Helical Rays Research Articles

Quantitative ray methods for scattering by tilted cylinders

After a review of ray methods and phenomena associated with high frequency scattering by spheres and cylinders in water viewed broadside, generalizations to tilted cylinders relevant to spectral and imaging domains will be summarized. These extensions were found to be useful for meridional as well as helical ray backscattering enhancements associated with leaky (or supersonic) waves on shells [F. J. Blonigen and P. L. Marston, J. Acoust. Soc. Am. 112, 528–536 (2002)]. For such enhancements, Fermat’s principle is useful for identifying ray paths of interest. In the case of helical waves (and in the broadside special case) the scattering amplitude can be expressed in terms of a Fresnel patch area where the guided wave is excited on the shell. Fresnel patches also give insight into the relatively large magnitude of meridional ray contributions. The coupling coefficient is proportional to the radiation damping of the leaky wave. For some shells it is necessary to take into account the anisotropy of the phase velocity. Computational benchmarks include scattering into the meridional plane by tilted infinite cylinders. Related phenomena include time-frequency domain features and enhancements from waves crossing truncations and from subsonic and negative group velocity guided waves. [Research supported by ONR.]

Modelling of EM propagation in simplified jet turbine structure using helical rays

To explain the variations in field strength measurements seen when measuring radio-frequency (RF) propagation in jet turbines, a model based on helical propagating waves is presented. The model uses a simplified model of the jet turbine based on a cylindrical structure and is compared with both electromagnetic (EM) simulations and measured values. Measurements and simulated values show good correspondence and show that a sum of helical propagating waves could be a useful tool to model the expected field variations.

Boundary enhanced coupling processes for rotated horizontal solid aluminum cylinders: Helical rays, synthetic aperture sonar images, and coupling conditions

Experiments with solid aluminum cylinders placed near a flat free surface provide insight into scattering processes relevant to other flat reflecting boundaries [J. R. La Follett, K. L. Williams, and P. L. Marston, J. Acoust. Soc. Am. 130, 669–672 (2011); J. R. La Follett, Ph.D. thesis, WSU (2010)]. This presentation concerns the coupling to surface guided leaky Rayleigh waves that have been shown to contribute significantly to backscattering by solid metallic cylinders [K. Gipson and P. L. Marston, J. Acoust. Soc. Am. 106, 1673–1689 (1999)]. The emphasis here is on horizontal cylinders rotated about a vertical axis away from broadside viewed at grazing incidence. The range of rotation angles for which helical rays can contribute is limited in the free field by the cylinder’s length [F. J. Blonigen and P. L. Marston, J. Acoust. Soc. Am. 112, 528–536 (2002)]. Some examples of surface enhanced backscattering may be summarized as follows. In agreement with geometrical considerations, the angular range for coupling to helical rays may be significantly extended when a short cylinder is adjacent to a flat surface. In addition, the presence of a flat surface splits synthetic aperture sonar (SAS) image features from various guided wave mechanisms on rotated cylinders. [Work supported by ONR.]

Quantitative ray methods for scattering by tilted cylindrical shells

Starting with a review of ray methods and phenomena associated with high frequency scattering by spheres and cylindrical shells in water viewed broadside, generalizations to tilted shells will be summarized. These extensions were found to be useful for meridional as well as helical ray backscattering enhancements associated with leaky (or supersonic) waves on shells [Morse and Marston, J. Acoust. Soc. Am. 112, 1318–1326 (2002); Blonigen and Marston, J. Acoust. Soc. Am. 112, 528–536 (2002)]. For such enhancements Fermat’s principle is useful for identifying ray paths of interest. In the case of helical waves (and in the broadside special case), the scattering amplitude can be expressed in terms of a Fresnel patch area where the guided wave is excited on the shell. Fresnel patches also give insight into the relatively large magnitude of meridional ray contributions. The coupling coefficient is proportional to the radiation damping of the leaky wave on the shell and in some cases it is necessary to take into account the anisotropy of the phase velocity. Computational benchmarks include scattering into the meridional plane by tilted infinite cylinders. Related phenomena include enhancements from subsonic guided waves and applications to sonar imaging and time-frequency analysis. [Work supported by ONR.]

Boundary enhanced helical ray coupling to high‐frequency modes of solid aluminum cylinders.

Surface guided leaky Rayleigh waves have been shown to provide a significant contribution to the backscattering by a solid metallic cylinder [K. Gipson and P. L. Marston, J. Acoust. Soc. Am. 106, 1673 (1999)]. Free field observations of the backscattering by solid aluminum cylinders confirm that both meridional and helical rays contribute to the backscattering for different angles of cylinder tilt [K. Baik, Ph.D. thesis, WSU (2008)]. Those studies show that a previously developed criteria, based on the length of the cylinder, limits the range of tilt angles for which helical rays can contribute [F. J. Blonigen and P. L. Marston, J. Acoust. Soc. Am. 112, 528 (2002)]. Experiments with solid aluminum cylinders placed near a flat reflecting boundary suggest that the presence of a boundary can increase the number of helical ray paths that contribute to the backscattering; this increase can lead to increased backscattering amplitudes and an increased range of angles for which coupling to helical rays can occur. To investigate this effect, solid aluminum cylinders were suspended below the free surface of a water tank. Targets were illuminated from below at grazing incidence, and the backscattering was measured as a function of target rotation. [Work supported by ONR.]

Interpreting free‐field scattering by metallic truncated cylinders and spherical shells.

When investigating the complications to the scattering caused by proximity of a target to a boundary, a detailed ray‐based interpretation of free‐field scattering is often helpful. For tilted metallic cylinders having flat ends, when ka exceeds about 10, strong contributions to the backscattering have been identified with the radiation of sound by guided elastic waves associated with meridional rays, helical rays, and face‐crossing rays [K. Gipson and P. L. Marston, J. Acoust. Soc. Am. 107, 112–117 (2000)]. These features also affect the frequency response as a function of tilt and bistatic synthetic aperture sonar (SAS) and acoustic holographic images [K. Baik, Ph.D. thesis, WSU, 2008]. When considering the response of shells to transient insonification, it can also be helpful to construct the time‐frequency response and to interpret the response using ray‐based methods [D. H. Hughes, Ph.D. thesis, WSU, 1992; S. F. Morse and P. L. Marston, J. Acoust. Soc. Am. 111, 1289–1294 (2002)]. Correct interpretation of the time‐frequency response requires detailed consideration of the dispersion and attenuation of elastic waves guided by the shell and the shape of the radiated wavefronts. When considering low‐frequency scattering, the inertia of the target significantly influences the scattering. [Work supported by ONR.]

Interfacial and elastic contributions to the backscattering of a right circular cylinder: Experiments and interpretation.

Synthetic aperture sonar (SAS) images of elastic targets near an interface contain features resulting from the interaction of incident and scattered sound with the interface in addition to those associated with the free-field dynamical response of the target. To improve the understanding of these features, a right circular aluminum cylinder was suspended in a water tank through an air interface and insonified at a grazing incidence. Monostatic SAS images were obtained by scanning the source (and receiver) along a horizontal line. Backscattering measurements were also made as the distance from the target to the interface was varied. Some of the features could be explained using a previously developed ray-based theory [K. Gipson and P. L. Marston, J. Acoust. Soc. Am. 106, 1673–1680 (1999);, 107, 112–117 (2000)]. Among these features were responses due to guided generalized Rayleigh waves such as meridional rays and helical rays. Also, ray path calculations for edge diffraction contributions (as a function of target-interface distance) were used to identify certain echoes in the SAS images. [Research supported by ONR.]

Tilted aluminum cylinder acoustic scattering properties and holographic and synthetic aperture sonar (SAS) images

A solid bluntly‐truncated aluminum cylinder in water is a convenient target for exploring the relationship between free‐field scattering properties of short tone‐bursts and acoustic images. These experiments concern situations where the product ka of the acoustic wave number and the cylinder radius exceed about 12. There are several prominent features in the tilt‐angle dependence of the time‐resolved backscattering. Some of these features are also present in prior measurements for tilted stainless steel cylinders using relatively long tone bursts [K. Gipson and P. L. Marston, J. Acoust. Soc. Am. 106, 1673–1680 (1999); K. Gipson and P. L. Marston, J. Acoust. Soc. Am. 107, 112–117 (2000)]. Prominent features include responses caused by guided generalized Rayleigh waves associated with meridonal rays, helical rays, and face crossing rays. These features are easily observed for tilted aluminum cylinders in the scattering of short tone bursts for cylinders having length‐to‐diameter ratios of 3 and 5. Guided Rayleigh waves also influence the appearance of the imaged cylinders in holographic and synthetic aperture sonar (SAS) images constructed using far‐field bistatic data. This influence on the images is explained using ray theory. [Supported by ONR.]

Diffraction of a Gaussian beam in a three-dimensional smoothly inhomogeneous medium: an eikonal-based complex geometrical-optics approach

We present an ab initio account of the paraxial complex geometrical optics (CGO) in application to scalar Gaussian beam propagation and diffraction in a 3D smoothly inhomogeneous medium. The paraxial CGO deals with quadratic expansion of the complex eikonal and reduces the wave problem to the solution of ordinary differential equations of the Riccati type. This substantially simplifies the description of Gaussian beam diffraction as compared with full-wave or parabolic (quasi-optics) equations. For a Gaussian beam propagating in a homogeneous medium or along the symmetry axis in a lenslike medium, the CGO equations possess analytical solutions; otherwise, they can be readily solved numerically. As a nontrivial example we consider Gaussian beam propagation and diffraction along a helical ray in an axially symmetric waveguide medium. It is shown that the major axis of the beam's elliptical cross section grows unboundedly; it is oriented predominantly in the azimuthal (binormal) direction and does not obey the parallel-transport law.

Helical rays in two-dimensional resonant wave conversion

The process of resonant wave conversion (often called linear mode conversion) has traditionally been analyzed with a spatially one-dimensional slab model, for which the rays propagate in a two-dimensional phase space. However, it has recently been shown [E. R. Tracy and A. N. Kaufman, Phys. Rev. Lett. 91, 130402 (2003)] that multidimensional rays have a helical structure for conversion in two or more spatial dimensions (if their dispersion matrix is generic). In that case, a one-dimensional model is inadequate; a correct analysis requires two spatial dimensions and, thus, four-dimensional phase space. A cold-plasma model is introduced in this paper which exhibits ray helicity in conversion regions where the density and magnetic field gradients are significantly nonparallel. For illustration, such regions are identified in a model of the poloidal plane of a deuterium-tritium tokamak plasma. In each conversion region, characterized by a six-sector topology, rays in the sector for incident and reflected magnetosonic waves exhibit significant helicity. A detailed analytic and numerical study of helical rays in this sector is developed for a “symmetric-wedge” model.

Leaky helical flexural wave backscattering contributions from tilted water-filled cylindrical shells.

Helical flexural waves on a bluntly truncated tilted water-filled cylindrical steel shell in water are found to give large contributions to the backscattering above the coincidence frequency. The presence of the water inside the shell increases the damping of the leaky wave when short tone bursts are used. The magnitude of the scattering is found by modifying a ray analysis developed for empty shells. When longer bursts are used, some of the internally radiated energy (corresponding to the case of one internal chord) is superposed on the ordinary helical ray backscattering. This occurs as a consequence of the internal excitation of helical rays.

Scattering by tilted cylindrical shells in water above the coincidence frequency

Experimental, theoretical, and computational results indicate that flexural waves on bluntly truncated tilted cylindrical steel shells are a major contributor to the backscattering of sound at high frequencies over a wide range of tilt angles. Important contributions are associated with meridional rays and with helical rays [S. F. Morse et al., J. Acoust. Soc. Am. 103, 785–794 (1998); S. F. Morse and P. L. Marston, J. Acoust. Soc. Am. 106, 2595–2600 (1999); S. F. Morse and P. L. Marston, IEEE J. Ocean. Eng. 26, 152–155 (2001)]. Successful development of a ray model for helical ray scattering contributions required the computation (at high frequencies) of the anisotropy of the flexural-wave velocity and radiation-damping parameters [F. J. Blonigen and P. L. Marston, J. Acoust. Soc. Am. (accepted for publication)]. In backscattering experiments with short and long tone bursts, the associated meridional and helical ray contributions are typically much larger than the scattering estimated for diffraction by a rigid tilted cylinder of the same size. The importance of the elastic response of shells was recognized in the early work of Junger [M. C. Junger, J. Acoust. Soc. Am. 24, 366–373 (1952)]. [Work supported by ONR.]

Backscattering from empty and water-filled tilted cylindrical shells due to leaky helical flexural waves: Comparison and ray theory

For tilt angles smaller than the meridional ray coupling condition previously investigated [S. F. Morse et al., J. Acoust. Soc. Am. 103, 785–794 (1998)], helical rays on empty and water-filled steel shells can significantly enhance the backscattering. These contributions are compared and modeled in the present work. Such contributions in the water-filled case are found to be weaker in amplitude, which is to be expected since additional energy is lost through radiation leaked to the inner fluid. A modified ray theory taking into account the increased radiation damping was compared with experimental results for the earliest helical wave arrival, yielding satisfactory agreement. However, the energy lost to the interior of the shell may be at least partially recovered, as internal rays can excite additional leaky waves on the shell. Contributions from rays with one internal chord inside the cylinder were superposed with those of the first helical wave, and the resulting backscattering amplitude was nearly that of the air-filled case for the first helical wave alone. This shows that we may regard the ray theory for the empty shell as an upper bound for corresponding amplitudes observed in the water-filled case. [Work supported by ONR.]

Leaky wave scattering contributions for tilted thick cylindrical shells in water above the coincidence frequency: Ray theory and computational models

Scattering measurements [Morse, Marston, and Kaduchak, J. Acoust. Soc. Am. 103, 785–795 (1998)] and sonar images [Kaduchak, Wassmuth, and Loeffler, J. Acoust. Soc. Am. 100, 64–71 (1996)] demonstrate that above the coincidence frequency, the responses of tilted cylindrical shells associated with leaky antisymmetric Lamb waves can give major contributions to the scattering of sound. Thin-shell dynamics is not applicable for the modeling of the shell’s response and an approximate ray formulation has been developed for situations where exact solutions are unavailable. Relevant contributions include meridional [P. L. Marston, J. Acoust. Soc. Am. 102, 358–369 (1997)] as well as helical [P. L. Marston, J. Acoust. Soc. Am. 102, 1628–1638 (1997)] leaky rays. The present research emphasizes computational tests for situations where helical antisymmetric Lamb wave contributions are important. These include situations where contributions from the broad meridional ray enhancement interfere with helical ray contributions. The results are helpful for interpreting features of the scattering. [Work supported by the Office of Naval Research.]

Meridional and helical ray contributions to backscattering by tilted cylindrical shells: High-frequency tone burst and wide bandwidth measurements and interpretation

When viewed in the frequency-angle domain, the backscattering by truncated thick cylindrical shells in water exhibit various ridges of high backscatter. These enhancements are associated with leaky waves excited on the shell and can be described in terms of meridional and helical rays. The enhancements are not present for a rigid scatterer of the same shape and may be important for interpreting high-frequency sonar echoes. Previous work [Morse et al., J. Acoust. Soc. Am. (to be published)] has shown that these enhancements are not limited to near broadside incidence but can extend to end-on incidence in the coincidence frequency region. Results of broadband and tone burst measurements of these enhancements are presented. High-frequency tone bursts may be used to separate the meridional ray enhancement from the generally weaker helical ray enhancement. The general angular width of the meridional ray enhancement may be explained using an approximate ray theory [Marston, J. Acoust. Soc. Am. 102, 358–369 (1997)]. The amplitude of the backscattered meridional ray depends on the leaky wave reflection coefficient at the end of the cylinder which appears to be affected by mode conversion near mode thresholds. [Work sponsored by the Office of Naval Research.]

Windowed displays of broadband impulse response measurements for finite cylindrical shells

Previously interpreted angle-frequency domain displays of impulse response measurements for thick nd thin finite cylindrical shells [S. F. Morse et al., J. Acoust. Soc. Am. 101, 3043(A) (1997)] reveal backscattering enhancements due to elastic effects on the cylinder over a range of target tilt angles. Presently, these signatures are examined in the angle-time and time-frequency domains. For selected angles, time-frequency analyses are presented as well as synthetic aperture images for selected apertures. From these it is possible to identify the spatial and angular locations of many of the important elastic contributions. Features associated with a meridional ray, as well as helical rays, have the appearance of originating from the back end of the cylinder. One major elastic reponse observed at large tilt angles is found below the coincidence frequency. [Work sponsored by the Office of Naval Research.]

Backscattering enhancements due to retroreflection of leaky Rayleigh waves at the truncation of a finite-solid cylinder: Measurements and test of a theory

Leaky surface waves (such as Rayleigh waves) once launched on an elastic object will undergo partial reflection at the truncations of the object. Under certain conditions, such reflections can radiate locally flat acoustic wavefronts with a reversal of the acoustic-wave vector, and thus a large backscattered signal is produced. In the present experiments, a solid stainless-steel cylinder was ensonified through an angular region, which included the conditions for launching meridional and helical Rayleigh waves, and the resulting backscattering enhancements are consistent with hypothesized reflection processes. For the enhancement associated with a meridional ray, the observed peak magnitude of the retroreflected signal is in general agreement with approximate theoretical predictions from a convolution formulation [P. L. Marston, J. Acoust. Soc. Am. 102, 358–369 (1997)]. Helical ray contributions are also observed, but these are generally weaker at ultrasonic frequencies.

Analytic Solutions of the Eikonal Equation for a GRIN-rod Lens 2. Skew Rays

Abstract Analytic solutions of the eikonal equation for a skew ray in a GRIN-rod lens are derived together with those of the ray equation by the perturbation method. It is algebraically shown that the derived solutions agree with the exact ones for a meridional ray in the sech profile and for a helical ray in the helical exact profile. The numerical accuracy is also examined by the Runge-Kutta-Gill method for the sech profile.

Sound radiation from a cylindrical duct. Part 1. Ray structure of the duct modes and of the external field

This paper determines the ray structure of a spinning acoustic mode propagating inside a semi-infinite circular cylindrical duct, and thereby determines the ray structure of the field radiated from the end of the duct. Inside the duct, but outside of a caustic cylindrical surface, the rays are piecewise linear helices; on striking the rim of the end-face of the duct, these rays produce ‘Keller cones’ of diffracted rays. The cones determine the structure of the radiated field: for example, no rays penetrate two cone-shaped far-field quiet zones centred on the duct axis; two rays pass through each point in a forward loud zone; and one ray passes through each point in a rearward loud zone. The two rays through each point in the forward loud zone interfere to produce an oscillatory directivity pattern. One quarter of the rays on each cone point back inside the duct and produce the reflected field. Thus the rim of the end-face of the duct acts as a ‘ring source’, in which the radiated and reflected fields have their origin. Every propagating duct mode determines a polar angle and an azimuthal angle; these are taken as parameters specifying the mode and are used to calculate the positions and angles of all the rays. The mathematical method on which the paper is based is Debye's approximation for the Bessel function which appears in the expression for the duct modes; the approximation shows also that the duct contains a region of smooth helical rays on which the field consists of inhomogeneous waves: this region is the inner cylinder, lying inside the annulus of piecewise linear helical rays. The results of the paper are very promising for the application of Keller's geometrical theory of diffraction to detailed calculations of the sound radiated from aeroengine ducts. An alternative description of the field, using Cargill's meridional rays, is summarized.

Microbending effects on monomode light propagation in multimode fibers

The effects of periodic axis deformations on propagation in multimode optical fibers with single-mode excitation are investigated numerically and experimentally. The numerical study, based on ray theory, deals with helical rays in the presence of sinusoidal axis deformations for various shapes of index profile. The corresponding experimental observations and results, carried out on tubular modes, confirm the existence of resonance effects between the helical ray period and the fiber axis deformation. This technique permits the observation of mode-to-mode power transfer and provides a sensitive tool to investigate the mode-coupling mechanism in optical fibers.