Showing papers on "Reflection (physics) published in 1978"

Nonlinear phenomena in the ionosphere

Abstract: 1. Introduction.- 1.1 Data on the Structure of the Ionosphere.- 1.2 Features of Nonlinear Phenomena in the Ionosphere.- 1.2.1. Nonlinearity Mechanisms.- 1.2.2. Qualitative Character of Nonlinear Phenomena.- 1.2.3. Brief Historical Review.- 2. Plasma Kinetics in an Alternating Electric Field.- 2.1. Homogeneous Alternating Field in a Plasma (Elementary Theory).- 2.1.1.Electron Current-Electronic Conductivity and Dielectric Constant.- 2.1.2.Electron Temperature.- 2.1.3.Ion Current-Heating of Electrons and Ions.- 2.2. The Kinetic Equation.- 2.2.1. Simplification of the Kinetic Equation for Electrons.- 2.2.2. Transformation of the Electron Collision Integral.- 2.2.3. Inelastic Collisions.- 2.3. Electron Distribution Function.- 2.3.1. Strongly Ionized Plasma.- 2.3.2. Weakly Ionized Plasma.- 2.3.3. Arbitrary Degree of lonization-Concerning the Elementary Theory.- 2.4. Ion Distribution Function.- 2.4.1. Simplification of the Kinetic Equation.- 2.4.2. Distribution Function.- 2.4.3. Ion Temperature, Ion Current.- 2.5. Action of Radio Waves on the Ionosphere.- 2.5.1. lonization Balance in the Ionosphere.- 2.5.2. Effective Frequency of Electron and Ion Collisions-Fraction of Lost Energy.- 2.5.3. Electron and Ion Temperatures in the Ionosphere.- 2.5.4. Heating of the Ionosphere in an Alternating Electric Field.- 2.5.5.Perturbations of the Electron and Ion Concentrations.- 2.5.6. Artificial lonization of the Ionosphere-Heating of Neutral Gas.- 3. Self-Action of Plane Radio Waves.- 3.1. Simplification of Initial Equations.- 3.1.1. Nonlinear Wave Equation.- 3.1.2. Nonlinear Geometrical Optics of a Plane Wave.- 3.2. Effect of Nonlinearity on the Amplitude and Phase of the Wave.- 3.2.1. Self-Action of a Weak Wave.- 3.2.2. Self-Action of a Strong Wave.- 3.2.3. Self-Action of Waves in the Case of Artificial lionization.- 3.3. Change of Wave Modulation.- 3.3.1. Weak Wave.- 3.3.2. Change of Amplitude Modulation of Strong Wave.- 3.3.3. Phase Modulation.- 3.3.4. Nonlinear Distortion of Pulse Waveform.- 3.4. Generation of Harmonic Waves and Nonlinear Detection.- 3.4.1. Frequency Tripling.- 3.4.2. Nonlinear Detection.- 3.5. Self-Action of Radio Waves in the Lower Ionosphere.- 4. Interaction of Plane Radio Waves.- 4.1. Cross Modulation.- 4.1.1. Weak Waves.- 4.1.2. Strong Perturbing Wave.- 4.1.3. Resonance Effects near the Electron Gyrofrequency.- 4.2. Interaction of Unmodulated Waves.- 4.2.1. Interaction of Short Pulses.- 4.2.2. Change in the Absorption of a Wave Propagating in a Perturbed Plasma Region.- 4.2.3. Generation of Waves with Combination Frequencies.- 4.3. Radio Wave Interaction in the Lower Ionosphere.- 4.3.1. Cross Modulation.- 4.3.2. Fejer's Method.- 4.3.3. Nonstationary Processes in the Interaction of Strong Radio Waves.- 5. Self-Action and Interaction of Radio Waves in an Inhomogeneous Plasma.- 5.1. Inhomogeneous Electric Field in a Plasma.- 5.1.1. Fundamental Equations.- 5.1.2. Distribution of Density and Temperatures in Plasma.- 5.2. Kinetics of Inhomogeneous Plasma.- 5.2.1. Kinetic Coefficients. Elementary Theory.- 5.2.2. Kinetic Theory.- 5.2.3. Fully Ionized Plasma.- 5.3. Modification of the F Region of the Ionosphere by Radio Waves.- 5.3.1. Modification of the Electron Temperature and of the Plasma Concentration.- 5.3.2. Radio Wave Reflection Region.- 5.3.3. Growth and Relaxation of the Perturbations.- 5.4. Focusing and Defocusing of Radio Wave Beams.- 5.4.1. Nonlinear Geometrical Optics.- 5.4.2. Defocusing of Narrow Beams.- 5.4.3. Mutual Defocusing.- 5.4.4. Thermal Focusing in the Lower Ionosphere.- 6. Excitation of Ionosphere Instability.- 6.1. Self-Focusing Instability.- 6.1.1. Spatial Instability of a Homogeneous Plasma.- 6.1.2. Instability in the Wave-Reflection Region.- 6.2. Resonant Absorption and Resonance Instability.- 6.2.1. Langmuir Oscillations in an Inhomogeneous Plasma.- 6.2.2. Excitation of Plasma Waves.- 6.2.3. Resonance Instability.- 6.2.4. Absorption of Ordinary Radio Waves.- 6.3. Parametric Instability.- 6.3.1. Langmuir Oscillations of a Plasma in an Alternating Field.- 6.3.2. Parametric Excitation of Langmuir Oscillations.- 6.3.3. Parametric Instability in the Ionosphere.- 6.3.4. Dissipative Parametric Instability.

An angular‐spectrum approach to contrast in reflection acoustic microscopy

Abstract: The scanning acoustic microscope in the reflection mode has proved to be a rather simple and direct means for monitoring the elastic properties of a solid surface. When smooth surfaces of crystalline material are examined in a liquid with a highly convergent sound beam they exhibit a distinct response. This characteristic response, which can be treated as a ’’signature’’, is obtained by recording the output of the microscope as the spacing between the acoustic lens and the object is varied. An angular‐spectrum approach is used to derive an expression for this output in terms of the reflectance function. This function has an angular dependence determined by the bulk constants of the material itself. The expression resulting from this treatment can be used to explain the source of contrast in acoustic images.

Observation of amplified reflection by degenerate four-wave mixing in atomic sodium vapor.

Abstract: This Letter is the first report of the observation of amplified reflection and self-oscillation produced by degenerate optical four-wave mixing by using a resonantly enhanced Kerr nonlinearity in atomic sodium vapor. Phase-conjugated reflected waves with intensities 100 times greater than input-signal intensities were achieved with only 40-kW/cm2 pump intensity.

Nonlinear phenomena in the ionosphere

Abstract: 1. Introduction.- 1.1 Data on the Structure of the Ionosphere.- 1.2 Features of Nonlinear Phenomena in the Ionosphere.- 1.2.1. Nonlinearity Mechanisms.- 1.2.2. Qualitative Character of Nonlinear Phenomena.- 1.2.3. Brief Historical Review.- 2. Plasma Kinetics in an Alternating Electric Field.- 2.1. Homogeneous Alternating Field in a Plasma (Elementary Theory).- 2.1.1.Electron Current-Electronic Conductivity and Dielectric Constant.- 2.1.2.Electron Temperature.- 2.1.3.Ion Current-Heating of Electrons and Ions.- 2.2. The Kinetic Equation.- 2.2.1. Simplification of the Kinetic Equation for Electrons.- 2.2.2. Transformation of the Electron Collision Integral.- 2.2.3. Inelastic Collisions.- 2.3. Electron Distribution Function.- 2.3.1. Strongly Ionized Plasma.- 2.3.2. Weakly Ionized Plasma.- 2.3.3. Arbitrary Degree of lonization-Concerning the Elementary Theory.- 2.4. Ion Distribution Function.- 2.4.1. Simplification of the Kinetic Equation.- 2.4.2. Distribution Function.- 2.4.3. Ion Temperature, Ion Current.- 2.5. Action of Radio Waves on the Ionosphere.- 2.5.1. lonization Balance in the Ionosphere.- 2.5.2. Effective Frequency of Electron and Ion Collisions-Fraction of Lost Energy.- 2.5.3. Electron and Ion Temperatures in the Ionosphere.- 2.5.4. Heating of the Ionosphere in an Alternating Electric Field.- 2.5.5.Perturbations of the Electron and Ion Concentrations.- 2.5.6. Artificial lonization of the Ionosphere-Heating of Neutral Gas.- 3. Self-Action of Plane Radio Waves.- 3.1. Simplification of Initial Equations.- 3.1.1. Nonlinear Wave Equation.- 3.1.2. Nonlinear Geometrical Optics of a Plane Wave.- 3.2. Effect of Nonlinearity on the Amplitude and Phase of the Wave.- 3.2.1. Self-Action of a Weak Wave.- 3.2.2. Self-Action of a Strong Wave.- 3.2.3. Self-Action of Waves in the Case of Artificial lionization.- 3.3. Change of Wave Modulation.- 3.3.1. Weak Wave.- 3.3.2. Change of Amplitude Modulation of Strong Wave.- 3.3.3. Phase Modulation.- 3.3.4. Nonlinear Distortion of Pulse Waveform.- 3.4. Generation of Harmonic Waves and Nonlinear Detection.- 3.4.1. Frequency Tripling.- 3.4.2. Nonlinear Detection.- 3.5. Self-Action of Radio Waves in the Lower Ionosphere.- 4. Interaction of Plane Radio Waves.- 4.1. Cross Modulation.- 4.1.1. Weak Waves.- 4.1.2. Strong Perturbing Wave.- 4.1.3. Resonance Effects near the Electron Gyrofrequency.- 4.2. Interaction of Unmodulated Waves.- 4.2.1. Interaction of Short Pulses.- 4.2.2. Change in the Absorption of a Wave Propagating in a Perturbed Plasma Region.- 4.2.3. Generation of Waves with Combination Frequencies.- 4.3. Radio Wave Interaction in the Lower Ionosphere.- 4.3.1. Cross Modulation.- 4.3.2. Fejer's Method.- 4.3.3. Nonstationary Processes in the Interaction of Strong Radio Waves.- 5. Self-Action and Interaction of Radio Waves in an Inhomogeneous Plasma.- 5.1. Inhomogeneous Electric Field in a Plasma.- 5.1.1. Fundamental Equations.- 5.1.2. Distribution of Density and Temperatures in Plasma.- 5.2. Kinetics of Inhomogeneous Plasma.- 5.2.1. Kinetic Coefficients. Elementary Theory.- 5.2.2. Kinetic Theory.- 5.2.3. Fully Ionized Plasma.- 5.3. Modification of the F Region of the Ionosphere by Radio Waves.- 5.3.1. Modification of the Electron Temperature and of the Plasma Concentration.- 5.3.2. Radio Wave Reflection Region.- 5.3.3. Growth and Relaxation of the Perturbations.- 5.4. Focusing and Defocusing of Radio Wave Beams.- 5.4.1. Nonlinear Geometrical Optics.- 5.4.2. Defocusing of Narrow Beams.- 5.4.3. Mutual Defocusing.- 5.4.4. Thermal Focusing in the Lower Ionosphere.- 6. Excitation of Ionosphere Instability.- 6.1. Self-Focusing Instability.- 6.1.1. Spatial Instability of a Homogeneous Plasma.- 6.1.2. Instability in the Wave-Reflection Region.- 6.2. Resonant Absorption and Resonance Instability.- 6.2.1. Langmuir Oscillations in an Inhomogeneous Plasma.- 6.2.2. Excitation of Plasma Waves.- 6.2.3. Resonance Instability.- 6.2.4. Absorption of Ordinary Radio Waves.- 6.3. Parametric Instability.- 6.3.1. Langmuir Oscillations of a Plasma in an Alternating Field.- 6.3.2. Parametric Excitation of Langmuir Oscillations.- 6.3.3. Parametric Instability in the Ionosphere.- 6.3.4. Dissipative Parametric Instability.

Some Microphysical Properties of an Ice Cloud from Lidar Observation of Horizontally Oriented Crystals

Abstract: Lidar observations of a winter ice cloud in the zenith gave a very high reflection but a very small depolarization. When the lidar was tilted more than 0.5° away from the zenith, the reflection amplitude fell to 3% of it's zenith value, but the depolarization increased. The above properties proved unambiguously that reflection was occurring from the specular surfaces of horizontal crystals. These properties were used to estimate some cloud microphysical properties. At a selected time, the estimates gave a mean “diameter” of 74 μm for the horizontal faces, a crystal number density of 0.78 l−1, and a maximum departure of the crystal axis from the horizontal of 0.5°. The fraction of the total crystal cross section which was specularly reflecting was estimated as unity.

Interaction of electromagnetic waves with a moving ionization front

Abstract: The problem of an electromagnetic wave packet, incident on a relativistically moving plane ionization or recombination front in a stationary gas, is considered. The frequency of the reflected wave packet is found to obey the usual double Doppler shift relation. However, the reflection coefficients and the physics can differ significantly from the case of reflection from moving material objects. In the unmagnetized case, the ratio of the energy in the reflected wave packet to that in the incident wave packet is found to be er*/ei*=ωi*/ωr* for an oncoming overdense ionization front, for which ωi*<ωr* (where ωi* and ωr* are the incident and reflected wave frequencies in the laboratory frame). For an oncoming ionization front in the presence of an applied magnetic field normal to the front, er*/ei* can considerably exceed ωi*/ωr*. For a retreating recombination front (ωi*≳ωr*), in the overdense unmagnetized case, er*/ei*=ωr*/ωi*. These results have important implications for the production of sub‐millimeter w...

Interaction of magnetostatic waves with a current

Abstract: It is shown that the interaction of magnetostatic waves with a current can be characterized by a coupling constant analogous to, but much greater than, the piezoelectric coupling constant. The theory is applied both to surface and forward‐traveling volume magnetostatic waves, and the problems of excitation, reflection, and absorption by a single microstrip and by a narrow‐band interdigital transducer are treated. It is found that the coupling constant is close to 0.5 for any reciprocal wave, and is greater for any nonreciprocal wave, when the transducer is in contact with the magnetic medium. These values are too large to allow effective signal processing, since the emitted waves react strongly on the transducer; narrow‐band transducers must be lifted above the surface so as to weaken the coupling.

Observation of amplified phase‐conjugate reflection and optical parametric oscillation by degenerate four‐wave mixing in a transparent medium

Abstract: We report on the observation of amplified reflection and optical parametric oscillation via degenerate four‐wave mixing in a nonresonant medium. The process is mediated through the third‐order nonlinear susceptibility in a transparent liquid medium, CS2. A collinear mixing geometry is utilized to obtain long interaction lengths and polarization discrimination is used to separate the pump and signal fields.

Structure of the East Pacific Rise crest from multichannel seismic reflection data

Abstract: Computer-processed multichannel seismic reflection profiles from the crestal zone of the East Pacific Rise near the Siqueiros Fracture Zone have revealed distinct layering in the structure not previously observed in single-channel reflection records. After processing, which included common depth point gathering, normal move-out and stack, time-varying predictive deconvolution, and wave equation migration, three distinct crustal layers emerge as coherent events across the rise crest. The base of the third layer appears to be at a depth of about 2 km below the sea floor and may represent the top of a low-velocity magma chamber, as postulated by Orcutt et al. (1975, 1976) from the results of ocean bottom seismometer refraction data. This layer appears to thicken away from the rise crest toward the edge of the raised axial block that characterizes the East Pacific Rise (Anderson and Noltimier, 1973). Illustrated in this article is the ability of modern processing techniques as applied to multichannel seismic data to enhance weak arrivals, to minimize sound source bubble-pulse reverberations, and to remove the diffraction patterns caused by rough topography.

The reflection, refraction, and diffraction of waves in media with an elliptical velocity dependence

Abstract: Assuming media having a velocity dependence on angle which is an ellipse, we have confirmed previously reported time‐distance relations for reflections from single interfaces, for reflections from sections of beds separated by horizontal interfaces, for refraction arrivals, and added the expression for diffractions. We also have derived expressions for plane‐wave reflection and transmission coefficients at an interface separating two transversely isotropic media. None of the properties differs greatly from those for isotropic media. However, velocities found from seismic surface reflections or refractions are horizontal components. There seems to be no way of obtaining vertical components of velocity from surface measurements alone and hence no way to compute depths from surface data.

Radar anisotropy of sea ice due to preferred azimuthal orientation of the horizontal c axes of ice crystals

Abstract: Results of impulse radar, ice crystal c axis, and subice current measurements on the fast ice near Narwhal Island, Alaska, are presented The crystal structure of the ice was found to have a horizontal crystal c axis with a preferred azimuthal orientation This orientation was found to align with the direction of the current at the ice-water interface Impulse radar reflection measurements revealed that the preferred orientation of the sea ice crystal structure behaved as a microwave polarizer It was observed that when the antenna E field was oriented parallel with the c axis of the crystal platelets, a strong reflection of the radar signal from the bottom of the ice was obtained However, when the antenna E field was oriented perpendicular to the c axis, no bottom reflection was detected The results of this study fully support earlier reports of sea ice inhomogeneity and anisotropy in reference to both structure and electromagnetic energy transmission

Glorified optics and wave propagation in nonplanar structure

Abstract: Waves propagating in varying nonplanar structure can produce many interesting phenomena, such as focusing, caustics, and triplications. A high-frequency technique based on the first-motion approximation, referred to as glorified optics, has been developed to generate synthetic seismograms for these types of problems. The technique, in its simplest form, uses the spreading rate of a beam with transmission and reflection coefficients along each possible ray path. The time behavior of each arrival is either that of the original pulse or its Hilbert transform depending on the position of caustics. The geophysically interesting structure of a soft basin over a half-space is investigated in detail by this method. Synthetic seismograms appropriate for various locations are compared with the results of finite difference and finite element methods. The technique appears rich in insight and should prove very useful in modeling problems.

Symmetries in the reflection and transmission of elastic waves

Abstract: Summary. The symmetry relations between the reflection and transmission coefficients for plane elastic waves incident upon an arbitrary horizontally stratified medium are derived by a novel approach. Previous results, particularly for a single interface, are obtained as special cases of this treatment. In addition, for perfectly elastic media, projection operators for travelling and evanescent waves are introduced and used to derive a number of new relationships between the reflection and transmission coefficients.

High-efficiency pulsed 10.6-Mu m phase-conjugate reflection via degenerate four-wave mixing.

Abstract: We present the first reported observation in the infrared of nonlinear phase-conjugated reflection. This was achieved via degenerate four-wave mixing in polycrystalline germanium. The observation was facilitated by taking advantage of the counterpropagating (strong) waves internal to a pulsed CO2 laser cavity. The measured effective reflectivity was 2% with a 10-mm interaction length. This simple intracavity technique is generally applicable to any nonlinear material transparent at the wavelength of the laser into which it is inserted.

Electromagnetic wave propagation in inhomogeneous plasmas

Abstract: The absorption and transformation of electromagnetic waves in the vicinity of electron cyclotron harmonics is considered. The results of a simple WKB theory and a full wave theory are compared. It is found that for waves perpendicularly incident on a resonant surface (where the wave frequency equals an integer multiple of the electron cyclotron frequency), the full wave treatment is necessary to determine reflection and mode conversion coefficients, whereas, the WKB theory and full wave theory predict the same transmission coefficient. As the angle of incidence is decreased from 90°, the reflection and mode conversion coefficients become unimportant so that the full wave theory and WKB theory agree. The relevance of these processes to the heating of toroidal plasmas with high frequency waves is discussed.

Ellipsoid-conic radiation collector and method

Abstract: Disclosed is a radiation collector apparatus and method primarily for counting and analyzing a flow of dilute particulate material, such as blood cells, sperm cells and the like, through the use of light detection. The radiation collector apparatus comprises a reflector chamber having an ellipsoidal reflector surface with a pair of elipsoidal foci defining a first focus, f 11 , and second focus, f 12 , and a second reflector surface with a primary focus, f 21 , positioned at the same point as focus f 12 , and a secondary focus, f 22 . The second reflector surface has the configuration of one of the conic sections of revolution. In operation the radiation collector apparatus is provided with an intensifed beam of light and a stream of particulate material aligned to intersect the intensifed beam of light at focus f 11 . Detectable light signals, after two reflections, are received in a focused beam by a photosensitive detector.

Optical behavior of a metal island film

Abstract: Calculations for optical reflection, transmission, and attenuated-total-reflection of a two-dimensional distribution system of metal particles are carried out in two different ways; one is the usual way utilizing a fictitious plane-parallel film model and the other is the integration method of scattered waves. Both results are compared and discussed.

X-ray photoabsorption of solids by specular reflection

Abstract: A method to obtain the X-ray photoabsorption spectra of a bulk sample based on the specular reflection of continuous radiation on its surface is proposed.

Exact effect of surface roughness on the reverberation time of a uniformly absorbing spherical enclosure

Abstract: We assess the validity of the Sabine, Eyring, and Kuttruff reverberation‐time expressions, and of their underlying mean‐path‐length formula 〈l〉=4V/S, by determining the exact reverberation time and asymptotic sound distribution for a uniformly absorbing spherical enclosure with any amount of air absorptivity and with a surface that can be continuously varied from nonabsorbing to completely absorbing and from specular to randomly reflecting. In disagreement with an extensive literature that presents the Eyring formula as a correction to that of Sabine, we find, on the one hand, that Sabine’s formula and the applicability of 4V/S are vindicated in this application under Sabine’s stated conditions of weak absorptivity and (any nonzero amount of) irregular reflection, while, on the other hand, we find that Eyring’s and Kuttruff’s expressions are less accurate than Sabine’s unless the roughness and absorptivity of the surface exceed certain levels which are evaluated. Related concepts are discussed in some detail. Geometrical acoustics is assumed throughout, and its limitations are not considered here. The analysis is equally applicable to light in LEDs (light‐emitting diodes) and to mechanical particles which stick or experience elastic reflection at the surface of their container.

A new method for determining thin‐film refractive index and thickness using guided optical waves

Abstract: We have developed a new method for determining thin‐film refractive index and thickness. In this technique, a sharply converging laser beam is coupled into the film by a prism. Dark mode lines characteristic of the waveguide structure are observed in reflection independently of absorption or scattering loss in the film. A particularly promising application is the rapid sampling of dielectric films in semiconductor technology.

Reflection of X rays by neutron star Surfaces

Abstract: A neutron-star surface should act as a good reflector for photons up to hard X-ray energies, due to the high density of the magnetic surface matter. Reflectivities are high, especially in spectral bands above the plasma and cyclotron frequencies depending on the photon polarisation. X-ray reflection may be important for the beaming of the binary neutron star Her X-1 and similar objects.

Multiple-reflection diffuse-scattering model for noise propagation in streets

Abstract: The sound field generated by an omnidirectional point source in an infinitely long, straight street is considered. The field is assumed to be the sum of a multiply‐specularly reflected field and a diffuse field that is fed from scattering at the walls at each reflection of the specular field. It is shown that scattering is important close to the source. The sound level depends on the width of the street and the height of the walls and on the reflection and scattering coefficients of the walls.

Contrast in reflection acoustic microscopy

Abstract: Although the reflection acoustic microscope has long been used to study solid surfaces, the contrast that has been observed in images of complex i.c. devices were not well understood until fairly recently. Our recent article where we calculated the acoustic response of different materials suggested that the observed contrast was a strong function of the elastic properties of the surface being investigated. This letter is devoted to explaining the analysis used in our calculations.

Synthetic seismic sections of typical petroleum traps

Abstract: Interpretation of seismic reflections remains a key problem in seismic exploration because reflections have complex behavior, especially near geologic structures. One method to gain an understanding of this complex behavior is to study synthetic seismic sections of models of typical petroleum traps as computed by zero‐offset ray tracing for primary P‐waves only. These synthetic sections have features of significant interpretative value to the practicing geophysicist, such as variations in reflection amplitudes and complexities in reflection‐time geometries. Asymptotic ray theory was applied to calculate reflection amplitudes, accounting for mode conversion and three‐dimensional geometric divergence of ray tubes in the presence of curvilinear interfaces. This suite of synthetic seismic sections illustrates the difficulties in making correct seismic interpretations of geologic structures and suggests three conclusions: (1) The customary assumption that seismic sections are simple images of geologic cross‐se...

Surface‐wave rays in elastodynamic diffraction by cracks

Abstract: A geometrical diffraction theory has been worked out to analyze the fields generated by diffraction of high‐frequency waves by cracks. The theory accounts for curvature of incident wavefronts, curvature of crack edges, and finite dimensions of the crack by providing first‐order corrections to the results for a semi‐infinite crack. The diffracted fields include direct diffractions from the crack edges as well as well as diffractions of signals which travel via the crack faces. On the faces of the crack the main contributions to the diffracted fields come from rays of surface waves. The directions of these surface‐wave rays, and the amplitudes, wavelengths, and phases of the associated surface‐wave motions have been related to the corresponding quantities of the incident body‐wave rays. Reflection and diffraction of surface‐wave rays by the edge of a crack have also been analyzed. As an example, diffraction by a penny‐shaped crack of a plane longitudinal wave under normal incidence has been considered in some detail. Explicit expressions are given for the diffracted fields. In these expressions a correction was introduced to extend the validity of the results to the normal axis through the center of the crack, which is a caustic axis. A simple expression for the scattering cross section is presented.