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Ionospheric transfer function (ITF) algorithms determine the effects of the ionosphere on an electromagnetic (EM) radio-frequency (RF) signal as it propagates through. In this report, the Snell’s law shell (slab) model is outlined.
An integral conservation law for wave numbers is considered. In order to test the validity of the proposed conservation law, a complete solution for the reflection and transmission of an acoustic wave impinging normally on a material interface moving at a constant speed is derived. The agreement between the frequency condition thus deduced from the dynamic equations of motion and the frequency condition derived from the jump condition associated with the integral equation supports the proposed law as a true conservation law. Additional comparisons such as amplitude discontinuities and Snells' law in a moving media further confirm the stated proposition. Results are stated concerning frequency and wave number relations across a shock front as predicted by the proposed conservation law.
Ionospheric transfer function (ITF) algorithms determine the effects of the ionosphere on an electromagnetic (EM) radio-frequency (RF) signal as it propagates through. In this report, the Bouger’s law shell model is outlined. This ITF is very similar to the Snell’s Law Shell ITF; however, in this formulation, plasma parameters in the ionosphere are allowed to change radially. This algorithm is expressed in the frequency domain. In this way, it is applied as linear time invariant (LTI) filter function. Signals in this report are assumed to have only a single component (i.e. x, y or z in a rectangular coordinate system). Multi-component signals can be treated simply by applying the specific ITF to each component separately.
In this work, we present a rigorous theoretical framework underpinning the technique of spin-echo modulated small-angle neutron scattering (SEMSANS), and show how the technique can be extended in order to generate spin-textured neutron beams with orbital angular momentum (OAM) via birefringent neutron spin-polarization devices known as magnetic Wollaston prisms. Neutron OAM beams are mathematically characterized by a “cork-screw” phase singularity e iℓΦ about the propagation axis where ℓ is the OAM quantum number. To understand the precise relationship between the emergent OAM state and the variety of spin textures realized by various setups, we have developed a path-integral approach that in the interferometric limit makes a judicious use of magnetic Snell's law. We show that our proposed technique produces a complex two-dimensional pattern of spin-OAM entangled states which may be useful as a probe of quantum magnetic materials. We compare our path-integral approach to the well-known single-path Larmor precession model and present a pedagogical derivation of magnetic Snell's law of refraction for both massive and massless particles based on Maupertuis's action principle.
A light ray, incident at about 5 deg to the normal, is geometrically plotted through the drawing of the cross section of a soybean leaf using Fresnel's equations and Snell's law. The optical mediums of the leaf considered for ray tracing are: air, cell sap, chloroplast, and cell wall. The ray is also drawn through the same leaf cross section with cell wall and air as the only optical mediums. The values of the reflection and transmission found from the ray tracing tests agree closely with the experimental results obtained using a Beckman Dk-2A Spectroreflector.
A light ray, incident at 5 deg to the normal, is geometrically plotted through the drawing of the cross section of a soybean leaf using Fresnel's Equations and Snell's Law. The optical mediums of the leaf considered for ray tracing are: air, cell sap, chloroplast, and cell wall. The above ray is also drawn through the same leaf cross section considering cell wall and air as the only optical mediums. The values of the reflection and transmission found from ray tracing agree closely with the experimental results obtained using a Beckman DK-2A Spectroreflectometer. Similarly a light ray, incident at about 60 deg to the normal, is drawn through the palisade cells of a soybean leaf to illustrate the pathway of light, incident at an oblique angle, through the palisade cells.
A light ray, incident at about 5 deg to the normal, is geometrically plotted through the drawing of the cross section of a soybean leaf using Fresnel's equations and Snell's law. The optical mediums of the leaf considered for ray tracing are air, cell sap, chloroplast, and cell wall. The above ray is also drawn through the same leaf cross section considering cell wall and air as the only optical mediums. The values of the reflection and transmission found from ray tracing agree closely with the experimental results obtained using a Beckman DK-2A spectroreflectometer.
The radiance and color of the twilight sky are calculated for single scattered radiation with the use of spherically symmetric models of the earth's atmosphere. Spherical geometry is used throughout the calculations with no plane-parallel approximations. Refraction effects are taken into account through fine subdivision of the atmosphere into spherical shells of fixed index of refraction. Snell's law of refraction is used to calculate a new direction of travel each time that a photon traverses the interface between layers. Five different models of the atmosphere were used: a pure molecular scattering atmosphere; molecular atmosphere plus ozone absorption; and three models with aerosol concentrations of one, three, and ten times normal together with molecular scattering and ozone absorption. The results of the calculations are shown for various observation positions and local viewing angles in the solar plane for wavelengths in the range from 0.40 to 0.75 micron.
Acoustic time delays across a free-jet shear layer are measured and compared with predictions based on (1) ray paths refracted abruptly across a cylindrical vortex sheet and (2) ray paths traced through a more realistic diverging flow model. The close agreement between measurement and theory confirms that Snell's law provides an accurate prediction of wavefront refraction or angle changes across a diverging shear layer. Microphones are placed on calculated ray paths to determine the coherent transmission and internal reflection characteristics of the shear layer and also the scattering of sound by the shear-layer turbulence. The transmission data essentially verify the proposed, theoretical calibration factor which forms part of a computational procedure that is being developed to convert model jet data from a free-jet facility to inflight conditions.
Optical measurements of range and elevation angle are distorted by the earth's atmosphere. High precision refraction correction equations are presented which are ideally suited for surveying because their inputs are optically measured range and optically measured elevation angle. The outputs are true straight line range and true geometric elevation angle. The 'short distances' used in surveying allow the calculations of true range and true elevation angle to be quickly made using a programmable pocket calculator. Topics covered include the spherical form of Snell's Law; ray path equations; and integrating the equations. Short-, medium-, and long-range refraction corrections are presented in tables.
Theoretical models of ionospheric whistler phenomena are reviewed and compared with experimental data. Whistlers were characterized as lightning discharges through a dispersive medium in 1919, with the first observed appearance of whistler noises detected in telephone communications. Magneto-ionic theory is used to characterize whistlers, with the Appleton-Hartree equations applied to the wave fields arising from lightning interactions with ionospheric plasma. Large values of the refractive index or slow propagation speeds give rise to the whistler mode, i.e., propagation of the wave through plasmas of any density. Propagation through the ionosphere is examined with the Snell's law, and account is taken of absorption and the necessity of obtaining full-wave solutions. Finally, theories are under development to explain the occurrence of ducting, i.e., guiding of the whistler wave by field-aligned plasma density irregularities.
General Optical Systems Evaluation Program, GENOPTICS, aids analysis and evaluation of optical systems that employ lenses, mirrors, diffraction gratings and other geometrical surfaces. GENOPTICS uses geometrical ray tracing based on Snell's law.
Results of a two-dimensional ray tracing computer code, based on Snell's law, for electrostatic wave propagation in a dipole magnetic field are discussed. A survey of possible ray paths varying a wide range of parameters is conducted for low-harmonic Bernstein modes in a high-density plasma. It is shown that the ray paths exhibit similarity with radial distance and that there exists the possibility of two classes of wave statistics of the equator: a broad emission region extending to about + or - 4 deg and a class of events restricted to the smaller region of 1-2 deg about the magnetic equator. The regulating parameter between these two types of events is the transition energy from the isotropic background electrons to the unstable distribution of superthermals. Ray paths for propagation in the magnetic equatorial plane are considered and an explanation is given for ray focusing in the equatorial plane based on electron gyroradius considerations.
Surface roughness and diffraction are two factors that have been observed to affect the accuracy of flaw characterization with scanning laser acoustic microscopy. In accuracies can arise when the surface of the test sample is acoustically rough. It is shown that, in this case, Snell's law is no longer valid for determining the direction of sound propagation within the sample. The relationship between the direction of sound propagation within the sample, the apparent flaw depth, and the sample's surface roughness is investigated. Diffraction effects can mask the acoustic images of minute flaws and make it difficult to establish their size, depth, and other characteristics. It is shown that for Fraunhofer diffraction conditions the acoustic image of a subsurface defect corresponds to a two-dimensional Fourier transform. Transforms based on simulated flaws are used to infer the size and shape of the actual flaw.
Near infrared leaf reflectance modeling using Fresnel's equation (Kumar and Silva, 1973) and Snell's Law successfully approximated the spectral curve for a 0.25-mm turgid oak leaf lying on a Halon background. Calculations were made for ten interfaces, air-wax, wax-cellulose, cellulose-water, cellulose-air, air-water, and their inverses. A water path of 0.5 mm yielded acceptable results, and it was found that assignment of more weight to those interfaces involving air versus water or cellulose, and less to those involving wax, decreased the standard deviation of the error for all wavelengths. Data suggest that the air-cell interface is not the only important contributor to the overall reflectance of a leaf. Results also argue against the assertion that the near infrared plateau is a function of cell structure within the leaf.
Surface roughness and diffraction are two factors that have been observed to affect the accuracy of flaw characterization with scanning laser acoustic microscopy. Inaccuracies can arise when the surface of the test sample is acoustically rough. It is shown that, in this case, Snell's law is no longer valid for determining the direction of sound propagation within the sample. The relationship between the direction of sound propagation within the sample, the apparent flaw depth, and the sample's surface roughness is investigated. Diffraction effects can mask the acoustic images of minute flaws and make it difficult to establish their size, depth, and other characteristics. It is shown that for Fraunhofer diffraction conditions the acoustic image of a subsurface defect corresponds to a two-dimensional Fourier transform. Transforms based on simulated flaws are used to infer the size and shape of the actual flaw.
Acoustic measurements collected during the Propfan Test Assessment (PTA) program in 1989 had demonstrated that cruising Propfans have undesirably high far-field noise level. This study attempts to investigate the source of these noise disturbances generated by a single-rotating propeller with supersonic tip speed. Various atmospheric and propagation effects responsible for shaping the far-field acoustics will be examined. A methodology for analyzing Propfan propagation noise is proposed based on the theory of geometrical acoustics. This approach allows acoustic rays to be traced into the far-field via Snell's law. Amplitude variation along the rays are evaluated by applying the Blokhintzev energy invariant principle. Additional terms are appended to the energy invariant principle to account for propagation and atmospheric effects. A FORTRAN program is written to analyze the rays emitted from a helicoidal source path traced by supersonic helical speed blade tip. Earlier papers by Myers and Farassat have demonstrated the presence of Mach envelopes that lead to formation of caustics. Linear geometric acoustics is used to ascertain the existence of these nonlinear caustics in the far-field. The importance of caustics to Propfan noise generation is then studied.
A consistent solution of the radiative transfer equation characterizing photon transport in a semi-infinite medium of refractive index greater than or equal to one is obtained following the method of Sobolev. Fresnel specular reflection, Snell's law and isotropic scattering are assumed. An algorithm is developed and its accuracy is demonstrated. A numerical Laplace transform inversion leads to an efficient evaluation for the interior flux and source function distributions.