A new simple interferometer for obtaining quantitatively evaluable flow patterns

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Title:
A new simple interferometer for obtaining quantitatively evaluable flow patterns
Series Title:
NACA TM
Physical Description:
62 p. : ill ; 27 cm.
Language:
English
Creator:
Erdmann, S. F
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

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Subjects / Keywords:
Aerodynamics   ( lcsh )
Interferometers   ( lcsh )
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federal government publication   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
The method described in this report makes it possible to obtain interferometer records with the aid of available schlieren optics by the addition of very simple expedients. Under certain conditions, the interferograms need not be inferior to those obtained by other methods. (However, one fundamental drawback of the method compared to the Mach-Zehnder interferometer lies in a relatively very poor light output.) The method is based on the fundamental concept of the phase-contrast process developed by Zernike but which has been enlarged to such an extent that it practically represents an independent interference method. The two light beams causing the interferences are not separated until immediately before photographing and up to that point are subject to the same effects. The theory is explained on a purely physical basis and illustrated and proved by experimental data. A number of typical cases are cited and some quantitative results reported.
Bibliography:
Includes bibliographic references (p. 34).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by S.F. Erdmann.
General Note:
"Report date November 1953."
General Note:
"Translation of "Ein neues, sehr einfaches interferometer zum erhalt quantitativ auswertbarer strömungsbilder." Appl. Sci. Research, vol. B 2."

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University of Florida
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Full Text
VCA Tv)-1%3










NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS


TECHNICAL MEMORANDUM 1365


A NEW SIMPLE INTERFEROMETER FOR OBTAINING

QUANTITATIVELY VALUABLE FLOW PATTERNS*

By S. F. Erdmann


SUMMARY


The method described in the present report makes it possible to
obtain interferometer records with the aid of any one of the available
schlieren optics by the addition of very simple expedients, which funda-
mentally need not be inferior to those obtained by other methods, such
as the Mach-Zehnder interferometer, for example. The method is based
on the fundamental concept of the phase-contrast process developed by
Zernike, but which in principle has been enlarged to such an extent
that it practically represents an independent interference method for
general applications. Moreover, the method offers the possibility, in
case of necessity, of superposing any apparent wedge field on the den-
sity field to be gaged, hence to produce more favorable evaluation con-
ditions and greater accuracy.

The theory is explained on a purely physical basis and illustrated
and proved by experimental data. A number of typical cases are cited
and some quantitative results reported.

It was found that this development reacts comparatively little to
disturbing acoustic or mechanical oscillations. This is probably due
to the fact that the two light beams causing the interference are not
separated until immediately before photographing and up to that point
are subjected to the same effects. As regards the special possibilities
which eventually might result with the use of white light and the use of
auxiliary cameras, no systematic investigations have as yet been made.


1. INTRODUCTION


Interest in interferometry for the visualization and the quanti-
tative evaluation of air flows at high subsonic and supersonic speeds
has increased considerably within the last few years. This holds true
for almost all laboratories which earlier were contented with schlieren

"Ein Neues, Sehr Einfaches Interferometer zum Erhalt Quantitativ
Auswertbarer Stromungsbilder." Appl. Sci. Research. vol. B 2.







2 IJACA TD 1565


photoraplhs. ,T Mach-Zehnder interferometer has itself proved to be
very expensive, t-sptcially for comparatively great images and hence
frequently as exorbitant. Furthermore, it is v.ry receptive to outside
disturbances, especially to mechanical and acoustic oscillations, and,
whti used in combination with the necessarily very careful ad.jstment,
is so difficult that highly trained personnel are required to operate
it. These drawbacA.z do not exist in the present method and its possi-
bilities are little if at all inferior to those of the Mach-Zehnder
interferometer in the majority of uses coming into question. This new
method in its present enlarged form makes it possible to change any
schlieren apparatus into an interferometer by a few manipulations and
a minimum of auxiliary equipment. The only fundamental drawback of the
method compared to the Mach-Zehnder interferometer lies in a relatively
very poor light output. But this need not signify a difficulty in
principle if the flow to be studied is itself sufficiently steady.

The method to be described here is a considerably extended elabo-
ration of the phase-contrast method suggested by Zernike in microscopy
and so successfully used in the study of astronomic mirrors refss. 1
to 5). The suggestion of investigating the suitability of this method
for quantitative studies of flows is due to Professor Burgers of Delft.

The principle of the described method is based largely upon the
fact that a certain part of the light passing through the plane of the
object experiences a special treatment in the plane of the light source
image, so that this light, in unison with the other unaffected light,
produces interference in the plane of the object image. Under certain
assumptions, the thus produced interference figure gives a true repro-
duction of the existing density field, by showing lines of equal density
(similar to the Mach-Zehnder setting for infinite band width), or
apparent density fields of constant gradients, termed wedge fields here-
after, can be superposed (similar to the band fields of the Mach-Zehnder
interferometer), or linear systems of bands of equal density increase in
a certain direction referred to a variable level can occur, for which
the Mach-Zehnder interferometer knows no analogue. The fact that the
separation of the last interfering light beams takes place immediately
before the picture is formed, ensures that this method is scarcely more
receptive to outside disturbances than any sensitive schlieren apparatus.

The aim of Zernike's phase-contrast method consists in rendering
minute phase differences even of small fractions of a wave length,
caused by an object, visible, and not by forming bands, that is, black-
white effects, but by differences in brightness (contrast). Even phase
fluctuations can be identified and even measured if necessary, which
normal interferometers no longer indicate. It can also be used to phase
differences of several wave lengths when pronounced fields of disturbance
are involved, that is, when small areas of greatly changed phase are
present in a relatively great field of undisturbed light. However, the







NACA TM 13o6


method fails when the strong disturbance zones occupy a large portion
or cover the field of vision completely. The assumptions necessary for
a successful application of the phase-contrast method can be formulated
correctly by the condition that the amplitude of the center of gravity
vector of the amplitude-phase-vector diagram of the object field must
remain of the same order of magnitude as the amplitude of the light
passing through the object field. (The amplitude-phase-vector diagram
will be discussed later.) The aim of the investigations described here
was to eliminate these restraining conditions, so that the method could
be applied to any field and to replace the tedious photometric brightness
measurement in the image field by a band field with easily valuable
black-white effects. The possibility of superposition of any wedge
fields in the course of this development entails no loss of sensitivity.

In the further course of this treatment, it is attempted to explain
the principle and the details of the method as simply as possible but in
enough detail for understanding and correct application. The experi-
mental data by Gayhart and Prescott (ref. 6) obtained with a schlieren
apparatus using a very narrow light slit are also explained.

At this point I want to mention duly the eager cooperation of
Mr. A. W. Meijer who helped indefatigably with the preparation (fre-
quently requiring much patience) and performance of the tests, the
evaluations and the providing of illustrations for this report.

2. PHYSICAL TREATMENT OF THE THEORY OF THE METHOD

(a) Image Forms of Simple Optical Systems


The explanation of the interferometric phenomena in question is,
naturally, based upon the wave theory of light.

Proceeding from the well-known Frauenhofer diffraction phenomena
from a slit, which is illuminated by vertically incident, parallel light,
the light is propagated in all directions perpendicular to the slit
according to Huygen's principle. Now, angles aZ can be identified
(fig. 1) for which, by reason of their different wave length, a second
beam can be found for each light beam which shows a phase difference of
a half wave length h/2 to it. When these beams are coincident in the
focal point of a lens, they extinguish each other. On a screen S, the
image then appears with a brightness distribution such as represented
qualitatively by figure 2.

The beam directions for the extinction of the light can be
described, according to figure 1, by the relation

sin az = z with z = 1, 2, 3 (1)

with Z = width of slit.






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Along the incident light, that is, z = 0, the brightness is
maximum. The point of extinction on the screen referred to the point
of symmetry, is expressed by the equation

dz = R tan a2 (2)


where R denotes the distance of the screen from the lens, in this
case, the focal length. Limited to small angles a as is generally
permissible, in case I is not extremely small, since the refractions
of higher order can be disregarded,


AR
dz z-


by reason of


tan a4 sin ao


The total width of the central intensity maximum from zero passage
to zero passage for the slit is then


D, 2d = 2 (3)


Considering the same phenomenon but with a circular diaphragm of
radius 7 instead of the slit, the picture on the screen shows then a
similar but coaxial intensity distribution. The central intensity
maximum, frequently termed diffraction disk, follows then as


R R(a)
D = 2.4 = l.2 -- (3a)
27 7


Thus the parallel light mentioned at the beginning can be visual-
ized as originating from a point source of light in the focal point of
the lens and the described diffraction pattern on the screen is then
the image of this light source. From the relation (3a), it is seen
further that this image shrinks more and more coaxially in proportion
as the lateral dimensions of the transmitted parallel beam, that is,







NACA TM 1363


the diaphragm radius, increases. Theoretically, it is not even neces-
sary that a real diaphragm is presented. The final dimension of the
image forming lens can already be regarded as such. A clear point
reflection of the light source would be obtained only when the diameter
of the diaphragm or the lens increases beyond all measures.

Continuing a step farther and replacing the screen by a lens set up
behind so as form a picture of the diaphragm in a new plane (fig. 5), a
sharp picture of it is obtained when a sufficient number of diffraction
maximums can enter in this lens. But if the aperture of the lens is
continuously narrowed, the haziness of the diaphragm picture increases.
When only the central intensity maximum is able to pass through, the
result is a washed out spot of light of approximately the size of the
original diaphragm picture with outwardly continuously decreasing bright-
ness. If the aperture is restricted to a point, the result is, according
to Huygens' principle, that instead of the diaphragm picture the entire
image plane is lighted up to infinity with brightness decreasing monoto-
nously from the center.

For a thorough understanding of the subsequently described method,
these phenomena and the consequences connected with it are of such
decisive importance that it is deemed appropriate to discuss the last
described mental experiment again in reversed order.

If, from the diffraction image of the light source in the focal
plane, only a very small quasi-pointlike sector of the center is admitted
for image forming, it results in an infinitely great lighted area instead
of a diaphragm image. When this sector is enlarged a little in every
direction, the additional light increases the central brightness of the
image and interferes with the border zones, so that the brightness in
the central ir- e is more evenly increased and the brightness decrease
in the border zones becomes more spontaneous. With it, the central
field becomes more and more defined and stands out more. Now if the
entire diffraction image of 0 and 1st order are admitted for image
forming, the diaphragm image already begins to become sharply delineated
until the picture becomes more and more perfect as further diffraction
orders are admitted.

If the same experiment is made with a variable slit instead of a
circular lens aperture of variable diameter, the result is an identical
blurring process in the image plane but limited to the direction perpen-
dicular to this slit. The form of the light source and the shape of the
diaphragm representing the object field play no part in theory.

The same statement made for the diaphragm in parallel light relative
to the diffraction image of the light source in the focal point of an
inserted lens and the image of the diaphragm with the aid of a second
lens applies to the arrangement according to the coincidence method, that







NACA TM 1565


is, a light source in the vicinity of the center of curvature of a
spherical mirror. (See fig. 4.) Its reflection is also the same dif-
fraction figure symmetrical to the position of the light source in the
vicinity of the curvature center. If the mirror forms only a small
spherical sector, the same relation


D = 1.2 or Ds = 2
7 I


applies again, where R is the curvature radius and 7 the radius of
the mirror, and I the width of a slit placed on the mirror.

The theoretical equality of both phenomena follows from the fact
that, in both cases in the critical section, that is, the diaphragm in
one case, the spherical mirror on the other, the light shows equal phase.

The kernel of the discussion is briefly as follows:

In the discussed optical system, any diaphragm, even where the
finite dimensions of the employed lenses or mirrors can action as such,
forms a diffraction figure of the light source. Furthermore, the dif-
fraction orders used for illustrating the plane of the mirror, lens, or
diaphragm define the type and quality of this image. Lastly, it is
emphasized again that the diffraction center, that is, the diffraction
of 0 order, considered as independent light source, covers the entire
image of the object plane comparatively evenly and even beyond on the
surrounding field, although with considerably less brightness.


(b) The Object Field in Vector Representation and Its

Interferometric Representation

By way of illustration, the simple optical arrangement of the
coincidence method shown schematically in figure 4 is to serve as basis;
all the phenomena described with it occur in completely similar manner
when parallel light is used, so that a separate representation of the
two cases is superfluous. The individual addition of a diaphragm is
also omitted, but the border of the mirror, or its total diameter for
the parallel beam, is considered as characteristic diaphragm quantity I
and an arbitrary plane in the parallel beam, respectively; the mirror
plane or the plane extended immediately in front, in the coincidence
representation, is designated as object plane.

Supposing that on a mirror to be regarded as ideal, several trans-
parent models as indicated in figure 5(a) are present which exert an







NACA TM 1363


influence on the phase of the light but not the amplitude. These models
on the mirror surface are numbered and the assumedly corresponding light
vectors are represented in an equally numbered vector diagram. (See
fig. 5(b).) The vector length indicates the amplitude, its direction,
the phase. Owing to the assumed transparency of the models, the end
points of all these vectors lie on a circle around point 0. These light
vectors can be visualized as being split in two components, one to con-
sist of a unit vector for all of which it is assumed that it represents
a portion of the light covering the entire mirror surface evenly.
According to Zernike (ref. 1), there is such a one which is found by
forming the center of gravity of the vector diagram (fig. 5(b)), after
adding the lighted up area to each end point of the plotted vectors.
This center of gravity is represented by A in figure 5(b).

The light represented by the vector OA forms then as image of the
light source a diffraction picture as described earlier, with a central
intensity maximum of


D = v with 2.0 < v < 2.4 (b)


depending on type and shape of the object field. In this central inten-
sity maximum, the total light represented by the vector length OA can be
visualized concentrated, in first approximation. This plainly follows
from the intensity distribution of the diffraction picture of the light
source (fig. 2(b)) wherein the higher orders of diffraction are negli-
gible within the scope of a first approximation relative to the 0 order.

Letting this diffraction center experience a special fate by which
it becomes distinguished from all others, the assumptions for the desired
interferometric effects can be produced. This can be effectuated in
various ways. The "dark-field effect" long used in microscopies can be
created by completely covering the diffraction center, or the "phase-
contrast effect" can be produced by adding a phase disk as suggested by
Zernike, which varies the phase of the central light abruptly and,
if necessary, reduces the amplitude too, or the entire remaining field
can be made strongly absorbent while leaving only the diffraction center
unaffected. This is termed "field absorption."

According to the foregoing, the light emitted from the diffraction
center is comparatively evenly distributed over the entire area of the
image to be formed by the mirror. Thus, after taking one of the cited
steps, the central light interferes like a veil over the image field.
Without partial interference of the diffraction picture of the light
source, an image of the object plane true in amplitude and phase would
result. The eye would see no differences in this case because the
amplitude was assumed identical in every point and does not respond to







NACA TM 1563


phase differences. But with special treatment of the central diffrac-
tion image, it results in a picture which can be best explained on the
basis of the vector diagram. (See fig. 5(b).) By plotting, as example,
three vectors (1, 3, and 4) with and without interference in the dif-
fraction image, the results shown in figure 6 are obtained. The original
vector is shown as chain dotted line, that as seen by eye as solid line.
The following is manifest:

(1) The dark-field method is equivalent to a shrinkage of the
center of gravity vector OA to length zero and thus to a shift of the
original zero point 0 to the new zero point OD at A.

(2) The addition of a phase disk corresponds to a rotation of the
OA vector about A, that is, about the angle by which this changes the
phase. Thus, figure 5(b) shows the new zero points Op', Op", and Op
corresponding to phase rotations through o = 900, 1800, and 2700 or,
respectively, -900.

(3) The writer's proposal (field absorption) is equivalent to a
relative increase of the vector OA, hence to an effective displacement
of the zero point leftward by an amount that is defined by the measure
of the intended absorption. If the absorption of the material sur-
rounding the diffraction center indicates a permeability of 1/2, 1/3,
or 1/4 of the amplitude, the new effective zero points OE2, 0E3, or
OEL, shown in figure 5(b) are produced. This corresponds to absorptions
referred to intensity, hence, the square of the amplitude in the same
sequence of 75 percent, 88.9 percent, or 95.7 percent.

As a result, the different vectors show different lengths in the
image of the object plane and are visible to the eye as differences in
brightness. By way of contrast, the object plane perhaps manifests
opaque objects (such as a model in the flow, for example) whose image
would coincide with point 0 in the vector diagram, no longer as dark
but with a brightness corresponding to the distance 00x, where 0x
indicates the newly created zero point depending upon the chosen method.

It should be clear from the aforegoing that the light concentrated
in the diffraction center can actually be regarded as a component of the
local light vectors common to the entire image of the object field. How-
ever, this component is not an arbitrary one of the infinite number of
imaginable components; it must be the center-of-gravity vector OA corre-
lated to the particular vector diagram, as is readily proved by a simple
approximate energy consideration. The phase-contrast method is particu-
larly suitable for this demonstration, inasmuch as none of the light is
suppressed by absorption, hence that according to the energy principle
it can and must be postulated that the sum of all the light in the image






NACA TM 1365


of the object field must remain the same independent of the affected
manipulations.

At the personal suggestion of Mr. Greidanus of the N.Y.I. Laboratory,
this demonstration is described in the following form:

Supposing the vectors of the zones 1, 2, 5 are expressed by

en = cos Cn + i sin cp with n = 1, 2, 5 .

and lenl = 1; the component OA common to all vectors assumed concen-
trated in the diffraction center to be

p = p cos(qC0 + i sin p0)

Then the individual vectors follow as sum of the two components

en = bn + p


en = (cos pn p cos c0) + i(sin Pn p sin 0o)] +


p cos qp + ip sin CPO


Next, on the basis of experimental experience, it is assumed that
an interference in the diffraction center has practically no effect on
the first component bn, but merely on the second component p
imagined as being concentrated there. When the latter'is rotated
through angle in the phase, p becomes

S= oFos 0 + i) + i sin( 0 + )p

and after reuniting the components in the image of the object field,
e becomes

en = bn +


n = cos n cos n0 + p cos,(0c + *) + i in p -

p sin p0 + p sin(cp + *)






NACA TM 1565


From this, it can be easily computed that the local image inten-
sity becomes

2
leni2 = 1 + 2p2(l cos ) 2p cos(q p0)(l cos *) +

2p sin(hn pn)sin *


Now the initially formulated energy relation can be written in the


form


S en 2 Fn = leni2 Fn


or en 2 Fn -len2 Fn = 0


with the premise that this condition is fulfilled independent of the
effected phase rotation *. The factor Fn indicates the areas or
zones of the object image related to the corresponding light vectors
in which they occur. Insertion of the above squared expression, while
allowing for the fact that leni2 = 1, gives

2p2( cos +) > Fn 2p(l cos C) O- cos(pn pp)Fn +


2p sin l sin(On P)Fn = 0


The stipulated independence of is then fulfilled only by the
conditions

Scos (n 0P)Fn = p Fn


and


T sin(Pn p0)Fn = 0


which, as is seen, represent directions for p0
the vector OA = p as center-of-gravity vector.


and p. which define


This consideration is rigorously valid theoretically. Its approxi-
mately close but not rigorous proof by experiment rests, in this case,







NACA TM 1363


not on the inadequacy of the theoretical treatment but on the experi-
mental impossibility of realizing a factually rigorous separation of
components, since fundamentally both cover the entire plane of the dif-
faction image. The bn components can be considered as spread so
generally over the entire field that the portion coincident with the
diffraction center of the evenly lighted-up mirror is practically of no
significance. But the center-of-gravity vector in the diffraction
center shows a concentration high enough to realize the desired effect
with sufficient approximation.

The fact that, by phase rotation or some other interference of
this center, the higher diffraction orders of the center-of-gravity
vector cannot be included has an effective not completely uniform
decrease of the average brightness of the object field as a result and
as compensation the radiation of the remaining light over the image
borders.

The quantitative reliability of the interference pattern obtained
with it does not suffer, at least as long as the magnitude of the
affected area remains within the dictates imposed by equation (3b).


(c) Discussion of Possible Interference Formations

From the foregoing, it is seen that the resultant center-of-gravity
vector OA is, after its special treatment, the carrier of the inter-
ferometric effects.

Beginning with the phase-contrast method, the exact conditions can
be easily illustrated on the basis of figure 7, where the locus of all
possible end points OprAr, that is, after -900 phase rotation, is
plotted. The result is a straight line starting at an angle of 450
with respect to the original vector OA from the original zero point 0.
The subscripts 7 at the points Opr and Ar indicate corresponding
vectors, whereby 7 indicates the amount of the vector referred to
the radius I of the principal vector diagram. Bearing in mind that
the occasional points Opr then denote the new zero points to which all
light points of the object field in the illustration are to be referred,
it is apparent that at first the contrasts to be achieved increase with
increasing 7, but only up to 7 % 0.7, where OpO.7 then lies on the
circle, hence contrasts of double brightness up to absolute black can
appear.

A further increase of 7 shifts the new zero point beyond the
circle, so that greater brightness but no complete extinction is possi-
ble. The proportion of extreme brightness can then no longer become
zero or infinite, but must remain finite. To prevent this, Zernike






NACA TM 1565


(ref. 1) suggested to cover the phase disk with a more or less strongly
absorbing layer. This reduces the slope of the Opr straight (dashed),
so that, if appropriately covered, the new zero point falls on the
circle again and hence renders extreme effects possible. A total absorp-
tion returns the zero points Opr to the relative points Ar and yields the
known dark-field effect, which thus proves to be only a special case of
the phase-contrast method.

After these deliberations, it is clear that a really well valuable
contrast-rich image is obtainable only when the length of the center-of-
gravity vector is no less than about 7 = 0.4. But this means that only
objects with relatively small phase variations in the object field must
be involved, or that in the presence of greater irregularities they are
limited to a small area compared to the total field. This might be the
main reason why this method has not been applied to flow investigations,
which usually deal with very severe disturbances frequently spreading
over the entire field.

The difficulties encountered by the dark-field and phase-contrast
method for a too small vector OA, can, however, be avoided by the
suggested absorption of the field surrounding the center of diffraction,
since it, as already stated, is equivalent to an arbitrary relative
vector increase, depending upon the degree of chosen absorption. Thus
satisfactory bright-dark effects can be obtained also with transmission
of the vector diagram, if the center-of-gravity vector OA is very small.
The extent of the absorption must be so chosen that the new zero point
Op falls about on the external beam of the vector diagram. In practice,
up to now a field absorption of about 95 percent has generally proved
very favorable. This method fails theoretically only in the practically
nonexisting cases of zero vector OA.

The discussed interference of the diffraction center are effectu-
ated by some auxiliary means such as cover plate, phase disk, or free
passage in the absorbent field. All these auxiliary means are here-
after gathered under the collective terms as interference plate, inter-
ference slit, or interference circle, depending upon shape.

Thus far, only the interference of the central diffraction image of
a quasi-point source of light was in question and the application of
interference disks associated with it. But in theory, an interference
slit spanning the entire aperture of the lens can also be used instead.
The result is, as briefly stated before, that the central light is then
dispersed only perpendicular to the slit and distributed over the object
field rather than toward all sides. Admittedly it is true, even though
with limitations discussed in section (3b) what was stated in the fore-
going regarding the vector diagram and its center-of-gravity vector,
but in a somewhat different form. In this case, a vector diagram and







NACA TM 1363


a center-of-gravity vector is no longer representative for the whole
object field; fundamentally there exist an infinite number of such for
every object field, each of which is characteristic for one section of
the object field. The consequence is, that fundamentally, aside from
exceptional cases, one assumption is no longer sufficient for a complete,
quantitative determination of the object field, but two with mutually
shifted interference slits are necessary.

Compared to the interference disk, the use of an interference slit
has four direct practical advantages:

(1) The extent of the light source which in principle must be
smaller than the diffraction center of its figure is limited in one
direction only. So, instead of a quasi-point source of light, a slit-
like source of light can be used and much more light made available.

(2) In fact, it requires just such treatment to make superposition
of wedge-shaped fields possible, which proves very desirable in many
cases for increasing the measuring accuracy.

(5) Excellent light-dark effects are attainable scarcely inferior
to those of a normal interferometer.

(4) In object fields with two wedge fields of different directions
superposed, as is frequently the case in open jet supersonic wind tun-
nels with glass walls, which manifest wedge errors, these wedge errors
can be eliminated and satisfactory test data obtained.

These are the advantages which give in the majority of cases the
use of the interference slit the preference.

Lastly, there is yet a third possibility which embodies advantage
(1) cited above and the advantage accruing from the use 'of the inter-
ference disk, namely, of obtaining a quantitatively completely deter-
minable result with a single photograph. (For the sake of completeness,
it should be stated that, in general, just as with every normal inter-
ferometer, one or two schlieren photographs are required in order to
determine whether a sequence of lines about density increase or decrease
are involved.) This third possibility is the use of a circular source
of light with a corresponding interference circle. The light-dark
effects obtainable with it are, however, not good save in exceptional
cases, and wholly inadequate in many instances. This rules out the
superposition of a wedge field.

The conditions are exemplified on two examples, with exception of
the superposition of wedge fields to be discussed later, by the inter-
ference figures (figs. 8 and 9). For comparison, the photographs
obtained with a Michelson interferometer in the Zeeman laboratory at







NACA TM 1363


Amsterdam have been included. Since the image field there was consider-
ably smaller than the objects, the field had to be made up from a series
of partial photographs. The possibilities of displacement of the model
were so primitive that a slight misalinement was unavoidable. It
explains the poor fitting of one sector to the next on the interference
band.

The evaluation of the figures obtained with interference slit are
discussed in detail in section (4). As to the photographs in fig-
ures 8 and 9, it may be stated that in fact the combination of two photo-
graphs made with mutually shifted interference slit affords a picture on
a par with the Michelson interferometer photographs. (Compare section 5,
fig. 24.)


3. REPRESENTATION OF LIGHT SOURCE AND IMAGE FORMATION

ON SOME SPECIAL OBJECT FIELDS

(a) Wedge-Shaped Density Variation


A rectangular object field is assumed which in depth, that is, in
direction of the transmitted light, is bordered by two parallel glass
plates of distance t. The object field is of width 1; air exists
between the glass plates, the density of which shows a constant increase
grad p = p' = constant in the direction in which I is measured. The
light passes through uneven optical wave lengths in various sections
perpendicular to this gradient. As a consequence, the light in gradient
direction is no longer in phase. This signifies on the one hand, con-
sidered from the point of view of the method treated here, that the light
vector rotates uniformly in the vector diagram of the object plane. On
the other hand, by reason of the fundamental relationship between a
prism and the field discussed here (termed wedge field hereafter), it
results in a deflection of the light. In order to be able to judge the
consequences associated with it exact, a quantitative examination is
indicated.

The refractive index for air is given by the relation (ref. 7)

n = 1 + ap

with the density p and a constant a dependent on the wave length
of the employed light. Its value is computed too from the relation


( 1)106 = 272.643 + 1.2288?J2 + 0.05555-4






NACA TM 1356


where ? = wave length of light expressed in p, and n the index of
refraction at standard atmosphere, hence, at a temperature of 150 C and
a pressure of 760 mm Hg.

For the two principal lines of the mercury spectrum, it is

Il = 0.546| (green line), a1 = 0.002226

N2 = 0.365i (ultraviolet line), a2 = 0.002280


Introducing the length coordinate x in direction of the density
gradient and shifting its zero point in the center of the object field
where the density pm prevails, the variation of the refractive index
follows as

n = 1+ a(m + p'x) (4)


and the optical path length within the object field of depth t as


s = nt = t 1+ a(Pm + p'x (5)


The total optical path difference made nondimensional by the wave
length of the employed light is equal to the total phase difference
Z = Ap/2it and follows as

As = tap'Z = = Z (6)
A h 21c

1 = width of object field.

The center-of-gravity vector of the vector diagram of the object
field follows, with stipulation of cp = 0 for x = 0, as


JOAI = cos cp dx (7)
x=0

since, on account of the symmetry to the abscissa c = 0, the components
perpendicular to it cancel out. This transforms with the phase rela-
tion (6) in general f.:'m

S= 2r ta x
AN






16 NACA TM 1363


into


|OA| = 2 1/2 cos(2 ta-p' xdx= sin tap'l
I -I cos 2' x dx = sin
IQ Xh ) ir tap'


Thus, the center-of-gravity vector is given as


|OAl sin lZ
iZ

in relation to Z = Atp/2n and plotted in figure 10(a). The center-of-
gravity vector is zero for all integral Z = 1, 2, 3 ., which is
readily apparent because it signifies that the vector diagram is exactly
Z-times run through, hence, is in equilibrium in the zero point.

Such a state forms one of the very rare situations occurring in
practice in which the method in question is unable to form interference.
In all other cases, the resulting vectors OA whose length, that is, the
brightness representing it, are considerably affected by Z and, in
general, tendency to such an extent that the momentary maximums of the
individual intervals decrease substantially with increasing Z. Hence,
it is possible to secure interference figures, but with the dark-field
and phase-contrast method only when Z is restricted to small values,
but unlimited with field absorption.

Comparison of figure 2(a) with figure 10(a) indicates that both fig-
ures are strikingly similar; moreover, bearing in mind that relation (6)
through

tap' a (8)

becomes equation (1)


a=z-


for small angles a, the suspicion suggests itself that, fundamentally,
the phenomena are alike in the sense that identical diffraction figures
of the light source occur in both cases, whereby that of the last
described test relative to the diffraction at the slit is laterally
displaced as a result of the diffraction by the wedge field. The suppo-
sition is extremely plausible and signifies that, in consequence of the
displacement due to the wedge field in place of the diffraction center a






NACA TM 1565


higher diffraction order is represented on the geometrical axis of
symmetry of the optical system. This assumption is confirmed by the
subsequently adduced proof that the relation defined in equation (8)
actually exists as a result of the diffraction of light by the wedge
field.

This proof is based on equation (4) which in conjunction with the
relationship between index of refraction and the ratio of local velocity
of light c to that in vacuum co

CO
n =
c

gives for the former


c=
CO
1 + a(p + p'x)

which, since a(pm + p'x) << 1 for air can be written in the form


c = Co 1 a(pm + p'x


Thus the wave front rotates about point x0 in which c would be
c = 0. This results in the conjunction

1 apm
x0 =
ap'

and, by reason of apm < 1


1
"0 ap'


The light, which as is known is propagated perpendicular to this
wave front, is rotated therefore by the same amount as it. As the
density field that produces this rotation is to have the depth t, the
light itself is deflected in its entirety through the angle


= = tap'
x0






NACA TM 1563


The agreement of this result with the definition (8) furnishes the
desired proof, namely, the suspected quasi-identity of the two appar-
ently different cases involved.

A density gradient produces, accordingly, a displacement of the
optical zero point of the diffraction figure relative to the geometrical
zero point of the placement in the plane of the light-source figure,
without entailing any subsequent variations. This result furthers the
expectation that it should be possible to produce a band field in the
plane of the object image by a specific displacement of the interference
slit from the geometrical zero point of the system perpendicular to the
slit, even in a field without gradient, and thus create the impression
of the existence of a wedge field in the plane of the object. That this
expectation proves correct is borne out by the photographs in figure 11,
obtained for a field without any gradient by a continuous shift of the
interference slit from the zero position.

The quantitative relationship between this simulated gradient and
displacement E follows as


e = OR = tap'R or p' =-~ (9)
taR

as is readily apparent from figures 1 and 4.

Obviously, the reversed process of the gradient of an existing
wedge field can equally be determined the same way by measuring its
displacement e with respect to the geometrical zero point, which is
necessary to let the object field appear as free from gradient. With
that, an arbitrary but known wedge field can be superimposed on each
object field by a corresponding displacement of the interference slit.
This is the second extremely decisive extension of the possibilities of
this method.

But if it is desired to get rid of the length measurement E, which
is rather inconvenient in practice, the gradient to be determined,
whether actually present or artificially superposed, can also be defined
interferometrically. This is accomplished by measuring those interfer-
ence bands per length h in the object field in direction of the dis-
placement, that is, perpendicularly to the interference slit

(1) which are produced by creation of a wedge field on the object
without gradient,

(2) those which became additive or diminished at superposition of
the wedge field on an arbitrary object field, or which remain
after removal of the latter, such as in flow measurements
with still air before or after each test,






NACA TM 1363


(5) those that originate after removal of the object, in the case
where compensation of an existent wedge field took place by
displacement,

(4) those arising in an auxiliary field without gradient which has
been fitted at an appropriate point, as for example, obtained
by hollowing out the model in direction of the light passage,
in flow measurements.

The gradient follows by equation (6) with h instead of I as


P' (10)
tah

In order to be able to determine the prefix of the gradient, the direc-
tion of the displacement must be considered, while bearing in mind that
the light is always refracted in direction of the rising gradient. In
other words, the gradient is positive in the direction in which a dis-
placement of the interference slit is accompanied by a reduction of the
number of bands per length.

At this point, a question arises that may intrude itself upon many
readers, namely, how the artificial production and the superposition of
ostensible wedge fields could be reconciled with the concept of the
vector diagram of the object plane discussed in section 2(b) and its
interference by modification of the center of gravity. The energy con-
siderations at the end of section 2(b) had proved that, if interference
phenomena are to be produced in the described manner, it can be accom-
plished only by way of the center-of-gravity vector of the vector dia-
gram. But this is, as stated, reflected in the diffraction center of
the uniformly lighted mirror. How can this concept be reconciled with
the production of ostensible wedge fields which precisely calls for a
migration from this center of diffraction?

The aforementioned energy consideration had proved that the exist-
ence of a phase-contrast effect was contingent upon the coaction and
the interference of the center-of-gravity vector of the related vector
diagram. On the other hand, the experiment has proved that such an
effect could be secured not only with an interference disk or slit in
the focal point of the system but also outside of it, even with pure
phase rotation. In these cases, the use of field absorption had usually
proved more favorable for obtaining better contrasts, owing to the then
usually small vector, although in principle a pure phase rotation also
yields interference figures. From the comparison of these two facts,
the general conclusion can be drawn that the light in each space point
can be regarded as center-of-gravity vector of all vector diagrams that






NACA TM 1356


can be constructed on spherical shells around this point, or on cylin-
drical shells, by the use of an interference slit, that is, in an
included angle perpendicular to the interference disk or slit, defined
by the relation (5b) in the form


S3= v with 2.0 < v <2.4
R D

where D denotes diameter or width of interference disk or slit. In
practice, it means that a displacement of the interference slit is auto-
matically followed by a change in the correlated vector diagram in such
a way that a lateral displacement in the plane of the original diffrac-
tion figure causes the effective object field to rotate correspondingly,
which in first approximation is equal to a wedge field and in case of
displacement toward the object field or away from it is equal to a
reduction or increase of the effective curvature radius R of the
object field, hence to a more or less concave development of it. (See
fig. 10(b).) The last phenomenon can be of great practical significance
insofar as it provides the possibility of compensating eventual concave
or convex errors of the glass plates closing off the object field by a
simple displacement of the interference disk or slit perpendicular to
the plane of the diffraction figure, just as wedge errors can be elimi-
nated by a lateral displacement in this plane. Moreover, there is no
change involved, in these cases, in the discussed mode of consideration
of the vector diagram and the possibility of its interference for
obtaining interference figures of the correlated object plane.


(b) Edge Effect and Band Formation

The aforementioned possibility of band production by displacement
of the interference slit from the zero position is easily proved experi-
mentally in its simplest form. Figure 11 shows such a photoseries in
which the object field is formed from a carton by a rectangular sector.
By successive shifting of the vertical interference slit in horizontal
direction, one, two, and more vertical bands are produced.

If the light and interference slit were replaced by a point source
of light and an interference disk, the position relationship between
diffraction of zero order and interference disk would always be une-
quivocally defined. In consequence, a certain shift from the zero
position would always reveal bands of identical width and direction,
independent of the form of enclosure of the object field.

But, if an interference slit is used, the unequivocal position
relationship is lost. It is then no longer a second quasi-point source
of light that produces the interference, but an unevenly covered slit






NACA TM 1565


of light whose individual points exhibit variable coordinates relative
to the center of diffraction. The fact whether the original light
source itself is a point or parallel to the interference slit is of no
significance since, of course, only coherent lights can interfere with
one another.

Because the distribution of light of the diffraction pattern is
markedly dependent on the form of the enclosure as well as on the
density variation in the object field, it also applies to the light
distribution that falls on the interference slit. Hence it is to be
expected that the ensuing interference figures themselves are affected
by the form of the borders of the object field. This is confirmed by
figure 12 which was obtained by interference slit displacement on a
circular object field, and yet shows no vertical straight interference
bands.

For a better understanding of this phenomenon, two examples are
discussed, although limited to the rough effects of first approximation,
since an exact treatment would in all probability be rather extensive
and does not appear to be absolutely necessary at the present state of
development. To define the concept, it is assumed that the interference
slit is fitted vertically and shifted horizontally.

The first example is illustrated on a "Swiss Cross" (fig. 15) made
as vector from cardboard and in which various ostensive wedge fields
are to be produced by displacement of the interference slit. For this,
two superposed diffraction figures are obtained. One results from the
comparatively long central crossbeam A and forms the diffraction maxi-
mums of distance 5 of the higher orders indicated as black dots in
figure 14. The two short projecting arms B, only 1/5 as wide as the
longitudinal, form diffraction maximums at distance 58 of the higher
orders, indicated by squares in figure 14. It follows that, at dis-
placement of the slit from its zero setting, first one, then two bands
are produced in the long transverse beam, while the short beam section
reveals nothing yet. Much better than figure 14 for a quantitative
definition is the brightness distribution of the two superposed dif-
fraction figures reproduced in figure 15, which shows the various slit
settings corresponding to the various photographs (fig. 16) of the
"Swiss Cross." With the aid of this representation, all photographic
phenomena can be explained practically without comment, even such minor
details as that at zero slit setting the borders of the A-zone are
illuminated much brighter than those of the B-zones (fig. 16(a)). This
is due to the fact that, through the slit in the A-diagram, much more
light overlaps than in the B-diagram (fig. 15). Furthermore, the bands
in the B-fields of figures 16(e) and (h) indicate better contrasts than
those visible in figures 16(d), (f), and (g). The reason for this is
also directly apparent from figure 15. It is seen that, in the cases
conforming to figures 16(d), (f), and (g), the slit with sector coincides






NACA TM 1563


with the B-brightness distribution, where this increases or decreases
and hence relatively less light is available for interference formation
than in those of figures 16(e) and (h), where sectors from brightness
maximums are presented.

It is of interest to note that this example reveals, for the first
time, the surprisingly visible, separistic behavior of different hori-
zontal zones, in this case, the boundary lines between A and B, whose
existence with the use of an interference band had already been repeat-
edly asserted in the foregoing.

The example indicates, further, very plainly that the first band
is not produced at a distance equal to an optical path difference of
0.5?h from the zone boundary as a superficial inspection might indicate,
but at a distance that may vary between 0.5X and l.O0, conformably to
the two passages of the diffraction figures of first order through zero,
shown in figure 15. This is the sole remaining and unavoidable inaccu-
racy inherent to the interferometric principle in general, not so much
to this special method alone. By the phase-contrast method, this range
would, as consequence of the phase rotation, be about 0.125\ to 0.625\
but with, usually, very much weaker contrasts for the previously cited
reasons. An illustrative example for it is given in figure 17, which,
of course, shows no rigorously symmetrical figure.

However, this uncertainty in the boundary zones detracts nothing
from the fact that on more than one existing band their mutual distance
exactly corresponds to an optical path difference of one wave length.

The second example is illustrated by means of a square set on its
tip (rhombus) so that the edges slope at 4i0 with respect to the
interference slit. Following the experimental proof of the separistic
behavior of the various zones perpendicular to the slit in the first
example, it is attempted to derive the interference figures to be
expected for displacement of the slit from the zero position and to
illustrate it in figure 18.

For a clear outline of the situation involved, figure 18(a) shows
the normal figure of the lines of equal density for the particular field.
The solid lines indicate blackening, the dashed lines the regions of
maximum brightness. In principle, it is immaterial whether a real or
fictitious wedge field is involved, such as can be obtained with an
interference disk shifted from its zero position or with a Mach-Zehnder
interferometer by tilting one of the mirrors with respect to the other.
Now the question is what kind of an interference figure would occur
with an interference slit under the same conditions?

In order to make the discussion clearer and facilitate mutual
reference, the fields of figure 18(b) and (c) are divided into zones by






NACA TM 1363


dotted horizontal lines in such a way that these dotted lines indicate
exactly, at the same time, whole multiples of a wave length as optical
path differences. Along these lines, no interference formation is
likely to be possible for a wedge field, when it actually starts along
this linear from the center-of-gravity vector of the vector diagram
(fig. 5(b)), since it is zero at that point.

Holding to this concept of interference formation for the time
being, the following relationship prevails:

If the total optical path difference from one border to another is
less than one wave length, the new zero point OE would lie in the
vector diagram (fig. 5(b)) for this zone at the left of the original
zero point 0, that is, the center-of-gravity vector OA and with it
OEA is in phase also with the vector of the zone center. However, if
the total path difference is more than one but less than two wave
lengths, OE lies to the right of 0, that is, OA is in opposite
phase to the zone center. At a difference of more than two, OE lies
left again, at more than three, to the right, etc. So, if this concept
of the cause of interference through an unequivocally defined center-
of-gravity vector proves true, it means that in the first instance the
zone center should show maximum brightness, in the second, maximum dark-
ness, then bright again, etc. (See fig. 18(b).) The result would be
dark line elements whose contrasts would be greatest in the center and
decrease toward the outside and become zero upon reaching the first-
following dotted line, since OA itself becomes zero. The anticipated
contrast yield is accordingly indicated by the local thickness of
blackening.

An identical interference figure but with unchangeable contrasts
in the individual blackening elements, would result if the object had
the step contour shown in figure 18(c) instead of the rhombus form.
After the results with the "Swiss Cross," it can be stated that in this
case the represented figure would be actually obtained. However, it is
to be assumed that this figure does not appear at once on the rhombus
field, because the direction of diffraction is, as known, perpendicular
to the wave front, hence perpendicular to its edges on the bounded field.
If these are parallel, the principal direction of diffraction and with
it the intersecting of it with the interference slit is unequivocally
defined, and it is this point that produces the interference figure
occurring as independent source of light. On the other hand, if the
two lateral zone borders are not parallel, two principal directions of
diffraction perpendicular to these borders result, besides that of the
zone flanks directed parallel to the interference slit, hence, two inter-
section points with the interference slit. Thus these two independent
radiating light sources cause two independent band systems parallel to
the field fringes. The center-of-gravity vector concept is therefore
ruled out and its validity is thus restricted to very special cases.






NACA TM 1565


In order to arrive at a quantitative conception, the situation may
perhaps be imagined to be such that the rhombus field is obtained from
a superposition of the step contour with a number of small, positive
and negative triangles of light bounded by the dotted line and the con-
tour in figure 18(c). What is meant by this is that these triangles of
light represent additional light sources, whose phase structure repre-
sents the continuation of the rectangular field in the positive case,
and as being in opposite phase with it in the negative case. Accord-
ingly, step contour and rhombus form merely do not differ at the half
zone heights. There the superposed light sources are zero, so that it
is assumed that the same interference occur there also in both cases
and that the existing blackening on the rhombus represents the inter-
section points of the two fringe systems. The final result to be
expected on the rhombus is the interference figure of figure 18(d),
which in many respects is confirmed by the experiment (fig. 19(a)).
The latter shows, in fact, band systems shifted parallel to the edges
toward the inside (compare figs. 12 and 20), but, contrary to the expec-
tation cherished according to the previous speculation, does not seem to
spread over the entire field. This divergence may be due to two causes.
Either the interference slit was not small enough for the total field
width so that the light was not uniformly enough distributed from both
sides over the field, or else it is the result of the likewise partial
light of diffraction falling on the interference slit, which, starting
from the corners of the rhombus is distributed in all directions within
450. Even the photographs on the small rhombus (fig. 19(b)) fail to
give definite particulars, for there too the same effect is noted on
few bands. Only the last figures with many bands create the impression
of complete agreement with the theoretical network of figure 18(d).

The appearance of two continuous systems had proved that the asser-
tion voiced in section 2(c) actually does not prove correct in general.
According to that statement, the interference phenomena with interfer-
ence disks or slits differed only in the interference being due to a
resultant vector for the whole field in the first case, and the center-
of-gravity vector related it being responsible in each horizontal band
in the second case. If this were correct, the interference figures
would likely be different, as demonstrated with figures 18(a) and (b),
but would have to be problematic, which obviously is not the case in the
two cited examples. Nevertheless, the fact remains that this unequivo-
calness occurs only in the horizontal sections with parallel side walls
(compare fig. 12) and even then only for the pure wedge field, not
generally. Furthermore, unequivocalness occurs only in the zones in
which the two systems intersect. In all other cases, the possibility
of double or ambiguity can be counted on. The latter may happen when
several such fields lie close to one another so that the light from the
slit can cover them.






NACA TM 1563


However, this phenomenon need not entail difficulties in practice,
since in general fields the contrast yield of one system so predominates
that the other scarcely appears; if it does, it is almost always possi-
ble to identify the two systems as such and to distinguish them. The
quantitative interpretation of the field can then be made by proceeding
with one of the systems and disregarding the other completely. If
desirable, a change-over from one to the other can be effected in the
intersection points, while still preserving the connection. It follows
that the quantitative interpretation can be made without it being abso-
lutely necessary to have both systems available. For this reason, a
further investigation into the final cause of the minor discrepancy
between theory and experiment, as indicated in figures 18(d) and (e),
was omitted for the time being.

The general conclusion is that when the interference slit is used
the interference figures do not indicate lines of equal density but
lines of equal density increase or decrease referred to the particular
border of the examined zone, measured normal to the interference slit.
However, these lines are likely to vary angularly and abruptly within
a width of 0.5h, even with continuous object field and density variation,
owing to the zonal separistic behavior in conjunction with the fluctu-
ation width of 0.5h illustrated on the "Swiss Cross" for the appearance
of the first band at the zone boundary. (Compare section 4.)

This result was confirmed on various random fields. (Compare
figs. 8 and 9.) It was also found that, in the sense considered here,
not only opaque object field boundaries are feasible, but also discon-
tinuities of the density within the object field, such as compression
shocks in supersonic flows, and areas of maximum and minimum density or
reversal points, for example. One such phenomenon at extreme densities
from the appearance of different band systems is particularly percep-
tible in figure 21(d).


4. INTERPRETATION OF INTERFERENCE FIGURES

(a) Preliminary Remarks


There is no intention of going into details about the general
theory of interpretation of interference figures, since ample literature
on the subject is already available. The same applies to the special
interpretation of rotationally symmetrical systems which also have been
treated extensively elsewhere, nor is it intended to submit rational
interpretation schemes. The purpose is rather to devote particular
attention to the characteristic peculiarities for the present method
and hence to guarantee the junction with the conventional methods. The
problem is therefore simply an attempt to ascertain how figure of lines






NACA TM 1365


of equal density can be constructed from interference records obtained
by interference slit with or without artificial superposition of a wedge
field. Everything else comes within the scope of ordinary interferom-
etry and can therefore be discounted.

At the end, several patterns of various interference records are
added, some without detailed discussion, simply with the intention of
giving an idea what this method is already able to do at the present
state of development.

Unfortunately, the writer had no Mach-Zehnder interferometer at his
disposal, hence was unable to compare the records made by both methods
under otherwise identical test conditions. It would be gratifying if
such opportunity presented itself some way or another in the near future.


(b) General Density Fields

As a rule, the interpretation of a completely unknown field
requires four photographs, two interference photographs, I and II with
mutually rotated slit and two schlieren photographs, I(a) and II(a) with
schlieren edges whose direction is equal to that of the employed slits.
In the majority of cases, it will be advisable to let the slits form a
900 angle with one another. The necessity for two interference photo-
graphs results from the fact that each zone normal to the slit leads,
so to say, its own independent lift, and is no way dependent on the
adjoining zones. One of the photographs serves to connect all zones of
the other photograph in one arbitrary cross section. But, since the
band systems give no indications of whether the transition to the next
band was accompanied by a density increase or decrease, the schlieren
photographs responsive to density gradients must make the decision
regarding this possible.

Fundamentally, however, it likewise is possible to take interfer-
ence photographs I(b) and II(b) with slit displaced relative to setting I
and II instead of schlieren photographs. The local gradient is then
deduced from the fact whether the band spacings are greater or lesser.

If the qualitative gradient field in the interference images I
and II is known, it is advisable to fix one arbitrary line each as zero
line and proceeding from it provide the whole network of bands with suc-
cessively increasing or decreasing numbers n depending on the gradient.1

Next follows the measurement in image I of the abscissas x of the
different bands normal to the interference slit for as many ordinate

1Hereafter n denotes the number of bands and numbering, not the
index of refraction.






NACA TM 1365


values y as parameter, as seems necessary within the ambit of desired
accuracy. It is advisable to let the zero point of the abscissa coin-
cide with the ordinate so that the latter covers the entire field height
as much as possible and runs perpendicular to the interference slit II.

Then the obtained values TI, reduced by the eventual, artificially
superposed wedge field from gradient -iD/Ax = n'I, are plotted in the
form

n" = fi(x,y) = Hi(x,y) n'I (11)


and nOI = f(O,y) determined from the interpolation for x = 0.

Then the image II is evaluated on the basis of the same system of
coordinates; the task can be limited to measuring the ordinates y of
the identically numbered bands for x = 0, that is, nOII(0,y) and,
after subtraction of the eventual artificially superposed wedge field
from the gradient n'Ii in the form


nII = f2(0,y) = nII n'IIy (12)


represented graphically or in tabular form.

This then corresponds to the existing conditions; n0I is then
corrected so that noI + An0 = nII,.

So the final result is the desired variation of the lines of con-
stant density according to the relation


n(x,y) = ni(x,y) + Ano(0,y) (15)


The density field itself can then be determined by plotting the
field or by further treatment from the tabulated designs in the custom-
ary form of normal interference photograph.

If the object field presents discontinuous pressure jumps, as fre-
quently occurs in supersonic flows with the appearance of compression
shocks, each one of the areas separated by an unsteady density variation
must be treated separately. In many cases, the mutual connection can be
found by the use of white light with the aid of the so-called O-line.






NACA TM 1565


In each case, for determination of the absolute density level, as for
ordinary interference photographs, knowledge of the absolute density in
all separated fields not coordinate with known fields, at least in one
point, is required. What fundamental possibilities the use of white
light in this particular method affords, perhaps by fitting small auxil-
iary cameras next to or in the object field, has not been investigated
so far.


(c) Fields of Disturbance and Normal Flow Fields

Fields of disturbance are defined as such objects in which in all
zones perpendicular to the interference slit a great percentage is
covered by a constant density field, and a comparatively small portion
is taken up by a variable density field on which in no way the condi-
tion of small variation needs to be imposed. In such a situation, the
center of gravity of the vector diagram of all zones exhibits almost
the same direction, with the result that a normal picture of lines of
equal density appears, whose interpretation in this respect requires
no special comment. Hence, it is not necessary to take two photographs
with mutually rotated interference slits.

An identical or similar situation is frequently encountered with flow
photographs, especially on models and where often comparatively great
areas of exact or sufficiently approximate undisturbed and known flow
occurs. But, even if the areas of equal density are not large enough
to produce lines of constant density, it may prove superfluous to make
two photographs with rotated interference slit if a cross section can
be found that reveals constant density or a known variation and over-
laps the various zones. The interpretation is then made again the same
way as described above. At times, one of the cited situations can be
produced by appropriate overlap of a wedge field.


5. EXAMPLES OF INTERFERENCE PATTERNS WITH

SOME EVALUATION RESULTS


The subsequently described examples are intended to give an idea
regarding the quantitative feasibility of the method and the quality of
the records obtained so far in comparison with phase-contrast and field-
absorption records under all kinds of conditions as well as to demon-
strate the method used by the writer in manipulating the photographs
for obtaining quantitative data.

Moreover, it should be noted that the experimental equipment avail-
able was rather limited, that is, actually comprising only a normal







IJACA TM7 1363


schlieren optics with a spherical mirror subjected to errors of as much
as three wave lengths toward the rims. The wind-tunnel disks likewise
were subject to irregularities of up to several wave lengths and, on
top of that, were mounted in such a way that they were exposed to high
mechanical and thermal stresses which could be distinguished in each
test. Consequently, in principle, comparable results still showed
certain discrepancies which, in the writer's opinion, are solely due
to the secondary circumstances, not to the method. This also applies
not in the least to the first two subsequent examples on object glass
for the comparison of which Michelson interferometer photographs were
employed and which had been obtained under primitive conditions as
already stated elsewhere.

In order to eliminate every conceivable source of error or uncer-
tainty, the first phase of this method was carried out with two glass
plates as object. The first, the so-called "small object glass,"
2.6 x 3.4cm, is a sector of a normal microscope objective; the other,
the "large object glass," is a 3 x 6cm sector from an ordinary, cleaned-
off photographic plate. The comparative photographs were taken with a
Michelson interferometer of 2.5 X 2.5cm field of vision, followed by
synthesis of the partial figures to a unit.

Figures 21 and 22 represent the field absorption and schlieren
photographs with vertical and horizontal slit for both object glasses.
The appearance of interference lines in the bright areas of the small
object glass on the schlieren photographs is of interest. It is a kind
of one-sided dark-field effect which can be produced direct with every
schlieren optics if a suitably narrow slit is used as light source.
This is the same phenomenon described by Gayhart and Prescott (ref. 6).
Figure 21(d) is a typical example of two intersecting systems of lines,
which are easily separated in the lower half. The data used in the
evaluation were inked in and numbered, the second, by way of illustra-
tion, was added as dashed lines in the lower half. Another system,
inked in and numbered is shown in figure 22(c). Both photographs, fig-
ures 21(d) and 22(c), were measured during the evaluation over the entire
field and, according to the rule for evaluation in section 4, the inter-
polation values for defining An along the three plotted lines were
read from the graphical representation. In principle, the reading along
one line is sufficient; the purpose of reading along three, in this
instance, was to demonstrate the degree of accuracy of the An deter-
mination reproduced in figure 25. In figures 21(c) and 22(d), the
blackenings are numbered only along the three intersection lines.
Plotting whole line systems would have been impossible anyhow, in this
instance, because the abrupt transitions of adjacent zones are already
so pronounced that it is no longer a question of uniform system. How-
ever, no difficulties are entailed, as seen from the good agreement of
the An values in figure 23, for the three sections obtained by com-
bining the related n-values at the particular sections of figures 21(c),
21(d) and 22(c) and 22(d), in exact accord with the evaluation rule of






NACA TM 1565


section 4. A constant was added to the difference formation at the indi-
vidual sections in such a way that all three results have, in principle,
an arbitrary point in common; it is the same as fixing the absolute level
which, as stated elsewhere, is, in principle, indeterminate.

Again it is pointed out that the numbering of the lines and black-
enings must be effected in closest cooperation with the respective
schlieren photographs in order to recognize if and where it must be
increasing or decreasing. Incidental to the An values of figure 23,
it should be noted that the fact that they carry the fluid character of
a curve is merely to be taken as exception. In principle, the An vari-
ation can, of course, be abrupt, hence it is not justified to strike an
average by plotting a compensating curve. But it is well permissible
and even advisable to average the An values obtained for identical
abscissas along different sections.

The final result of the two evaluations on lines of equal density
is represented in figure 24 together with the corresponding photographs
obtained by Michelson interferometer in the Zeemann laboratory at
Amsterdam. The agreement may be regarded as very satisfactory, consid-
ering the aforementioned inadequacies of the Michelson interferometer
photographs.

A comparison of several photographs on a Laval nozzle for Ma = 5
and their evaluation is represented in figure 25. In view of the large
areas of practically constant density at the nozzle inlet and outlet,
this object is very appropriate for field absorption as well as for
phase-contrast photographs with vertical slit (figs. 25(a) and (b)),
but of course only on the assumption that these two areas differ in the
optical path length approximately by a whole multiple of a wave length.
If this is not the case, there is a twofold possibility of inferior or
totally useless figures. In the first place, if the interference slit
is narrow enough so that the entire field of vision is practically
evenly covered, the center-of-gravity vector can become very small, so
that field absorption alone produces good figures, or in the second
place, if the interference slit is too wide, each one of the two areas
produces an independent interference field with the result that the lines
in the central portion become vague or completely undefinable, even by
the field absorption method.

For the photographs with horizontal slit (fig. 25(c)) and for over-
lapping of a wedge field (fig. 25(e)), the field-absorption method is
definitely superior.

Schlieren photographs were not necessary in this evaluation,
since the gradient variation was sufficiently known from pressure-
distribution measurements.






NACA TM 1565


The numbering in figure 25(a) is monotonic. But in the interest of
image quality it is written out only there where the line density per-
mits this easily.

As regards figure 25(e), it is readily apparent that the density
variation plotted against nozzle length manifests an S-shaped character;
so the superposition of an oppositely directed wedge field produces two
areas in which the mutual gradients are inversely equal. The compara-
tively wide, fringeless areas appear. Since, after the overlapping,
the newly created apparent density field changes the sign of the density
with respect to the ends in the midportion between these areas, the
numbering of the fringes must count backwards.

As a consequence of a contour error of the nozzle, the supersonic
part of the flow, the compression shock, indicated by dashes in fig-
ures 25(a), (c), and (e), occurs, which in association with the nozzle
contour divides the supersonic parts in four zones: I, II, III, IV.
The An-relation for zone I follows from the fact that equal density
(almost that of the atmosphere) prevails in section A. The same holds
true for zone II, the measuring rhombus, if the density is everywhere
constant and the level can be regarded as known, which is imputed here,
by reason of the absence of lines (at zero slit setting). The An-relation
in section B for zones III and IV is obtained from a combination of fig-
ures 25(c) with (a) and (e). The asymmetry is due to the superposition
of a weak density field in figure 25(c). The level for these zones is,
by way of illustration, so determined that the levels of zones III and IV
are mutually equal in the objective point 0 of the four zones and equal
to the mean value of zones I and II in this point. Such averaging was
omitted in the present example because the small pressure increases
accompanying the compression shock fell within the degree of accuracy
desired in this experimental evaluation.

The result of the evaluations on lines of equal density is repre-
sented in figures 25(d) and (f). An originally intended direct compar-
ison with the theory was omitted for two reasons. In the first place,
it was found that the nozzle contour differed from the chosen form as a
result of a systematic measuring error in manufacturing and so produced
the compression shock. In the second place, the employed channel windows
show stated but qualitatively not yet accurately determined irregular-
ities, which in conjunction with the likewise unsatisfactory mirror pro-
duce errors of several wave lengths. As a result, the number of bands
is not exactly (5 to 10 percent discrepancy) agreeable with that expected
by theory; but the general variation of the lines is entirely satisfac-
tory with theory.

Also of interest are the differences of the two evaluations.
According to the writer's conception that obtained with superposed wedge
field is more accurate, although its departure from theory is greater as
regards number and position of bands, for it even shows slight flaws in
the glass structure in the basically gradient-poor zones, to which the






NACA TM 1563


zero setting already ceases to respond. From this, it can be concluded
that it is necessary to investigate eventually defective glass with
which one is forced to work with superposed wedge fields rather than
zero slit setting, to define the variation exactly and take it into
consideration in the evaluation. Furthermore, it is necessary to deter-
mine if and what temperature stresses occur during measuring, since it
was found that they could be considerable and a function of the meas-
uring time.

Figure 26 represents several variations of the method applied to a
supersonic airfoil at Ma = 2.1, along with the graphical presentation
of the lines for the purpose of evaluation of three different cases.
With slit turned through 900, the evaluation in first approximation
required no corresponding photographs, since the center-of-gravity vec-
tor in the entire figure is likely to be controlled by the comparatively
wide field of constant density before the airfoil and, even if this
should not hold true for the entire field, it would still be applicable
to the very small half profile height, to which the evaluation can be
limited. For the aim of the evaluation is not, as in the previous
example, to find the system of lines of equal density, but to define
the pressure distribution on the airfoil for the purpose of comparing
it with the theoretical variation. The results of figure 27 was dis-
cussed without going into further details. The theoretical curve is
shown as solid line. The test points indicate a very gratifying agree-
ment with theory up to the separation point on the airfoil. The greatest
number of points and the relatively little scattering indicate, as was
to be expected, the overlapping of the negative wedge field (in flow
direction), whose gradient is in the same direction as the field in
question. The superposition of a positive wedge field is entirely
unsuitable in the present case, since only two or three test points are
obtained then. The number of test points at zero slit setting is less
and scattering greater than with negative wedge field. Whether the
variation after separation of flow (chain-dotted line) was correctly
reproduced, seems problematic, since the extent to which the outer field
may be continued through the dead water between sound flow and profile
contour is not guaranteed. But this is an aerodynamic rather than an
optical problem and therefore not explored further.

The airfoil test data are comparatively very favorable as regards
accuracy and mutual comparability and theory. This is undoubtedly due
to the smallness of the object field, especially in contrast to the pre-
viously described nozzle measurement, with the result that the errors of
the mirror and the channel plates can be disregarded.

Lastly, figure 28 represents three complete sets of photographs of
airfoil flow measurements as illustrative examples. They require no
special comments, except one concluding remark. For economical reasons,
it proved expedient to combine the photographs in groups and to make






NACA TM 1563


collective productions from great enlargements, which, however, did not
benefit the visible contrasts any. The reader is therefore requested
to bear in mind that the originals show better contrasts than the copies
shown here.


6. EXPERIMENTAL SETUP


A brief summary of the experimental setup (figs. 29 and 30) follows.

Obviously the coincidence method is involved here. The light source
is a water-cooled Philips-maximum pressure-mercury lamp from which light
of a certain wave length (A = 0.541) was screened out for use with the
aid of a double monochromator; however, it was found that a Kodak-filter
No. 77 itself produced acceptable fringes.

After passing the monochromator, a picture of the monochromator
entrance slit is formed in the plane L with the aid of a condenser.
This is then the slit that serves as active light source of the actual
optical setup. Directly behind, a miniature mirror, 4 mm in diameter,
deflects the light 900 in direction of the object field or spherical
mirror. The latter is mounted in such a way that the light source is
slightly excentrical to its axis of symmetry at the distance of the
radius of curvature of this mirror. By this method, the diffraction
pattern B is formed at the same distance reflective to the axis of
symmetry, and where the schlieren edge is placed for the schlieren
method and the interference slit when the interference method is used.
The optics mounted directly behind it forms then the desired schlieren
or interference pattern of the object plane on the ground glass S.

It is true that the coincidence method has the drawback that a not
completely identical course of the reciprocal light beams may at times
result in double pictures or at least in reduced sharpness. Besides, the
glass plates bordering the object field produce very disturbing reflexes
occasionally. A parallel light in the object field and one passage of
light would be preferable, in principle, in view of the greater possi-
bilities for the proportions of the light source. In spite of that, it
still seemed necessary to apply the coincidence method, since, owing to
the smallness of the supersonic wind tunnel with its 3 x 3cm2 test sec-
tion available for flow investigations, a second passage of light was
necessary to assure a somewhat useful number of bands. With greater
working sections or by working with higher static pressure, as intended
in future tests, the conditions will be much better in this respect.
The other setup can then be used immediately.

For flow studies, the small supersonic tunnel (fig. 31) placed
directly in front of the spherical mirror, forms the object plane, while







NACA TM 1363


for the other more basic tests, the plane 0 (fig. 29) is used as such.
The image field varies in the various tests and may amount to 25cm diam-
eter corresponding to the mirror radius, which matches the corresponding
minimum dimensions of the light and interference slits, according to
equation (3b). Concerning the latter, it is quite conceivable, espe-
cially with a single passage of light through the object field that
eventually light and interference slits placed parallel at proper dis-
tance from one another will be used, because then there is no danger of
doubling. Thus, the brightness of the picture could be increased and
the exposure time kept short.


Translated by J. Vanier
National Advisory Committee
for Aeronautics




REFERENCES


1. Zernike, F.: Physica 9, nos. 7 and 10, 1942.

2. Zernike, F.: Roy. Astron. Soc., March 1934.

5. Burch, C. R.: Roy. Astron. Soc., March 1954.

4. Linfoot, E. H.: Proc. Phys. Soc. London 58, 1946 .

5. Kohler, A., and Loos, W.: Naturwiss. 29, Heft 4, 1941.

6. Gayhart, E. L., and Prescott, R.: J. Opt. Soc. Amer. 59, July 1949.

7. International Critical Tables, vol. VII (National Research Council
of the U.S.A.).






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NACA TM 1365


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NACA TM 1363


Figure 30.- Optical system.


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