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Advantage is taken in this paper of the parametric properties of families of curves to express in a simple manner several fundamental properties of moir fringes. Attention is called, in particular, to the necessary limitations on the angle of rotation of two gratings, and on the magnitude of their difference in pitch, to obtain an easily interpretable moir-fringe pattern.

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Two superposed gratings can produce many other moiré patterns in addition to the pattern commonly observed. They correspond to different forms of the indicial equation employed in parametric descriptions of moiré phenomena. An analysis of the Fraunhofer diffraction pattern of two superposed gratings by the methods of Fourier optics shows that the different moiré patterns can be separately observed by spatial-filtering techniques. Examples of additive and subtractive moiré patterns obtained individually over the whole field of two superposed gratings are presented. This method is combined with a previously developed method for obtaining partial derivatives from deformed gratings as a direct moiré pattern (rather than as moiré of moiré) to determine experimentally the whole field of cartesian shears and rigid rotations. The method is applied to a circular ring subjected to diametral compression.

The interferometric method of Nisida and Saito, the absolute retardation method of Post and the double-exposure holo-interferometric method of Hovanesian, Brcic and Powell are correlated and shown to give identical fringe patterns as regards intensity. Various discrepancies in Nisida and Saito's interpretation of their fringe patterns are pointed out, and the simultaneous presence and relative predominance of isochromatic and isopachic half-tones are discussed. The principles of photography and the concept of half-tones are used to explain the formation of concomitant isochromatics and isopachics and to give a new interpretation to these fringe patterns.

A physical explanation of photoelastic interferometry and holography is presented. Reasons for special properties and specific restrictions are argued in relation to this physical picture. The controlling intensity relationships for interferometry and holography are derived by simple physical analysis. The relationship between absolute retardation, isochromatic- and isopachic-fringe systems is clarified. Model material requirements for photoelastic interferometry and holography are shown to be identical.

The equations prescribing the gradient and inclination of fringes in moiré interferometry are derived from the basic laws
of diffraction and interference. A vectorial representation of three-dimensional diffraction employs incidence and emergence
vectors in the plane of the grating; the representation is especially well suited for this type of analysis. The corresponding
equations for geometrical moiré are derived by a remarkably direct vectorial method. The analyses prove that the patterns
of moiré interferometry and geometrical moiré are governed by identical relationships.

The analytical method of superposition is combined with the experimental technique of multiple-exposure holography to decrease the sensitivity of holographic measurement by at least an order of magnitude. Moir fringes of a lower frequency are produced which simultaneously extend the range of measurement to larger displacements. The method is demonstrated for the case of a clamped circular plate subjected to a concentrated load centrally applied.

It is shown in this paper how the whole field of displacement components and of their time and space derivatives (isothetics,
isotachics and isoparagogics) can be obtained individually, as separate patterns in a simple and precise manner using spatial-filtering
techniques. This result can be obtained even when crossed gratings are used on the deformed body. A method for achieving fringe
multiplication in moiré patterns produced by superposed, crossed gratings is also demonstrated. It is also shown that displacement
components and their time and space derivatives in directions diagonal to the crossed-grating lines can be obtained by proper
handling of grating transparencies and spatial-filtering techniques. Hence, the moiré equivalent of a whole field of rosette-strain-gage
measurements is obtained from a single photograph of a deformed crossed grating. A disk compressed between two wedges is used
as an example. Important applications will be found in the fields of dynamics, nonlinear elasticity and plasticity.

A particular variation of holographic imaging system is described which, when used as a multiple-exposure holographic interferometer, possesses advantages for applications in static and dynamic photomechanics. Large fields of view can be obtained. Rigid-body motions produced by loading are automatically eliminated. The holograms can be recorded on medium-resolution films which have high sensitivity. Specimens manufactured with readily available materials can be used for the determination of isochromatics and isopachics.
An attempt is made to describe this contribution in the background of previous developments in interferometry. Operation of the system is interpreted by showing the equivalence of the holographic interferometer to a combination of two systems presently in use in experimental stress analysis: a Fizeau interferometer and an optical spatial filter. The interpretation of isochromatics and isopachis as moiré phenomena is emphasized.
Isochromatics and isopachics are presented as illustrations of the applications of the method to the solution of static-stress problems, and they are used in the solution of some not yet solved dynamic-stress problems. Whole-field static isochromatics obtained as absolute-retardation interference are shown. Also shown are whole-field dynamic isopachics.

In grid and moir analyses, patterns developed in a deformed body are sometimes superposed to obtain the difference of their parameters, as when it is desired to obtain velocities by superimposing two isothetic patterns. In this operation, an inherent difficulty develops because, as a consequence of the deformation, not all points can be matched. An error follows which is analyzed here. Means to avoid or to compute the error are suggested.

A high-frequency phase grating on a specimen surface is illuminated symmetrically by two oblique beams. The diffracted beams emerge with wave front warpages that define both the in-plane U and out-of-plane W displacement fields. Contour maps of these wave fronts, with added carrier fringes, are obtained as a single photographic record. They are manipulated by moiré and optical filtering steps to yield whole-field fringe patterns of U and W. Sensitivities of 0.833 microm/fringe (32.8 microin./fringe) for in-plane displacements and 0.132 microm/fringe (5.2 microin./fringe) for out-of-plane displacements were demonstrated. Since data acquisition is experimentally simple, dynamic as well as static analyses are applicable.

It is shown that, when a certain space exists between a linear rectangular grating and a lenticular grating, the resulting moiré pattern is a linear rectangular grid. An analysis of the phenomenon is provided, and some of its applications to moiré analysis are demonstrated.

Whole-field contour maps of shear strains γxy
are derived from displacement fields obtained by moiré interferometry with 2400 ℓ/mm (60,960 ℓ/in.). Cross-derivatives of displacements are obtained by mechanical differentiation. They are summed by graphical additive moiré. The high sensitivity of moiré interferometry permits quantitative analysis in the small-strain domain.

A general mathematical technique for the solution of moiré patterns produced by the overlapping of two figures is presented. The technique is applied to combinations of figures involving parallel lines, radial lines, and concentric circles including those in which the spacing is variable. When an equispaced parallel line figure is overlapped on a parallel line figure whose spacing is variable, the resultant moiré pattern reveals the functional form of the variation.The theory of the measurement of refractive index gradients by the moiré technique is presented. Three arrangements of the positions of the figures with respect to the sample are analyzed. One compact arrangement gives exclusively the refractive index gradient. The interpretation of moiré patterns distorted by lenses is considered in terms of the properties of the lenses.The mathematical solutions of moiré patterns are, in many cases, identical with those arising in physical problems. Examples are given for a number of phenomena arising in physical optics, hydrodynamics, and electrostatics.