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National Advisory Committee for Aeronautics, Technical Notes - A Theory and Method for Applying Interferometry to the Measurement of Certain Two Dimensional Gaseous Density Fields

A theory and method are described for the application of interfer—
ometry to the measurement of certain two-dimensional gaseous density
fields. The theory includes the effects of optical refraction upon the
observed interference pattern. Exact equations denoting the relative
density difference corresponding to an observed interference~fringe
shift and the optical distortion caused by refraction are derived.
Corresponding approximations in the form of power series expansions are
developed to permit practical application of the theory. Expressions
are derived which permit calculation of the values of the power series
coefficients from.experimental data.

The approximations were applied to available interference data in
order to determine the density distribution in a boundary layer formed
by supersonic air flow along a flat plate. Good agreement was obtained
between the density distribution calculated from the interference data
and that obtained from pressure-probe measurements.

various methods for verifying the theory are considered and an eval—
uation process is outlined.

In investigations of high subsonic and supersonic gas flow, where
compressibility of the gas necessitates the treatment of density as a
variable, various optical methods (reference 1) have been utilized as a
means of density measurement. This is possible because density variations
in optical media such as air act in a measurable way on light. More-
over, optical methods of investigation possess the advantage of permitting
instantaneous recording of the flow without disturbing the flow. Of the
optical methods, quantitative investigations by the method of interfer-
ometry have proved most successful. Interferometric investigations of
flow fields have been conducted primarily with interferometers of the
Zehnder-Mach type. The principles of interferometry and the ZehnderéMach
interferometer have been described elsewhere (references 2 and 5).

The analysis by means of interferometry of density distributions
within optical media is based on the concept of "optical path differ-
ences” of interfering light wanes: the optical_path is defined as the
integrated product of the physical path traversed by a given wavefront
and the refractive index along that path. Thus, for convenience, optical
path differences will be expressed in terms of refractive index rather
than density. The two quantities are related by the Lorentz-Iorenz law
(reference 2). Also, for purposes of clarification, the mathematical
development is conducted on the basis of the ray theory rather than the
wane theory of light propagation. This is possible because light rays
are functions of the light waves in that a ray indicates the direction
of light propagation, which, in nonpolarizing media, is normal to an
advancing wayefront.