanbereducedfar
the performance of the panel in this case. For “thick” layers
the initial partof the panel is energized. The asymptotic expressions the free-field value.4rtificiallyrefractivemediafor
containonlythematerialconstants
of this part. The results are
microwaves and sound waves are well
interpreted physically. Numerical solutions of the reflection factor
This paper derives the differential equations for the
for highly refractive panels with exponentially varying materialconelectromagnetic
and acoustic inhomogeneous absorber.
stants are reported.
I. IYTRODGCTION
T
Some general conclusions for absorbers, which
are either
very thin or very thick in comparison t o X, are drawn.
The performance of gradual-transitionabsorbers
is
thenillustratedbyseveralnumericalsolutionsfor
highlyrefractivepanels,inwhichthematerialconstants vary according to exponential functions.
HE PROBLEM of absorbingelectronlagnetic
andacousticwaves
is important for manyapplications in measuring techniques, for the construction of anechoic chambers, and for camouflaging
targets in radar and sonar detection. Only plane waves,
which propagate in a direction perpendicular to a plane
11. THERICCXTI-DIFFERENTIAL
EQUATION
FOR THE
absorbing surface, shall be considered here. The cases
GRADUAL-TRANSITION
ABSORBER
of electromagnetic and acoustic waves can be treated
in
-4. ElectromagneticWaves
complete
analogy.
Assuming
normal
incidence
and
isotropic
media,
the
differential
equations
become
The theory of electromagnetic wave propagation in
identical in both cases. Various basic principles can be
a n inhomogeneous dielectric medium can be found in a
utilized in constructing absorbers. A good summary of paperbyBarrarandRedheffer.4Consider
a medium
these can be found in
a paper by Aieyer and Severin.’
inwhich
thematerialconstants
€(x) =€’(x) -je”(x)
One objective in absorber design is to achieve a low and p(x) =p’(x) - j p ” ( x ) dependononecoordinate
x
reflectionfactoroverafrequencyrange
as largeas
only. €(x) .and p ( x ) are, respectively, the relative dielecpossible and simultaneously to keep the thickness 1 of tric constant and permeability referred
to the free-space
theabsorbinglayerverysmallincomparisontothe
values € 0 and po. A linearly polarized plane wave of time
free field wavelength X. This paper considers the prindependence exp ( j w t ) is assumed to propagate in parallel
ciple of thegradual-transitionabsorber:
A layer of to the x axis. Then Maxwellian equations read
lossy material covers
a perfectlyreflectingplanesurdHz(x)
face. By varying the material constants within the
lossy
-- - jCIJ€oe(x)E,(x)
panel, a good impedance match between the free propaax
gationmediumandtheabsorbingstructurecanbe
achieved.Such a gradualtransitioncanberealized
either by arranging a homogeneouslossymaterialin
the form of wedges and pyramids or by using layers of
where
varying material constants.
If the absolute amount r of the amplitude reflection
&,(x) =electric field strength
factorfor a gradual-transitionabsorberisplottedas
H,(x)= magnetic field strength.
a function of the ratio ID, the reflection decreases with
1
* Receivedbythe PGAP, April 14,1960; revised manuscript
received, June 16, 1960.
iResearch Labs. Division, Bendix Corp., Southfield, Mich.
1 E.
RIever and H. Severin.
“AbsorDtionsanordnuneen
fur
elektromagn&scheZentimeterwellenund
‘ihre akustischcn .AnaPhys., col. 8, pp. 105-11-1; March, 1956.
logien,” 2. mgm~.
W. E. Kock, “1Metallic delay lenses,” Bell Sys. Tech. J.,vol. 27,
pp. 58-82; January, 1948.
3 IT.E. Kock and F. K. Harvey, “Refracting sound waves,”
J.
Acousb. SOC.
Ant., vol. 21, pp. 471481; September, 1949.
R. B. Barrar and R. M. Redheffer, “On nonuniform dielectric
ON ANTENNAS AND PROPAG.4TION, vol. AP-3,
media,” IRE TRANS.
pp. 101-107; July, 1935.
Authorized licensed use limited to: UNIVERSITY OF VICTORIA. Downloaded on October 14, 2008 at 16:33 from IEEE Xplore. Restrictions apply.
Walther: Reflection Factor of G!rudual-Transition Absorbers
1960
An electric wave impedance Z,(x) can be defined for
e\Fery position x within the layer:
609
density, referred t o t h e \ d u e s K O and po for the homogeneous medium in front of the absorber.A plane sound
wave of timedependenceexp(jot)shallpropagate
in
parallel to the x axis. Then the soundfield equations for
small amplitudes can be written
The minus sign in this definition is necessary because of
the choice of the coordinate system in Fig. 1. differential equation satisfied byZ,(x) can be derived by n1eans
of (1) :
r\ I
1 dH,
H, ds
dx
=
(3)
jqLop(x) - j W E O E ( X ) Z e y x ) .
where P(r)=sound pressure and V ( x ) = particle velocity.
The acoustic wave impedance Z a ( r ) is defined
P(2)
Za(S) = - -.
The amplitude reflection factor Re(x)is defined
v (x)
-
where 2 0 = dp,,/e0 = 377Q = wave impedance
space.Eq.
(3) maybetransformedintoaRiccatidifferential equation for X,(x) :5--i
of
free
A differentialequation
(6):
for Z , ( x ) can be derived from
= jupop(x) - ~ w K , ~ K ( s ) Z , ~ ( P ) .
(8)
The acoustic amplitude
reflection factor X,(X) is given
by
where X,
length.
= 2~/o
(~
~ p , J is
- ~the
i ~ electricfree space wave-
R,(x)=
-
ABSORBING
PANEL
z
Za(4 + z’
Za(.) -
where Z = d p O / ~ is
O the wave impedance of the homogeneous medium in front of the absorber. Eq. (8) may
be transformed into a Riccati-differential equation for
R a ( x ):
I
MEDIUM
I
E
x=o
E-!
Fig. 1-Inhomogeneous
X.1
X
E-0
absorbing panel of thickness 1.
B. A caustic Wuoes
The theory of acoustic wave propagation in a n inhomogeneousmediumcanbefound
i n a paperby
Bergman.8 The material constants K ( X ) = K’(x) $!’(x)
and p ( x ) =p’(x) - j p ” ( x ) of theacousticmediumare
assumed to depend on one coordinate x only. K(X) and
p ( x ) are, respectively, the relative compressibility and
R. M. Redheffer, ‘Reflection andtransmissionequivalences of
dielectric media,” PROC.
IRE, vol. 39, p. 503; May. 1951.
6 R.hI. Redheffer, “Sovel uses of functional equations,” J . Rat.
Xeech. Anal., ~ o l3,
. pp. 271-259; March, 1954.
R. %I. Redheffer.“Limit-periodicdielectricmedia,”
J . -4ppl.
P h ~ s . \-01.
,
2 i , pp. 1136-1140; October, 1956.
* P. G. Bergman, ‘The wave equation in a medium with a variable
index of refraction,’ J . ilcoust. SOC.dm.,col. 17, pp. 329-333; .%pril,
1946.
“electric”an
where X, = ~ T / ~ ( K O ~ O ) - ~is! ?the acoustic wavelength in
the homogeneous medium in front of the absorber.
C. The Analogy
Since ( 5 ) and (10) are identical in form, the reflection
factor X for the electromagnetic and acoustic cases can
be evaluated by the same differential equation:
between electric and acoustic
quantities
is used.
+
According to (4) and (9) a boundary condition R= 1
corresponds to a “magnetic” wall (surface of zero magnetic field strength) in the
electromagnetic
case
and
to
a rigid Ll.all in the acoustic case. ~h~ condition R = - 1
requires
plate)
wall (metal
Authorized licensed use limited to: UNIVERSITY OF VICTORIA. Downloaded on October 14, 2008 at 16:33 from IEEE Xplore. Restrictions apply.
or a pressure
IRE TRANSACTIONS ON ANTENiVAS
PROPAGATIOS
AND
610
Xmember
R , ( x ) a t x = x o f l corresponds to the reflectionfrom
a n inhomogeneouslayer,inwhichthewholeregion
~ 2 x 0
hasbeenreplacedbyahomogeneousmedium
of material constants f(x0) and g(x0) (Fig. 3 ) . Since in
general f ( x 0 ) and g(x0)Z 1, theintermediatepoints
R,(xo) according tothe Schelkunoffmethod
donot
111. SOME
G E N E R ~PROPERTIES
L
OF THE REFLECTIOS representphysicallyinterestingsolutionsfortheabFACTOR
FOR THE GFL~DCAL-TRANSITION ABSORBER
sorber problem. On the other hand, the intermediate
A . Interpretation of the Rejection Factor f o r a n Inhomo- points of the Redheffer solution (Fig. 2) may give useful information on the efTect of cutting away part of
genous Medium
the panel (reduction of layer thickness, effect of a n iniThegeometry of theproblemunderconsideration
tial step). For the point
x = I the resultsof both methods
is shown in Fig. 1. An absorbing panel is terminated a t are identical, if layers with no initial step are considx=O by a totally reflecting wall. A t the front surface
ered. An example of the Schelkunoff method for calcux = 1 the material constants shall be the same as for the lating the reflection factor of gradual-transition sound
homogeneous propagation medium, f (x) = g ( x ) = 1 for absorbers
can
found
paper
beaWe
in
by
x>l. Thegradualtransition
of materialconstants is prefer t o use the Redheffer method in this paper.
f(x) and
representedschematicallybythefunctions
g(x). 4 solution R ( x ) can be computed from (11) with
TERMINATING
the proper initial condition at x = 0. The point R(Z) is
the reflection factor of the total layer. A physical interpretation of the solution R ( x ) is possible also for a point
HOMOGENEOUS
xo#Z. T h e definition of R ( x ) in (4) is due to Redheffer.5
T h e Redheffer-reflection factor R(x0) represents the reflectionfrom a n inhomogeneouspanel,inwhichthe
region x. _<x51 has been replaced by
a homogeneous
medium of material constants f ( x ) = g ( x ) = 1 (Fig. 2).
7---releasesurfaceinthetwocases,respectivelq-.Interchangingf(x) and g(x) in (11) is equivalent to changing
the sign of R ( x ) . According t o (4) and (9) thiscorrespondstoreplacingimpedancebyadmittanceand
vice versa. This is the well-known principle of duality.
‘.
x=
TERMINATING
x=o
Fig. 3-Interpretation
x0
X=R
X
of the Schelkunoff reflection factor a t x = x o .
PANEL
f(x); g(x1
‘\
\
B. Behavior of Panels, Thin Compared to Wavelength
HOMOGENEOUS
MEDIUM
Integration of theRiccatiequation
(11) normally
leads tononelementaryfunctionsandhastobeaccomplishednumerically;suchresultsarereportedin
Section IV. Somegeneralconclusionscanbedrawn,
i
however,
for the limiting cases
of a layer thickness I
x=o x = x - x 0
R
X
which is either very small or very large in comparison
Fig. ?-Interpretation of the Redhefferreflectionfactor a t Z = X O . to the wavelength X.
1) Taylor-Series Solution of theRiccatiEquation:
For
the case 1/x<<1 the quantity r(x) = R(z) may be
A different approach to the problem
of wave propagacalculated
from (11) intheformofaTaylorseries,
tion in a n inhomogeneous medium may be found by destarting
at
the point x=O:
fining the reflection factor R,(x) :
I
I
instead of using (4). Kayg associates this approach with A similar method has been applied by Sampson.ll Resultsshallbegivenhereonlyfortwosimplecases
the name of Schelkunoff (see additionalreferencesin
(Table I). These can be characterized physically by the
his paperg) and compares several aspects of the “Redfacts:
Case 1 shows high energydissipationnearthe
heffer”and“Schelkunoff”methods.Thedefinitionin
terminating
surfacex = 0. In Case 2 a ZOW energy dissipa(14) leads t o a different Riccati equation for R,(x), requiring f ( ~ )and g(x) to bedifferentiable. A solution
9 A. F. Kay, “Reflection of a Plane
\\’ave by a Stratified Medium.” iLIcI\rIillan Lab., Inc.. Ipswich, Mess. Tech. Rept., Contract
AFlS(600)-1044; November 22,1954.
10 N. B. Miller, “Reflections from gradual transition sound absorbers,” J. Acoast. SOC.Am., vol. 30, pp. 967-973; October, 1958.
11 J. H. Sampson, “Some Absorbing Panels with \Tanable DielectricComposition,”kIcMillanLab.,Inc.,Ipswich,Mass.,Tech.
Rept., Contract 4F18(600)-10444;December 20,1954.
Authorized licensed use limited to: UNIVERSITY OF VICTORIA. Downloaded on October 14, 2008 at 16:33 from IEEE Xplore. Restrictions apply.
T-ABLE I
I
Case 1: High Energy Dissipation S e a r x = O
Case 2: Low Energy Dissipation S e a r x=O
i
Electromagnetic \\-aves
i
a) Electric losses-Magnetic wall
Magneticb)
losses-Electric
wall
a) Electric losses-Electric wall
b j RIagnetic losses-Magnetic
wall
Acoustic 1Vaves
~
~
~
~
a ) Compressibility losses-Rigidwall
bj Density (friction) losses-Pressure
l
release \\-all
tion
near
terminating
the
surface
x = 0 prevails. T h e
energy ;V dissipated per unit volume is proportional to:
for electromagnetic waves:
-
-I-,
E2
+
HS
f
T.72.
a) Compressibility losses-Pressure release wall
b j Density (friction) losses-Rigidwall
.h
r(x)
A
’
LOW ENERGY
DISSIPATION
for acousticwaves:
-I-,
N
.pz
I a) HIGH ENERGY
DISSIPATION
I n the following cases we assume that only one type of
loss mechanism is present, putting eitherf(x) org(s) = 1
0
i n (11). T h e possible “simple” cases are summarized in
x/x
Table 1.
Fig. 4-Initial behavior of the reflection factor ~ ( x )
The cases thatcan be realized most easily i n abfor “thin” absorbers: s/h<<l.
sorber design are Case 2a) for microwaves, Case la) for
waterborne sound, and Case2b) for airborne sound. The
Taylor-series expansion of (15) shall be given for Casesfied.
For ver)l ccthil17! layers energ). is penetrating the
la) and 2a) as
examples: f(x) = E “ ( x ) ,
full thickness I of the
absorber.
Therefore,
can
it be
ex-
Case l a ) :
pectedphysicallythatcertainaveragevalues
material
constantsf(x)
and
g ( rform
) in the
of the
of integrals
jif(x) .wl(r)dxand j i g ( x ) .w2(x)d.v are moreappropriate for the description of very “thin” layers. These
x
relations will now be derived.
2 ) W K B Solution of the Wave Equation: T h e WKB
.$+ . . . (16) perturbationmethod for obtainingapproximatesolutions of the inhomogeneous wave equation has proved
Case Za) :
useful inproblems
man~7
of mechanics
quantum
and
wave propagation.’? Wefollow a treatise by Zharkovskii
16a3
and Todes,13 but we consider a more general case. Sim‘V 1 - __ ~ ( 0 ) ~ 3
3x3
paper
approximation
ilar
a found
in methods
becan
by
Pipes.]?
The wave equationfor the electric field strength E(.$)
(17)canbederivedfrom
( l ) ,wherethecoordinatetransformation 4 = 1- x / l has been used (see Fig. 1):
These curves are shown in Fig. 4. For the case of high
d z E d(lnp(t)) dE
energy dissipation a sharp drop
of T ( X ) near x = 0 can be
-.( W W ) P ( t ) E ( t )= 0 , (18)
observed.Inthe
case of low energydissipationthe
dl’
dE
dt
curve r(r) startsfromthepoint,
r = 1, with a horizontal slope, the deviation being given bya third-order
See, e.g., references in C. 0. Hines, “Reflection of waves from
term.
media,”
varying
Quaut. J . Appl. Musth.,
pp.vol. 11,
9-31; April, 1953.
l 3 A. G. Zharkovskii and 0. M. Todes, “Reflection of waves from
It
be emphasized that the convergence Of the a n isotropicinhomogeneouslayer,‘! Smiet Phys. J E T P , vol. 1,pp.
series (15) is verg; poor. Moreover, this approach is not 701-703; J ~195;.~ ~ ,
very satisiactory for a physical interpretation,since’
li L. A . Pipes,
Computation
of the impedances of nonuniform
lines by a directmethod,”
T a n s . A I E E , vol. 75, pp. 551-554;
only functional values and derivatives a t x = 0 are speci- xovenlber, 1956.
Sa
.r(x) ‘v 1 - - €’l(O)X
+
Authorized licensed use limited to: UNIVERSITY OF VICTORIA. Downloaded on October 14, 2008 at 16:33 from IEEE Xplore. Restrictions apply.
where k l = 2 d / X . A solution of this equation
in theform:
is sought
+E--exp ( - j k l . l E F s ( t ) d t ) .
After some mathematical manipulations, including removal of multipleintegrations, (25) canberewritten
for thenormalizedinputimpedanceandmaybeinterpreted physically for simple cases.
Assuming p ( $ ) =l, theresult,includingterms
of
order (KZ)3, reads
(19)
The functionsf,($) and Fs(l) satisfy the following firstorder differential equations:
where the weighting function
Z L , I ( ~ )=
3(1 - t)';
solw(t)dt=
1
(22
has been introduced. If this result is written in the form
and
For the caseof very thin layerskl<<l we use asymptotic
expansions of the form:
i t is equivalent to the input impedance of a homogeneous layer with average dielectric constant.
m
fs(t)
=
C kmfs,m(t)
=
(22)
JO1€(t)W1(t)dt.
(29)
m=O
1
I n the case €(E) = 1, the expansion of the normalized
input impedance may be written.
and
oc
k"Fs,m(O.
Fs(E) =
(23)
m=O
Eqs. (20) and (21) are solved bya perturbation method.
An approximate solution E ( t ) satisfying the short-circuit boundary condition at $ = 1 can thus be derived:
In order t o show the physical significance of this resultmoreclearly,
we consider a layerinwhichthe
permeability ~ ( 5 differs
)
b37 only a small amount
v($)
from the homogeneous value p : E*(,$) =p+v($). For this
case we can rewrite (30) in the form
- =j k l .
where
where the error A
This expansion is correct including terms of order (KZ)3.
Byapplication
of hiaxwell'sequations
(1) thenormalized input admittance for the front surface E = 0 of
the layer may be derived:
is a second-order term inY (4) and can be neglected,whenever v(4) <<p. Thus, the almost homogeneous layer
canberepresentedwithin
a good approximation b y
means of two weighting functions:
I
1
w&) = 1
Hi
-= j k l - J o l ( c
YO
- p)&'
-j[l
(33)
+ 2(kZ)2
which are different for the two degrees of expansion in
(W.
Authorized licensed use limited to: UNIVERSITY OF VICTORIA. Downloaded on October 14, 2008 at 16:33 from IEEE Xplore. Restrictions apply.
Walther: Rejlectiorr Factor of Gradztal-Transition
1960
These analogies with the homogeneous case suggest
the definition of two average valuesof the permeability:
6 13
bsorbers
TERMINATING
HCYCGENEOUS
MEDIUU
Gi)?
=
6Jo1iuiC)
s
‘df’)Q‘f’dE.
(36)
Using these values, ( 3 0 ) may be rewritten
Zi
-
zo
jkrp
1
+ j.--.( k 0 (c)?
-
jkb.p’
+ki.7 + ...
( 37)
3
This is a well-known result for the homogeneous case,
where p = F = p .
Theweightingfunctions
w1([) through wa([) for a
short-circuit termination a t ( = 1 are plotted in Fig. 5.
A physical interpretation suggests that wl([) has to be
zero near f = 1, because in a region of low electric field
strengthadielectricconstant
of thematerialcannot
be effective. From (27) i t couldbeconcluded
that a
variation of dielectric constant
E(f)
,-
(1 -
( 38)
0 - 2
would result i n a constantvalue of the weighteddielectric constant e(<) .wl(l) throughout the layer and
should have favorable applications in absorber design.
Such a function (38) was considered by Jacobs.’j This
author arrives at ( 3 8 ) by attempting to find a function
E ( [ ) , for which the fractional change l/e-de:.lrl[ referred
to the L‘local” wavelength X(f)
is the same
small constant a for all positions [ within the layer:
=Xidm
1
__.-2irE(f)
a€
df
(f )
-
x
1
2ir
€3[2
d€
Q’t
=
a
<< 1.
x
This statement may be similar to our interpretation of
(38). Because of the singularity of ( 3 8 ) a t i = 1 we have
to be cautious, however, in drawing too many conclusions from perturbation theorp.
Fromasimilarphysicalreasoningtheweighting
function w2([)should be unityneartheshortcircuit
E = 1, since in a region of maximummagnetic field
strengththepermeabilitycanbefullyeffective.The
higherorderweightingfunction
w3([)shoulddecrease
with the distance from the terminating wall, since the
magnetic field strength decreases. The functions wl(E)
and w3(t) should thus have a n opposite trend.
Assunling frequency independent material constants
E and p , theimpedancecurvescorrespondingto
(28)
and (37) are shownschematically in Fig. 6. Forthe
case of electric losses (curve 11, the deviation from the
imaginary axis is given by a third-order term in w . In
the case of magnetic losses (curve Z), the impedance
15 I. Jacobs, “The nonuniform
transmission line as a broadband
termination,“ Bell Sys. Tech. J., vol. 37,pp. 913-924; July, 1958.
Fig. 4-Impedance curves for
(a)7,
< YC,
(b)
“thin” absorbers.
Fm = I:,,
(c)
7 , > I-..
Curve 1: electric losses, short-circuit termination.
- -.
Curve 2: magnetic losses. short-circuit termination; tan $ =p”/p‘.
Cuwe 3 : conductivity losses, short-circuit termination.
Curve 4: conductivity losses. open-circuit termination.
l’q =averaged value of the “area-conductance.’
Yc= critical valueof the “area-conductance. ’!
curve starts
at an angle
J/ off the imaginary axis, where
tan J/ = p 1 ” / 3 .
The reflection factors corresponding to (28) and ( 3 7 )
are
These functions are shown in Fig. 7, curves 1 and 2.
A comparisonwiththeinitialparts
of (16) and (17)
shows that the averaged values of e’’ and p” have been
substituted for the initial values a t 4= 1. [Instead of
(16) the dual case has to be considered.]
Authorized licensed use limited to: UNIVERSITY OF VICTORIA. Downloaded on October 14, 2008 at 16:33 from IEEE Xplore. Restrictions apply.
6 14
Fig. 8-Approximate equivalent circuit for the “thin’! conductive layer with short-circuit termination.
input admittance Yi/Yois written instead of Zi/Z, the
following resultcanbeobtaineddirectlyfrom
(37):
-
0
I/x
Fig.7-Initialbehavior
of the reflection factor IRI for “I :hin” absorbers: l/k<<l.
Curve 1: electric losses, short-circuit termination.
Curve 2: magnetic losses, short-circuit termination.
Curve 3: conductivity losses, short-circuit termination.
Curve 4: conductivity losses, open-circuit termination.
By inserting the valueE = E’ - j ( c r / u o ) the expansion including linear terms in frequency can be written.
yi
.-?.
I.’, - j Q [ s - < ! ! i ) 2
- 71,
(44)
-
Yo Yo
w
A different result is obtained if frequency dependent
losses aretakenintoaccount.A
loss mechanism,in
which the loss factor is proportional to u-l, has special
interestforverythinpanels
at low frequencies.Examples of such loss mechanisms are: 1) ohmic conductivity u in the case of electromagneticwavesand
2)
viscous friction in the case of acoustic waves (porous
absorbers). Only the first case shall be considered here.
Substituting the expressionfor Ell= ( T / u O into (28)
gives the following result for the short-circuit case:
where Q = ( w / c ) l is thenormalizedfrequencyand
=J;~‘(t)dt. Twoaveragedvaluesforthe“area-conductance” of the layer have been introduced:
y _- -a l ,
y
=
where
‘ii = Jolr(.$‘)dk
3 , where (F)?= 6
d
(45)
Jo’I.(O
~ t l d F ’ ) d l l d i . (46)
T h e normalized input impedance Z i / Z , corresponding
to (44) is plotted as curves 4a through 4c in Fig. 6. All
these curves start with a finite real value of the input
impedance Z i = Ro = 1/y0.The reactive behavior depends on thevalue of theaveragearea-conductance
I’lo with respect to a critical value
-
Here the normalized frequency Q = (w/c)I and the averaged “area-resistance”
= l / d of the layer have been
introduced,where a = j & u ( [ ) .wl([)dE is theaveraged
conductivity. Thereflection factor correspondingt o (41)
reads
xu
These relations are shown graphically as curve
3 in
both Fig. 6 and Fig. 7. The deviation of the impedance
curve 3 (Fig. 6) from the imaginary axis is given by a
second-order term in w and thus is much more effective
than in the case of curve 1. Fig. 7 shows that thereflection factor for curve 3 drops much faster with increasing l / x as compared to curve1. T h e low-frequency properties of a“thin”conductivelayerwithshort-circuit
termination
can
be
described
qualitatively
by
the
equivalent circuit of Fig. %.I6
Thecase of a conductivelayerwithopen-circuit
termination a t E = 1 gives considerably different results.
If €(E) is substituted instead of p ( [ ) and the normalized
l6
Ibid., p. 916.
Y , = Y o .4
7.
(4i)
Low conductive layers with f,> Y,show a capacitive
component; highly conductive layers with f 5 Y,are
inductive. In the case E?,= Y,, the first-order reactive
component is zero. From higher-order approximations
the frequency dependence of curves 4a through 4cin
Fig. 6 can be derived. The equivalent circuit of Fig. 9
qualitatively shows the same behavior if the resistance
R 2is varied with respect to the wave impedance
dLa/C2.
T h e reflection factor corresponding to(44) starts with a
finite value
(Rlo
-RZO- Zo!
I & + Zol
These facts are familiar from the theory of the Salisbury screen, for which the area-resistance is chosen t o
be R, = 377 Q in order to match the absorber. The opencircuit termination in the caseof electromagnetic waves
can be realized in
a resonance condition by placing a
resistive foil a t a distance of X/4 in front of a metal
plate. A “magnetic”wallcan
only beapproximated
Authorized licensed use limited to: UNIVERSITY OF VICTORIA. Downloaded on October 14, 2008 at 16:33 from IEEE Xplore. Restrictions apply.
TValther: Rejection Factor of Gradual-Transition Absorbers
INPUT
the terminating wall f
YRMINYON
= 1 can
615
thus be derived:
I T
Fig. +Approsinlate equivalent circuit for the “thin”
conductive la>-er with open-circuit termination.
roughly over a limited frequency bandbelow the microwaverange b>- usingmediawith
\ pI >>I e \ (forinstance, ferrites).
The analogous acoustic case would be a porous absorber i n waternearapressure
release surface.Such
porousmaterials(forinstance
fiberglass) arevery effective as absorbers for airborne sound. However, the
achievablefriction losses withporousframeworksor
similar structures for waterborne sound are much more
limited.l’.ls The reason for this is partlJ- that a motion
of theabsorbingstructurecannotbeavoided
a t low
frequencies, thus the achievable friction is limited. &\Ithough the acoustic analog of a “magnetic” wall can be
realized easily, for the reasons mentioned, the required
layer thickness of the absorber would be rather large.
Therefore the analysis for the “thin” layer cannot be
appliedinthiscase.
T h e normalized input admittance for the front surface E = O canbecomputedbymeans
of Maxwell’s
equations (1) :
Consideringalayerwithoutaninitialdiscontinuity,
E = , U = 1 for f = 0, (51) can be simplified:
-j-cotan ( k l .
s,
’dGdt)
C. Beha.zior of Panels, Thick Compared to Wavelength
Section 111-B dealtwiththetheory
of the“thin”
layer.Theanalysisandinterpretationhavetobe
rather detailed, because energy penetrates the full layer
thickness 1. If the other limiting case of panels, thick
compared to wavelength, is considered, we should expectphJ-sicallyasimplersituation.Inthiscaseonly
the initial part of the la>-er near the front surface f = 0
is energized. The multiple reflectionsfrom the terruinating wall 4 = 1 hecome more and more insignificant a s
t h e frequency increases.
The analysisof the “thick” layer startsi n a way t h a t
is analogous to the derivations
i n Section 111-B, 2).13
solution of the lvave equation (18) is sought i n the
form of (19), with ( L O ) and (21)being thesame.Instead of theasl;mptoticexpansions
(22) and (23) we
use
(49)
for the case kl>>l. T h e differential equations (20) and
(21) are solved by a perturbation method. ,\ solution
E(E) satisfying the short-circuit boundarl- condition at
E. Meyer and K. Tamm, “Breitbandabsorber fur Flussigkeitsschall.” Acustica, 1701. 2 , Beiheft 2, pp. AB 91LA3B109; 1952.
l a K. Tamm, “Broad-hand absorbers for water-borne sound,” in
E. G. Richardson, “Technical Aspectsof Sound,” Elsevier Publishing
Co., Kew York,X. Y . , vol. 2, ch. 6, pp.24Ck286; 1957.
where
T h e real and imaginary parts of p ( 4 ) and E ( [ ) have been
substituted from (12) and (13).
Consider the case of frequency independent material
constants e([) and p ( 4 ) first. Then the impedance curve
correspondingtothefirstterm
in (52) runs to zero
along a straight line in the complex plane a s w--t x . T h e
second term in (52) represents the short-circuit input
admittance of a transmission line with an averaged refractionindex:
According to our assumptions, no initial discontinuity
of waveimpedanceforthisequivalenttransmission
line is present at E=O. From transmission line theory
i t is well known that the corresponding impedance plot
is a spiral in the complex plane, approaching the point
IT;/ II’,= 1 i n a clockwise sense as a+ x . In (52) both
the real and imaginary part of the argument for the
cotan
function
increase
proportional
to
frequency.
Therefore, the distance between any point of the spiral
Y.:;I’o -- - j c otan(kI.j,,’dG@)
and the limiting point
Y i / Y o=
1 vanishes according to an exponential function of general form exp ( -const l / X ) . Since the first
+
1.
+
Authorized licensed use limited to: UNIVERSITY OF VICTORIA. Downloaded on October 14, 2008 at 16:33 from IEEE Xplore. Restrictions apply.
termin (52) onlyvanisheslikeaninversepowerin
( l / X ) , this is the significant part for the calculation of
the reflection factor R( in the limitingcase w--t x .
Thus the following result can be obtained:
1
The
impedance
values
of the
function
-j-cotan
(klj:,,/Fdt) are located on a closed limit-curve ( w - ’ ~ )
around the point +1, which is determined by the finite
imaginary part in ( S i ) . Since the real part in (57) increases with o,rotations along this limiting curve continue with increasing l / X . T h e reflection factor, according to the cotan-term in (52), does not approach zero,
but shows fluctuations between two finite values which
are determined by the distance from the limiting-curve
to the point +l. For these considerations see also the
papers of Lenz and Zinke.19,20
T h e reflection factor vanishes like (l,/A)-’ for l / X + = .
The only quantities of the layer which appear in this
equation are the first order derivatives
of the material
constants at the front surface
5 = 0. Reflections from the
terminating wall can be neglected completely, since no
I\,’. NL-MERICAL SOLUTIONS O F RI FORABSORBERS
energy enters this part of the absorber. By application
WITH EXPOSENTIALLY
VARYING
of Schwarz’sinequality,Kayghasshown,generally,
MATERIAL CONSTANTS
thatthe Schelkunoff-reflectionfactor Rs (14) cannot
In this section some numerical solutions for the absovanishasaweakerpowerthanforthelimit
h+O.
lute amount r of the reflection factor of gradual-transiEq. (54) breaks down for the case
tion absorbers shall be reported. These solutions have
(11) b y
4.9 = d o ,
(55) been calculated from the differential equation
means of a Bendix analog computer. Some check solubecause then the first term in (52) vanishes. However, tions on an IBA,I 650 digital computer have been comin this case i t is easy to derive an exact solution for R puted also. The accuracyof the calculations is estimated
from (11):
to be A r = 2 1 per cent.
Lenzgocarriedoutmanyanalogmeasurementson
electrical models of gradual-transition absorbers, in order to determine optimum functions for the variation
of
thematerialconstantswithinthelayer.Inthese
Thus, an exponential decay
of RI forall values I/X
measurements
ohmic conductivity losses were studied,
should be expected.
the
real
part
of
the dielectric constant being unity. A
Eq. (52) shows t h a t , in the general case, both effects
short
circuit
termination
of the layer was assumed. In
are present simultaneously. The qualitative course
of
theinvestigations of Lenz,layerswithexponentially
1 with increasing l / X can thus be described as fol- varying
conductivity losses show afavorableresult.
lows. T h e reflection factor RI shows damped oscillaThe
condition
r < 10 percentcanbesatisfiedforall
tions around a decreasing average value. The decrease
I,’X>0.35.
Layers
withthreesections
of piece-wise
of the average level is determined by the combination
constal~t
conductivity
even
show
a
slightly
better reof a n inversepower term and an exponential term
in
s
u
l
t
:
r
<
l
O
per
cent
for
all
l/X>0.33.
Kumerical
calcul / X . Thisalsocanbe
seen byplottingsome
of the
lations
of
Sampson”
on
some
highly
refractive
dielectric
numericalcurves of Section IV onadoubleorsemifor exponentially
logarithmic scale. In neither of the two cases straight panelsalsoshowsomeadvantages
varJ-ing
material
constants.
lines areobtained.hIoreover,
(52) shows that b?- inFor these reasons we choose exponential functions t o
creasing one typeof material losses the oscillatory term
describe
thevariations of materialconstants in the
can be damped out faster with increasing1 / X . However,
numerical
solutions. For the first partof the calculations
at the same time the average level represented by the
[Figs.
10
through
18(b), pp. 613-6193 we consider “siminverse power term is raised. Therefore, a compromise
ple”
cases
according
to Table I, forwhich eitherf(s) = 1
in choosing material losses is necessary according to the
or
g(x)
= 1 in (11).As an example the functionss ( x ) and
selectedperformancecriterionfortheabsorber.
The
t ( x ) and the boundary conditions
for R(r) at r = O are
numerical solutions in Section IV shall illustrate these
specified
i
n
Figs.
10 through 18(b). The physical interstatements.
In the case of ohmic conductivity losses (54) is not pretation of theresultsforeitherelectromagneticor
(52). -Assuming acoustic waves can be accomplished easily by means of
valideither.Thiscanbeseenfrom
Table I and the analogy giveni n Section 11-C, including
p ( [ ) = 1, the argument of the cotan-function can be apthe principle of duality. We consider the general formof
proximated for very high frequencies:
I
1
IZI
I
19 K. L. Lenz and 0. Zinke, “Die Absorption elektromagnetischer
\Yellen in Absorbern und Leitungen mit abschnittweise homogenem
. pp. 481-489; October, 1957.
Leitaertsbelag,” 2. angew. Phys., 1-019,
20 K. L. L e u , “Leitungenmitortsabhangiger
Damfung zur
reflesionsarmen Absorption elektromagnetischer It7ellen, ‘’ Z. angev.
Phys., vol. 10, pp. 17-25; January, 1958.
Authorized licensed use limited to: UNIVERSITY OF VICTORIA. Downloaded on October 14, 2008 at 16:33 from IEEE Xplore. Restrictions apply.
Walther: Rejection Factor of Gra~uaI-Tra.nsitior1-4 bsorbers
1960
617
constants are assumed. The
nlaxilnumvalues of s(xj
vary
from
1
in
Fig.
1
2
to
100
in Fig. 18. The parts (a)
s(s) =
(38)
of Figs. 12 through 18 satisfy the boundary condition
f(.rj = b(l-r,’!) - d.
(39) R(0)= -1. Thiscasecanberealizedu-ithelectric
losses and a short-circuittermination for electromagA morespecialform
of equation (59) is alsoused
netic waves. For parts (b)
of Figs. 12 through 18 the
throughout the computations:
boundary condition reads R(0)= + l . An example for
f(*) = c[a(l--s!l) - 11 = c [ s ( x ) - 11.
(60) waterborne sound waves can be realized with compresThe coordinate system of Fig. 1 is used here. T h e real sibility losses in front of a rigid wall. Some numerical
part of g(s) increases from a value s(Z) = 1 at the front solutions for this case have been reported by AIiller,lo
describing linear transitions in material constants and
surface x = l to a maximum value
s(0) = a at the terother functions.
minating wall .x= 0. T h e loss part t(x) may have an
Theresults in Figs.12(a)through18(b)showthat
initial step a t x = / , if d Z 1 in (59).
the
smallest l/X value for which r < 1 0 per cent is acIn Fig. 10 two solutions are plotted for comparison.
complished
for theoptimumcase
of Fig.12(a)with
T h e solidlinerepresentstheelectromagneticcase
of
s
(
x
)
= 1 (see also Lenz20). For highly refractive panels
conductivity losses (t,-l!w) with a short-circuit termination. The dashed line corresponds to frequency inde- (Figs. 16 through 18) oscillations duetomultiplereflections are observed. The “wavelength”of these oscilpendent losses. T h e differences between the two cases
lations
decreases with increasing s(x), as we would esfor small /;X have been discussed in Section 111-B, 2 ) ;
pect.
Xarrow-band
resonance
absorption
for
some
also compare Fig. 7. T h e solid curve in Fig. 10 is very
points
can
be
accomplished.
However,
judged from the
close toanexamplegivenby
Lenz.?O T h e 1 / X cutoff
r
<
1
0
per
cent
criterion,
these
curves
look
inferior to
values for r 5 10 per cent are located between 0.35 and
the
result
of
Fig.
12(a).
The
general
behavior
of these
0.40.
curves
has
been
discussed
in
detail
in
Section
111-C
of
The solutions in Fig. 11 are to illustrate the electrothis
paper.
If
only
one
loss
mechanism
is
present
If(.)
magnetic case of a conductive layer with an open-circuittermination:compareFig.
7, curve 4. T h e diffi- = 1 in Figs. 12(a) through 18(b)], increasing the losses
does not improve the performance of the absorber. The
culties in realizing this case physically have been menoscillations
are damped out by this method, but the
tioned in Section 111-B, 2). The curves in Fig. 11 start
“average”
level
is raised at the same time (52).
with finite r values for l : / X = O . Reflection factors below
An
improvement
in the performance of highly refrac10 per cent are not accomplished over a wide l!h range
tive
panels
can
be
achieved
if two types of losses are
for low frequencies. For low conductivities (solid line
present
simultaneously
(for
instance,
electric and magin Fig. l l ) , afairlygoodimpedancematchcanbe
netic
losses).
T
h
e
difficulties
for
realizing
the desired
achieved
for
I,:’h=O; however,
the
attenuation
for
condition
f(x)
= g ( x ) in absorber design for electromaghigherfrequencies
is insufficient ( S i ) . Therefore,the
netic and acoustic waves are mentioned
in the literareflection factor increases with /,!X. Increasing the cont
~
r
e
.
’
.
?
PottelZ3
~
points
out
some
possibilities
of realizductivity (dotted line in Fig. 11)lowers the reflection
ing
the
condition
E = p for some anisotropic media
in
factor for higherfrequencies;however,asevere
misthe
microwave
range.
I
n
artificial
dielectric
media
both
match is caused for small llh values.
electricandmagnetic
losses arepresentsimultaneAnother
interesting
possibility
of combining
the
ously,
although
generalll:
el
p~ in the microwave
principles of theSalisbury screen andthegradual,?3--26 Porousmaterials
for absorption of sound
transition absorber has been pointed out by Deutsch
waves i n air (for instance fiberglass) show high friction
and Thust.?’ These authors suggest placing a resistive
foil infront of agradual-.transitionabsorber in order losses and a small component of compressibilit): losses.26
to estend the lox-frequency absorption range by meansPlastic materials are used in underwater sound absorpof a resonance effect. Kurtze?’ was able to improve the tion. Complex compressibilities and densities of the materials can be adjusted i n wide ranges by either introlow-frequency
response
of porous
sound
absorbers
ducing
cavities filled with different lossy materials or
(wedge-type)byprovidinganairspacebetweenthe
by
loading
the material with heavy, rigid obstacles, for
wedges and the terminatingwall.
instance
metal
disks, etc.1,3~10.1i,1~
In order to study the
effect of highly refractive panels
functions g(x) =s(sj -j.t(x):
I >>I
for theconstructionofgradual-transitionabsorbers,
several numerical solutions of r have been calculated.
These are shown as a function of the ratio 1,:’X in Figs.
12(a)through18(b). Frequent!, independentmaterial
J.Deutschand
P. Thust,“Breitbandabsorber
fur elektromagnetische \Yellen,’ 2. angew Ph.ys.. vol. 11: pp. 453455; December, 1959.
G. Kurtze, “Cntersuchungen zur \Terbesserung der Auskleidung
schallgedampfter Raume,” Ac.ustica, vol. 2, Beiheft 2, pp. -4B 104AB 107:1952.
?3 I<. Pottel, “.\bsorption
elektromagnetischerZentimeterwellen
in kiinstlich anisotropen Medien,“ 2. axgew. Ph.ys., vol. 10, pp. 8-16;
January, 1958.
J. RI. Kelly, J. 0. Stenoien,and D. E. Isbell,“\Yaveguide
measurements i n the tnicrowave region on metal powders suspended
in paraffin was,” J. d p p l . Phys., vol. 24, pp. 258-261; Narch, 1953.
25 E. Meyer, H. J. Schmitt,and
H. Sererin,“Dielektrizitats3 cm Welkonstant; und Permeabilitat kunstlicher Dielektrika bei
lenlange, Z. angew. Phys., vol. 8, pp. 257-263; June, 1956.
C . \\:. Kosten, “Behaviour of absorbing materials,” in “Technic a l Aspects of Sound,” E. G. Richardson, ed., Elsevier PublishingCo.,
New York. N.Y.. vol. 1. ch. 4. DD. 49-104; 1957
Authorized licensed use limited to: UNIVERSITY OF VICTORIA. Downloaded on October 14, 2008 at 16:33 from IEEE Xplore. Restrictions apply.
618
1.0
1.0
0.8
0.8
a6
r
r
0.4
0.6
0.4
0.2
0.2
0
0
0.2
0.4
x/!-
h8
0.6
0
1.0
0
Fig. 10-Reflection factor P for layers with electric losses and shortcircuit termination.
Solid line: ohmic conductivitylosses with t-o-'.
Dashed line: frequency independent electriclosses.
62
a4
a2
0.4
0.8
1.0
1.0
0.8
0.6
r
0.4
a2
0
0
0.6
0.8
1.0
J/X
0
Fig. 11-Reflection
0.1
0.4
6.8
0.6
i.0
(b)
Fig. 13-Reflection factor r for layers with o m type of frequency independent losses: g ( x ) = s ( x j - j . f ( x j ;
= l . (a) Boundary condition R(0)= - 1. (b) Boundary condltlon R(0)= +l.
factor r for layers with ohmic conductivitylosses
and open circuit termination.
1.0
LO
0.8
0.8
0.6
0.6
r
r
0.4
0.4
0.2
0.2
0
0
a
0.2
66
0.4
0.8
0
i.o
0.2
1.0
(a)
(a>
1.0
1.0
0.8
0.8
0.6
0.8
0.6
a.4
-0=3
- 0 -
0.6
5
r
r
0.4
a4
0.2
0.2
0
0
0
Fig. 12-Reflection factor r for layers with one type of frequency independent losses: g(x) =s(x) - j . t ( x ) ; , f ( x ) = l . (a) Boundary condition R(0)= - 1. (b) Boundary condition R(0)= +1.
0.2
0.4
0.6
0.8
1.0
Fig. 14-Reflection factor r for layers with o m type of frequency independent losses: g ( x ) =s(x) - - j . t ( x ) ; f(s)r l . (a) Boundary condition R(0j = -1. (b) Boundary condition R(0)= +1.
Waltlther: ReJlection Factor of Gradual-Tramition Absorbers
1960
1.0
1.0
0.8
0.8
619
0.6
0.6
c-
-----
r
r
0.4
0.4
0.2
0.2
c
-- 0.3
0.5
0
0
0
0.2
0.4
0.8
0.6
1.0
0
1.0
0.2
0.4
0.8
0.6
1.0
1.0
0.8
-a=
0.6
7
-c
0.6
r
-0.3 ’
r
0.4
a4
0.2
a2
0
0
0.2
a6
0.4
0.8
0
1.0
0
a.2
0.6
0.4
//X
0.8
1.0
J/X
(b)
(b)
Fig. 15-Reflection factor r forla>-ers with one type of frequency
independent losses: g(x) =s(x) - j . t ( x ) ; fix) = l . (aj Boundary
condition R ( 0 )= -1. (b) BoundarJ- condition R(,01= +I.
Fig. 17-Reflection factor I for layers with one type of frequency independentjlosses:g(s)=s(s)--j.l(+-);f(s)-l.(a)Boundar?.
condition R(0)= - 1. (b) Boundary condition R(O)= + l .
1.0
0.8
c
0.6
=
05
r
r
0.4
0.2
0
0
0.2
0.I
0.6
0.8
1.0
0.2
0.4
0.6
0.8
1.0
1.0
0.2
04
0.6
0.8
0
//X
(a)
1.0
0.8
0.6
r 0.4
0.2
0
0
0.2
0. L
06
0.8
J/X
(b)
Fig. 16-Reflection factor r for layers with one type of frequent).
independent losses: g(x) =six) - j . t ( x ) ; f(s)= 1. (a) Boundary
condition R(0)= - 1. (b) Boundary condition R(0)= i-1.
Fig. 18-Reflection factor r for layers with one t)-pe of frequency independent losses: g(n) = s ( x ) - j . t ( x ) = 1. (a) Boundary condition
R(O)= -1. (b) Boundary condition R(Oj = + l .
Authorized licensed use limited to: UNIVERSITY OF VICTORIA. Downloaded on October 14, 2008 at 16:33 from IEEE Xplore. Restrictions apply.
These examples illustrate that generally two typesof
losses can be realized simultaneously. However, either a
condition
g ( or
gl may prevail. Figs. 19(a)
and 19(b) show that even under such conditions a considerable improvement in the performance of highly refractivepanelscanbeachieved.Forthemicrowave
case,Fig.19(a)mal-representapanelwith
highreal
part of dielectric constant s(0) = d ( O ) = 5 0 and a small
realpart u ( 0 ) =p‘(Oj =3.3 in permeability. I t canbe
seen t h a t for increasing magnetic losses v ( x ) the typical
oscillations of Fig. 18 are damped out very rapidly. At
the same time, the “average” level is lowered, since the
tendency is to approach the conditionf(x) = g(x) in this
1.0
case. Measurements of P 0 t t e 1 ~on
~ mixtures of paraffin
0.8
andcarbonylironpowder
a t frequencies around 4
kmc show that the material constants of Fig. 19(a) can
0.6
be realized approximately. I t should be mentioned t h a t
r
for the calculations in Figs. 19(a) and 19(b) the material
0. I
constants are assumed to be independent of frequency.
0.2
Insuchartificialdielectrics,however,themagnetic
losses vary with frequency according to a magnetic re0
0
a2
0.4
0.6
0.8
1.0
laxation process.2i
According to the criterion r 5 10 per cent, the curves
in Figs. 19(a) and 19(b) do not show advantages as comparedtotheresults
of Lenz20 (seeFig. IO). If, however,somewhathigher
reflection factorscanbetolerated, for instance 73520 per cent, the case of highly
refractive panels in Fig.19 shows adefiniteimprovement over the case s ( ~ =
) 1 (Fig. 10).
Experimental results for the reflection factor of gradLIST OF SYMBOLS3’
ual-transition absorbers show that, in practice, smaller
1 =thickness of absorbing layer.
Ijh cutoff values for r 5 10 per cent can be obtained
x , y , z = rectangular coordinates.
than those predicted from theor).. For wedge-type por4 =normalized length coordinate.
ous sound absorbers the condition r _ <10 per cent can
X =wavelength infreepropagation
be
realized
for
Z!X>O.2
or
even
slightly
smaller
medium.
values.22.2s For comnlercially available microwave abk = 27i-/X = wave number.
sorbers a similar situation prevails. Jleasurernents by
w = angular frequency.
HaddenhorsPg ondielectric wedge structures show t h a t
Q = (w/c)Z = normalized frequency.
for lossless dielectrics with moderate E’ values a wedge
c = w / k = phase velocity in
free propagastructure can be adequately described by the theory of
tion
medium.
an equivalent inhomogeneous layer. We feel, however,
~ “ ( 4( v)= 1, 2, 3) =weighting functions for thin abthat the one-dimensional theory of inhomogeneous laysorbers.
ers is not completely satisfactoryfor describing the perE
(
x
)
=electric
field strength.
formance of highly absorbent wedge or pyramid strucH
(
x
)
=magnetic
field strength.
tures.Forthesecasestheexacttwo-orthree-dimenE(.%) = ~ ‘ ( x -) j d ‘ ( x ) =relative dielectric constant.
sional boundary value problems have to be solved in
p(x) = p ’ ( x ) -jp”(x) =relative permeability.
order to get a better agreement with the experimental
eo = absolutedielectricconstant
of
results.
free space.
,uo = absolutepermeability
of free
space.
2T L. Lemin, “The electrical constants of a material loaded with
Zo = v ‘ G O = electric wave impedance of free
spherical particles,” J . I E E , pt. 111, vol. 94, pp. 65-68; January, 1947.
28 L. Z. Pronenko and -4. X. Rivin, “Absorbent linings of glass
space.
staple fibre for an acoustic test chamber,” Soeiet Plzys. Acoustics,
If1 >>I
1
lfi <<I
vol. 5, pp. 387-388; February, 1960.
p 9 H. G. Haddenhorst,
“Durchg$ygvonelektromagnetischen
Z . engml. Plzys., vol. 7 , pp.
R-ellendurchinhomogeneSchichten,
4 8 7 4 9 6 ; October, 1953.
30 Further symbols, which are used less frequently, are explained
in the text.
Authorized licensed use limited to: UNIVERSITY OF VICTORIA. Downloaded on October 14, 2008 at 16:33 from IEEE Xplore. Restrictions apply.
Ortner, Eqeland, and Hultqrid: d S e t c Sporadic Layer Providi.ng VLF Propagation
1960
621
zi/Z,= normalized input impedance.
P ( s ) =sound pressure.
u =ohmic conductivity.
V(s)= particle velocity.
R, = area-resistance.
K ( S ) = K’(.x) - j d ’ ( x ) =relative compressibilitJ-.
F,, = Ko-‘ = area-conductance.
p ( x ) = p ‘ ( x ) -jp”(s) =relative densit>-.
K~ = absolute
compres-!
“e!’ as a subscript = refers to
the
electromagnetic
case.
l of homogeneLL ,,
sibility
acoustic
subscript=refers
thea ato as
case.
‘ous medium.
p o = absolute density J
Z = d p x = acoustic wave impedance of hoACKNOWLEDGJIENT
mogeneousmedium.
Stimulatingdiscussions xvith Dr. W. E. Kockand
Z(x) =general wave impedance.
E. J . XlcGlinn on the general subject are appreciated.
R(xj = “Redheffer”-reflection factor.
Xssistance in numerical calculations was given by R. E.
RF(x)= “Schelkunoff ”-reflectionfactor.
T ( X ) = R(x)! =absoluteamount
of reflection Lord and J. M. Patterson, supervisors of the Bendix
Analog andDigitalComputerFacility.Theauthor
factor.
f ( s )= . ( x i - j v ( x ) =loss functions in absorbing ma- especially xvishes to acknowledge thecooperationof
Alrs. P. -A. Jacoby, who carried out most of the numerg(x) =s(x) -jt(x)
terial.
ical runs on the Bendix analog computer.
Lri!
= normalized inputadmittance.
I
-1
A New SporadicLayer Providing VLF Propagation”
J. ORTNER, A. EGELAND, mn B. HULTQVISTf
Summary-During two periods in May and July, 1959, following
Some of the evidence as to the effecton VLF propastrong solar flares, the signal strength
of receptions at K i m a of gation of the beams of corpuscular radiation emitted by
YLF transmissions from Rugby (16 kc)showed no or only slight
solar flares,which produce magnetic storms, is incondiurnal variation. It is proposed that the change of the diurnal variaclusive and in certaincasescontradictory
[ N ] . HOWtion is due to the production by solar protons of an ionized layer
thatovertransatlantic
very deep in the atmosphere, the electron density of which is suf- ever,severalworkersagree
ficient for reflectionof very long waves but too
low to cause measur- paths, the effect of a magnetic storm on \rLF propagaable nondeviative absorption in the HF band at geomagnetic latition is t o reduce field strength at night and to increase
tudes lower than approximately 60’.
i t by day [16], [3], [4], [ i ] .I n a n y case, the magnetic
storm
effect is not a spectacular one.
IKTRODUCTIOK
A very great effecton ITLF propagation, of a kind
OL-AR flares are known to influence I’LF propaganever before observed [lS], [Is], was produced by the
tion in several ways. One of the most spectacular
solar eruption of February 23, 1956.
effects of the ultraviolet radiation emitted by such
Coinciding u-ith the increase of cosmic ray intensity
flares is the “sudden phase anomaly” of \‘LF transmisa t a b o u t 0345 UT, a rapiddiminution of thesignal
sions,firstreportedbyBuddenandRatcliffe
[12],
strength of atmospherics in the I,’LF and lower L F
which permits good timing of the arrival of the electrobands was observed by 1Sarons and Barron [l ] (49 kc),
nlagnetic radiation upon the day side of the earth. The
Belrose, et al. [SI (22 kc), Ellison and Reid [ l S ] (24
suddenionosphericdisturbanceusuallydoesnotprokc), Gold and Palmer [IS] ( 2 i kc), Ehmert andRevellio
duce any appreciable changeof amplitude on very low
[14] (27 kc),andLauter,
et al. [ X ] ( 2 5 , 3 5 , and 40
frequencies, a t least over distances shorter than
1000
kc). -1similarly rapid and substantial decrease
of the
km [ l o ] , but there are indications that forlyavesrefield strength of transmissions from Rugby (GBR,
16
flected a t oblique incidence, the reflection coefficient of
kc) was found by Stoffregen [37] at Uppsala, Sweden;
the ionosphere is reduced during an SIDfor frequencies
and an analogous phenomenon was recorded
a t Munich,
less than 10 kc [17], [SI, and increased for frequencies
Germany, by Schumann [34] for signals from the U. S .
greater than 10 kc [13].
transmitter NSS (17.7 kc). 4 t the same time, a sudden
* Received by the PG-AP, June 30, 1960.
phase
anomaly occurred in the receptions of G B R at
f Kiruna G e o p h y . Obsen-atory of the Royal Swedish Academy
Cambridge, England [8]. Allan, et al. [2] in New Zeaof Science, Kiruna, Sweden.
s
Authorized licensed use limited to: UNIVERSITY OF VICTORIA. Downloaded on October 14, 2008 at 16:33 from IEEE Xplore. Restrictions apply.