Showing papers on "Plane wave published in 1983"

Field and wave electromagnetics

Abstract: 1. The Electromagnetic Model. Introduction. The Electromagnetic Model. Si Units and Universal Constants. Review Questions. 2. Vector Analysis. Introduction. Vector Addition and Subtraction. Products of Vectors. Orthogonal Coordinate Systems. Integrals Containing Vector Functions. Gradient of a Scalar Field. Divergence of a Vector Field. Divergence Theorem. Curl of a Vector Field. Stoke's Theorem. Two Null Identities. Helmholtz's Theorem. Review Questions. Problems. 3. Static Electric Fields. Introduction. Fundamental Postulates of Electrostatics in Free Space. Coulomb's Law. Gauss's Law and Applications. Electric Potential. Conductors in Static Electric Field. Dielectrics in Static Electric Field. Electric Flux Density and Dielectric Constant. Boundary Conditions for Electrostatic Fields. Capacitances and Capacitors. Electrostatic Energy and Forces. Solution of Electrostatic Boundary-Value Problems. Review Questions. Problems. 4. Solution of Electrostatic Problems. Introduction. Poisson's and Laplaces' Equations. Uniqueness of Electrostatic Functions. Method of Images. Boundary-Value Problems in Cartesian Coordinates. Boundary-Value Problems in Cylindrical Coordinates. Boundary-Value Problems in Spherical Coordinates. Review Questions. Problems. 5. Steady Electric Currents. Introduction. Current Density and Ohm's Law. Electromotive Force and Kirchoff's Voltage Law. Equation of Continuity and Kirchoff's Current Law. Power Dissipation and Joule's Law. Boundary Conditions for Current Density. Resistance Calculations. Review Questions. Problems. 6. Static Magnetic Fields. Introduction. Fundamental Postulates of Magnetostatics in Free Space. Vector Magnetic Potential. The Biot-Savart Law and Applications. The Magnetic Dipole. Magnetization and Equivalent Current Densities. Magnetic Field Intensity and Relative Permeability. Magnetic Circuits. Behavior of Magnetic Materials. Boundary Conditions for Magnetostatic Fields. Inductances and Inductors. Magnetic Energy. Magnetic Forces and Torques. Review Questions. Problems. 7. Time-Varying Fields and Maxwell's Equations. Introduction. Faraday's Law of Electromagnetic Induction. Maxwell's Equations. Potential Functions. Electromagnetic Boundary Conditions. Wave Equations and their Solutions. Time-Harmonic Fields. Review Questions. Problems. 8. Plane Electromagnetic Waves. Introduction. Plane Waves in Lossless Media. Plane Waves in Lossy Media. Group Velocity. Flow of Electromagentic Power and the Poynting Vector. Normal Incidence of Plane Waves at a Plane Conducting Boundary. Oblique Incidence of Plane Waves at a Plane Conducting Boundary. Normal Incidence of Plane Waves at a Plane Dielectric Boundary. Normal Incidence of Plane Waves at Multiple Dielectric Interfaces. Oblique Incidence of Plane Waves at a Plane Dielectric Boundary. Review Questions. Problems. 9. Theory and Application of Transmission Lines Introduction. Transverse Electromagnetic Wave Along a Parallel-Plate. Transmission Line General Transmission-Line Equations. Wave Characteristics on Finite Transmission Lines. Transients on Transmission Lines. The Smith Chart. Transmission-Line Impedance Matching. Review Questions. Problems. 10. Waveguides and Cavity Resonators. Introduction. General Wave Behaviors Along Uniform Guiding Structures. Parallel-Plate Waveguide. Rectangular Waveguides. Circular Waveguides. Dielectric Waveguides. Cavity Resonators. Review Questions. Problems. 11. Antennas and Radiating Systems. Introduction. Radiation Fields of Elemental Dipoles. Antenna Patterns and Antenna Parameters. Thin Linear Antennas. Antenna Arrays. Receiving Antennas. Transmit-Receive Systems. Some Other Antenna Types. Review Questions. Problems. Appendix A: Symbols and Units. Appendix B: Some Useful Material Constants. Bibliography. Answers to Selected Problems. Index. Back Endpapers.

Three-dimensional vector coupled-wave analysis of planar-grating diffraction

Abstract: Diffraction by an arbitrarily oriented planar grating with slanted fringes is analyzed using rigorous three-dimensional vector coupled-wave analysis. The method applies to any sinusoidal or nonsinusoidal amplitude and/or phase grating, any plane-wave angle of incidence, and any linear polarization. In the resulting (conical) diffraction, it is shown that coupling exists between all space-harmonic vector fields inside the grating (corresponding to diffracted orders outside the grating). Therefore the TE and TM components of an incident wave are each coupled to all the TE and TM components of all the forward- and backward-diffracted waves. For a general Bragg angle of incidence, it is shown that the diffraction efficiency can approach 100% for a lossless grating if either the incident electric field or the magnetic field is perpendicular to the grating vector. Maximum coupling between incident and diffracted waves is shown to occur when the incident electric field is perpendicular to the grating vector. In general, the diffracted waves are shown to be elliptically polarized. The three-dimensional vector coupled-wave analysis presented is shown to reduce to ordinary rigorous coupled-wave theory when the grating vector lies in the plane of incidence.

How can a particle absorb more than the light incident on it

Abstract: A particle can indeed absorb more than the light incident on it. Metallic particles at ultraviolet frequencies are one class of such particles and insulating particles at infrared frequencies are another. In the former strong absorption is associated with excitation of surface plasmons; in the latter it is associated with excitation of surface phonons. In both instances the target area a particle presents to incident light can be much greater than its geometrical cross‐sectional area. This is strikingly evident from the field lines of the Poynting vector in the vicinity of a small sphere illuminated by a plane wave.

Electromagnetic modeling of three-dimensional bodies in layered earths using integral equations

Abstract: An algorithm based on the method of integral equations to simulate the electromagnetic responses of three-dimensional bodies in layered earths has been developed. The inhomogeneities are replaced by an equivalent current distribution which is approximated by pulse basis functions. A matrix equation is constructed using the electric tensor Green's function appropriate to a layered earth, and it is solved for the vector current in each cell. Subsequently, scattered fields are found by integrating electric and magnetic tensor Green's functions over the scattering currents. Efficient evaluation of the tensor Green's functions is a major consideration in reducing computation time. We find that tabulation and interpolation of the six electric and five magnetic Hankel transforms defining the secondary Green's functions is perferable to any direct Hankel transform calculation using linear filters. A comparison of responses over elongate three-dimensional (3-D) bodies with responses over two-dimensional (2-D) bodies of identical cross-section using plane wave incident fields is the only check available on our solution. Agreement is excellent; however, the length that a 3-D body must have before departures between 2-D transverse electric and corresponding 3-D signatures are insignificant depends strongly on the layering. The 2-D transverse magnetic and corresponding 3-D calculations agree closely regardless of the layered host. (Author)

Reflection of elastic waves from periodically stratified media with interfacial slip

Abstract: A periodically stratified elastic medium can be replaced by an equivalent homogeneous transverse isotropic medium in the long wavelength limit. The case of a homogeneous medium with equally spaced parallel interfaces along which there is imperfect bonding is a special instance of such a medium. Slowness surfaces are derived for all plane wave modes through the equivalent medium and reflection coefficients for a half-space of such a medium are found. The slowness surface for the SH mode is an ellipsoid. The exact solution for the reflection of SH-waves from a half-space with parallel slip interfaces is found following the matrix method of K. Gilbert applied to elastic waves. Explicit results are derived and in the long wavelength limit, shown to approach the results for waves in the equivalent homogeneous medium. Under certain conditions, a half-space of a medium with parallel slip interfaces has a reflection coefficient independent of the angle of incidence and thus acts like an acoustic reducing mirror. The method for the reflection of P- and SV-waves is fully outlined, and reflection coefficients are shown for a particular example. The solution requires finding the eigenvalues of a 4 × 4 transfer matrix, each eigenvalue being associated with a particular wave. At higher frequencies, unexpected eigenvalues are found corresponding to refracted waves for which shear and compressional parameters are completely coupled. The two eigenvalues corresponding to the transmitted wavefield give amplitude decay perpendicular to the stratification along with up- and downgoing phase propagation in some other direction. Much of this work was performed while the author was at the Department of Geophysics and Planetary Sciences, Tel-Aviv University, Ramat-Aviv, Israel. The author is grateful for illuminating discussions with K. Helbig and K. Gilbert.

Kinematics of Turbulence Convected by a Random Wave Field

Abstract: Turbulent velocity spectra measured beneath wind waves show a large enhancement about the central wave frequency. A “5/3" frequency dependence can be seen both above and below the central peak, but with an apparent increase in spectral density at high Frequencies. We show that these features can be understood via a generation of Taylor's hypothesis to the case in which frozen, isotropic, homogeneous turbulence is bodily convected past a fixed probe by a combination of drift and wave orbital motions. In a monochromatic wave field turbulent energy is aliased into harmonics of the wave frequency fp. We show qualitatively how drift currents or a random wave field broaden these lines into a continuous spectrum, and present the results of direct calculations which demonstrate clearly the transition from “line-like” to a smooth “5/3" spectrum. We calculate the leading asymptotic behavior in the limit of large and small frequencies for an arbitrary wave-height spectrum. For wave orbital velocities larger...

Polarization Effects in the Diffraction of Electromagnetic Waves: The Role of Disclinations

Abstract: Three-dimensional diffraction patterns of monochromatic electromagnetic waves contain moving line singularities where the magnitude of the transverse field is zero, and its direction is therefore indeterminate. They are called disclinations, by analogy with the corresponding linear features in liquid crystals. A disclination in a vector wave is a natural generalization of a dislocation in a scalar wave. Where the scalar wave approximation in optics predicts a dislocation, or interference null, the full vector theory reveals a double helix structure: a disclination line in the electric field and another in the magnetic field winding around each other with a spacing of order (wavelength/2$\pi$). A perturbing plane wave causes this composite structure itself to become coiled. When there is a 'polarization effect' in the diffraction pattern a disclination in the electric field becomes a moving helix or, more generally, a coiled coil. As it moves it sweeps out a surface on which the polarization is everywhere linear. In optical experiments this observable surface is the most significant effect of disclinations. In general, however, the disclinations constitute elements of structure of the electromagnetic field, and their arrangement thus provides an effective way of describing the three-dimensional geometry of even very complicated diffraction fields, for example of microwaves.

Theory of a self-pumped phase conjugator with two coupled interaction regions

Abstract: We present a plane-wave analysis of a recently demonstrated self-pumped phase conjugator. This device uses four-wave mixing to produce the phase-conjugate replica of an incident optical wave. All the waves are derived from the single incident wave: there are no externally supplied pumping beams. We consider the case of four-wave mixing in two interaction regions coupled by simple reflection. We calculate the phase-conjugate reflectivity as a function of coupling strength, taking into account imperfect coupling between the two interaction regions, and show that there is a threshold coupling strength below which the reflectivity is zero and above which the reflectivity is multiple valued. We also compute the coupling strength per unit length for a photorefractive crystal of barium titanate.

Beam propagation in uniaxial anisotropic media

Abstract: Paraxial wave equations are derived for the propagation of beams in uniform uniaxial anisotropic media. The equations are generalized to the case of nonuniform media with weakly varying refractive indices. An ordinary wave beam is governed by a standard paraxial equation, whereas an extraordinary wave beam is governed by a paraxial wave equation, which involves both a displacement relative to the position of an ordinary wave beam and a rescaling of one transverse coordinate. The solution to the latter equation for a propagating Gaussian beam displays a distortion of both shape and phase front. Numerical results for diffraction by a uniformly illuminated circular aperture in a calcite medium display various anomalies ascribable to a loss of circular symmetry.

Modulational instability of finite amplitude dispersive Alfvén waves

Abstract: The stability of a finite amplitude plane monochromatic circularly polarized Alfven wave is studied by using an approximate two-fluid model obtained by performing an amplitude and dual time scale expansion in which temporal changes, observed when moving with the wave, are assumed slow. The lowest order dispersive effects and coupling terms to sound waves are of the same order. A large-amplitude Alfven wave obeying an approximate dispersion relation is an exact solution of the resulting model equations. Small perturbations of this solution consisting of two sideband Alfven waves and a sound wave are then introduced. The resulting dispersion relation, which is of fourth order, is examined analytically for small wave amplitudes of the parent wave. It reveals different regions of stability and instability in a plot of wave number k0 of the unperturbed wave versus β0, the ratio of sound speed to Alfven speed. For long wave lengths, the left-hand polarized mode is found to be stable for β0 > 1 and the right-hand polarized mode for β0 1. Decay instability is predicted near β0 = 0 and β0 = 1. Formulas for growth rates and unstable wave number bands are given.

The Korteweg–de Vries Equation, Posed in a Quarter-Plane

Abstract: An initial- and boundary-value problem for the Korteweg–de Vries equation is shown to be well-posed. The considered problem may serve as a model for unidirectional propagation of plane waves generated by a wavemaker in a uniform medium. Such models apply in regimes in which nonlinear and dispersive effects are of comparable small order.

First-order equivalent current and corner diffraction scattering from flat plate structures

Abstract: The equivalent current concept is used to compute the scattering patterns of flat plate structures. It is also used to obtain the broadside scattering lobe for any incidence plane. The essential feature introduced in this paper is that only the components of the equivalent current perpendicular to the incidence plane are used. No special treatment of the singularity in the plane wave diffraction coefficient (which is the basis of the equivalent current concept) is required. Instead, this choice of equivalent current components is such that the singularity at one edge segment is canceled by the singularity at the opposite edge segment. For modern day computers there is sufficient accuracy that the main scattering lobe can be obtained in the limit as one approaches broadside. The same results can be obtained for plate structures made of straight edges by using a new corner diffraction analysis. For certain cases where the observation angle is sufficiently removed from normal incidence to an edge, the corner diffraction analysis appears to yield more accurate results.

Double beta decay

Abstract: The double β decay is investigated by the use of the relativistic Coulomb wave function including the finite nuclear size effect. A general form of the V±A interactions is considered. It is shown that the decay rate is enhanced considerably in comparison with the case of the plane wave multiplied by the non-relativistic Fermi factor.

Coherent scattering of a spherical wave from an irregular surface

Abstract: The scattering of a spherical wave from a rough surface using the Kirchhoff approximation is considered. An expression representing the measured coherent scattering coefficient is derived. It is shown that the sphericity of the wavefront and the antenna pattern can become an important factor in the interpretation of ground-based measurements. The condition under which the coherent scattering-coefficient expression reduces to that corresponding to a plane wave incidence is given. The condition under which the result reduces to the standard image solution is also derived. In general, the consideration of antenna pattern and sphericity is unimportant unless the surface-height standard deviation is small, i.e., unless the coherent scattering component is significant. An application of the derived coherent backscattering coefficient together with the existing incoherent scattering coefficient to interpret measurements from concrete and asphalt surfaces is shown.

Receiver-aperture averaging effects for the intensity fluctuation of a beam wave in the turbulent atmosphere

Abstract: Using a Gaussian weighting function for the receiver aperture, we obtain a closed-form representation for the receiver-aperture averaging effect for the intensity fluctuation of a beam wave in the turbulent atmosphere. It is shown that, unlike for the plane-wave case, the power scintillations do not always decrease when the receiver aperture is increased. The reasons are that (1) the intensity fluctuations on the axis for a coherent beam-wave source are smaller than these off the axis and (2) the averaging effect cannot show up when the total beam is within a coherent patch (i.e., the coherence length is larger than the beamwidth).

Distortion of finite amplitude ultrasound in lossy media

Abstract: A form of Burgers’ equation is used to derive an algorithm for calculating harmonic generation by a continuous plane wave of ultrasound propagating in a nonlinear, lossy, nondispersive medium. The algorithm accounts for attenuation that is not quadratically related to the frequency of the wave. Attenuation strongly affects the rate of harmonic production. The effect of variations of the relationship between attenuation and frequency is shown. Biological tissue is an example of a highly lossy medium where the attenuation does not increase with the square of the frequency. Calculations for several types of tissue and biological fluids are presented that show, for certain conditions, finite amplitude distortion is possible.

Reflection from a periodically perforated plane using a subsectional current approximation

Abstract: The scattering from a zero thickness plane having finite sheet resistance and perforated periodically with apertures is calculated for arbitrary plane wave illumination. The surface current density within the unit cell is approximated by a finite number of current elements having rooftop spatial dependence. The transverse electric field is expressed in terms of the current, and the electric field boundary condition is satisfied in an integral sense over the conductor, generating a finite dimension matrix equation whose solution is the current density. Since the conductor shape is defined through the locations of subsectional current elements, arbitrary shaped apertures can be handled. The reflection coefficient and current distribution are calculated for square apertures in both perfectly conducting and resistive sheets, and for cross-shaped apertures. Finite resistivity is shown to cause the magnitude of the transverse magnetic (TM) reflection coefficient to decrease more rapidly and its phase to decrease less rapidly, as the angle of incidence approaches glancing. Through detailed plots of the current density, the current crowding around the apertures is made clearly evident.

Diffraction of Elastic Waves by a Penny-Shaped Crack: Analytical and Numerical Results

Abstract: The diffraction of time-harmonic stress waves by a penny-shaped crack in an infinite elastic solid is an important problem in fracture mechanics and in the theory of the ultrasonic inspection of materials Martin (Proc R Soc Lond A 378, 263 (1981)) has proved that the corresponding linear boundary-value problem has precisely one solution, and that this solution can be constructed by solving a two-dimensional Fredholm integral equation of the second kind However, this integral equation has a complicated matrix kernel and the components of its vector solution are coupled The main purpose of the present paper is to show how Martin's integral equation can be explicitly solved in terms of a sequence of functions, each of which satisfies a very simple scalar integral equation of the second kind; this simplification may be made for any incident wave For an incident plane wave, further simplifications are possible We show that the solution at an arbitrary angle of incidence can be derived from the solution at a particular angle of incidence, namely grazing incidence The resulting computational procedure is especially attractive if only the stressintensity factors or the far-field displacements are required Finally, we present some numerical results for the scattering of a P-wave at normal incidence and an SV-wave at oblique incidence, and compare these with those of other authors

Free-space wave propagation beyond the paraxial approximation

Abstract: Wave propagation in free space is considered without invoking the paraxial approximation. An explanation is provided for the discrepancy which arises when the general formalism is applied to the case of a Gaussian beam.

The viscoelastic reflection/transmission problem: Two special cases

Abstract: In the general problem of plane wave reflection and transmission at a boundary separating two linear viscoelastic media, the mathematical formulas for the reflection and transmission coefficients, the transmission angle, the attenuation vector, etc., are not easily interpretable because they cannot easily be expressed in terms of the basic input parameters ( Q , incidence angle, etc.). To gain further insight, we study two special cases in which mathematical simplifications occur. No low-loss approximations are involved. In the first case, the incident wave is homogeneous, and the Q values of the two layers are equal, and we find, among other things, that the reflection and transmission coefficients are the same as the ones for perfect elasticity (they do not involve complex velocities, etc., and are independent of Q ). In the second special case, the degree of inhomogeneity of the incident wave approaches its upper limit, and we find that the reflection and transmission coefficients approach constant (complex) values independent of the incidence angle, and that there is almost no ray-bending (refraction) upon transmission of the incident wave through the boundary.

Scattering of electromagnetic waves from a half space of densely distributed dielectric scatterers

Abstract: The scattering of a plane wave obliquely incident on a half space of densely distributed spherical dielectric scatterers is studied. The quasi-crystalline approximation is applied to truncate the hierarchy of multiple scattering equations, and the Percus-Yevick and the Verlet-Weis results are used to represent the pair distribution function. The coherent reflected wave is studied with these approximations. The incoherent scattered wave is calculated with the distorted Born approximation. In the low-frequency limit, closed-form expressions are obtained for the effective propagation constants, the coherent reflected wave, and the bistatic scattering coeficients. Results at higher frequencies are calculated numerically. The advantage of the present approach is that, in the low-frequency limit, it reproduces the effects of specular reflection, Fresnel reflection coefficient, Brewster angle, and Clausius-Mosotti relation. In addition to the classical results, the bistatic scattering coefficients are also calculated. The theory is also applied to match backscattering data from dry snow at microwave frequencies.

Ray analysis of two-dimensional radomes

Abstract: The fields transmitted from a line source through a two-dimensional curved dielectric layer of variable thickness are constructed by geometric-optical ray tracing that accounts for multiple reflections on the concave side, where the source is located, as well as for reflections between the layer boundaries. Moreover, internally trapped rays, excited by evanescent tunneling, are included when source and observer are near the layer boundaries but laterally displaced along it. By applying Poisson summation to the ray series, the multiple reflected contributions, for weakly tapered configurations, can be summed to yield trapped and leaky local modes guided along the layer, as well as a "collective" ray field that incorporates a plane layer transmission coefficient, with curvature and slope corrections, instead of the conventional coefficients for individual boundaries. Detailed calculations are performed for the special cases of a circularly curved layer of constant thickness and a tapered layer with nonparallel plane boundaries. For the former, the various ray-optically derived solutions agree completely with those obtained from a rigorous analysis. For the latter nonseparable configuration, no rigorous solution is available. With direct summation of conventional ray fields taken as a reference, extensive numerical results demonstrate the economies effected by the collective ray formulation and the importance of including the curvature or slope corrections in the equivalent slab transmission coefficients.

Wave scattering and guidance by dielectric waveguides with periodic surfaces

Abstract: The extended-boundary-condition method is used to solve the wave scattering and guidance by a stratified medium with a periodic surface. First, the transition matrix for two sets of Flocquet waves incident from above and below a periodic surface separating the two media is derived. The results are applied to the diffraction of an incident plane wave by a multilayered medium having a periodic surface. The theoretical results can easily be applied to the guidance problem for which the complex guiding constant of the periodic structure is obtained. Numerical results for both the scattering and the guidance problems are obtained. The theory is compared with experiment for metallic gratings with surface-plasmon excitations. The calculated guiding constants of a thin film with a periodic sinusoidal surface are also compared with other numerical methods.

Dispersion relation for propagation of light in cholesteric liquid crystals

Abstract: A general theory of the light propagation in periodic structures characterized by a uniform rotation of the dielectric tensor about a given axis is presented. Starting from a fundamental approach of Dreher and Meier, which is mostly numerical, an analytical solution of the characteristic equation has been found which can be used to calculate the wave vectors as a function of $\ensuremath{\omega}$ and of the incidence angle ${\ensuremath{\theta}}_{i}$. The electromagnetic wave is described as a superposition of elementary modes having the form of Bloch waves. Each elementary mode is represented by a sum of plane waves elliptically polarized, whose wave vectors are the roots of the characteristic equation. The analysis of the solutions of such an equation allows us to draw a more complete map of the stability and instability regions for light propagation in helical structures than the ones currently available in the literature. The coexistence of two distinct modes, with different polarization states, determines the shape of the stability map. Each mode presents a series of Bragg instabilities. Between the two Bragg instabilities of the same order a further instability exists which is common to both modes and does not satisfy the Bragg conditions. All instability bands, with the exception of only one of the first order, vanish at normal incidence. This occurs for any value of the optical anisotropy and is a peculiarity of perfectly ordered helical structures. The bandwidth increases with ${\ensuremath{\theta}}_{i}$, and overlapping may occur. Typical plots of dispersion curves and attenuation constants are reported. Finally, we compute the intensity and the polarization state of the light reflected from a thin film, in order to clarify the controversial point about the structure\char22{}doublet or triplet\char22{}of the higher-order reflection bands.

Spectral density of millimeter wave amplitude scintillations in an absorption region

Abstract: A detailed theory including the effect of water vapor fluctuations for the spectral density of the log amplitude scintillations of a radio wave propagating in an absorption medium is presented. The scintillation spectra obtained from links at 55.5 and 36.1 GHz on a common 4.1 km path are given together with the relevant meteorological data. Results show that the lower corner frequency predicted by Ott and Thompson, for the enhancement of the scintillations in an absorption region, is a good approximation.

Electromagnetic plane wave scattering by a system of two parallel conducting prolate spheroids

Abstract: By means of modal series expansions of electromagnetic fields in terms of prolate spheroidal vector wave functions, as exact solution is obtained for the scattering by two perfectly conducting prolate spheroids in parallel configuration, the excitation being a monochromatic plane electromagnetic wave of arbitrary polarization and angle of incidence. Using the spheroidal translational addition theorems recently presented by the authors which are necessary for the two-body (or multibody) scattering solution, an efficient computational algorithm of the translational coefficients is given in terms of spherical translational coefficients. The field solution gives the column vector of the series coefficients of the scattered field in terms of the column vector of the series coefficients of the incident field by means of a matrix transformation in which the system matrix depends only on the scatterer ensemble. This eliminates the need for repeatedly solving a new set of simultaneous equations in order to obtain the scattered field for a new direction of incidence. Numerical results in the form of curves for the bistatic and monostatic radar cross sections are given for a variety of prolate spheroid pairs having resonant or near resonant lengths.