The fine features of X-ray propagation both in an ultra-narrow collimator and at glancing reflection from a smooth surface can be described within the unified theory of trapped radiation propagation: surface channeling in μ-guides and bulk channeling in submicron/nano-guides.

The question on X-ray extreme focusing (smallest reachable spot size) brings us to the idea for using the wave features of X-ray propagation in media. As known, wave features are revealed at propagation in ultra-narrow collimators as well as at glancing reflection from smooth flat and/or strongly curved surfaces. All these phenomena can be described within the general formalism of X-ray channeling.

Technical requirements for elastic (metal) cylindrical shells include the knowledge of their natural frequency spectrum. These shells may be empty and fluid-immersed, or fluid-filled in an ambient medium of air, or doubly fluid-loaded inside and out. They may support circumferential waves, or axially propagating waves both in the shell material, and in the fluid loading. Previous results by Bao et al. (J. Acoust. Soc. Am. 105 (1999) 2704) were obtained for the circumferential-wave dispersion curves on doubly loaded aluminum shells; the present study extends this to fluid-filled shells in air. For practical applications, steel shells are most important and we have here obtained corresponding results for these. To find the natural frequencies of cylindrical shells, one may invoke the principle of phase matching where resonating standing waves are formed around the circumference, or in the axial direction if the cylindrical shell is terminated at both ends. In this way, we obtain (circumferential and axial wave) eigenfrequency spectra for water filled aluminum and steel shells, and also for brass shells (axial-wave resonances only).

We consider evacuated thin semi-infinite shells immersed in a fluid, which may be either of cylindrical shape with a hemispherical shell endcap, or formed two-dimensionally by semi-infinite parallel plates joined together by a semi-cylinder. The connected shell portions are joined in a manner to satisfy continuity but with a discontinuous radius of curvature. Acoustic waves are considered incident along the axis of symmetry (say the z axis) onto the curved portion of the shell, where they, at the critical angle of coincidence, generate Lamb and Stoneley-type waves in the shell. Computations were carried out using a code developed by Cao et al. [Chinese J. Acoust. 14, 317 (1995)] and was used in order to computationally visualize the waves in the fluid that have been re-radiated by the shell waves a the critical angle. The frequency range was below that of the lowest Lamb wave, and only the A0 wave (and partly the S0 wave) was observed to re-radiate into the fluid under our assumptions. The results will be compared to experimental results in which the re-radiated waves are optically visualized by the Schardin-Cranz schlieren method.

The T-matrix method initiated by Waterman was extended in a clever paper by Peterson and the Varadans in 1980 to the complicated problem of elongated elastic shells. There have been some questions on whether the method converges for thin shells due to the impression that one must be able to inscribe a spherical surface in the annular region of the shell. The method may be derived by making use of several of the constraining equations that arise naturally and we show that the expression derived in 1980 is generally correct at least for objects with mirror symmetry. We present some details of the theoretical development with some calculations.

Elastic waves propagating on thin shells may be classified like for the Lamb waves on plates (A0, S0, A1, S1...), and the Scholte-Stoneley wave (A) in the fluid loading. The present study deals with evacuated shells of semi-infinite extent and a uniformly curved front surface, on which acoustic pulses are incident head-on. The incident signals cause the generation of the mentioned shell waves at a critical angle of incidence; these may be observed by their reradiation into the fluid at the same critical angle. Our demonstration of the reradiated pulses consists in a numerical evaluation of the incident and reradiated fields, and a visualization of the corresponding pulses in a tank experiment employing the Schlieren method. While the A wave could not be observed because of its rapid decay following its generation, we were able to demonstrate by both methods the generation of reradiated pulses of the A0 and the S0 wave, at the same time verifying the value of the critical angle of their generation.

Axial coherent bremsstrahlung of type A (ACBA) has not been intensively investigated either theoretically or experimentally. Making use of the many-beam (two-dimensional quantum treatment) formalism for transversely bound electrons moving through crystal lattices, we have computed ACBA spectra for 17 MeV electrons passing through a 10 μm thick diamond (C) crystal. We found that the momentum transfer occurs in the plane perpendicular to the axis of interest. Only momentum transfers along the scan direction (electron transverse momentum direction) result in a photon emission in the forward direction. Two different scans have shown that the energies of the coherent bremsstrahlung peaks depend strongly on the direction of the electron transverse momentum. We also present a comparison of the first order Born approximation and the many-beam formalism.

The impact of an acoustic pulse on a submerged elastic shell (which we shall assume evacuated) generates three types of circumferentially propagating surface waves: those that are analogous to plate waves of type A0(A1,A2,...) and S0(S1,S2,...), and a Scholte–Stoneley wave of type A that propagates in the surrounding fluid. A computer program devised by Hui Cao et al. renders visualizations of the generation and propagation of circumferential pulses on spherically endcapped shells, visualized by the re-radiation into the surrounding fluid of A, A0, and S0 waves in sequential pictures. These surface waves are generated experimentally by us in the laboratory from an ultrashort-pulse source, at axial incidence on a hemispherically endcapped glass tube. Sequential visualizations of the re-radiated pulses are obtained by using the Schardin–Cranz Schlieren method. These observations lead to an experimental measure of the group velocity dispersion curves of the surface waves.

Exact calculations of the scattering of radar pulses from a perfectly conducting wire, employing the Einarsson expression for the wire scattering amplitude, have indicated (Y. Guo, Ph.D. thesis, Catholic University, Washington, DC) the appearance of multiple return pulses, related to back-and-forth reflections of traveling waves along the wire. These reflections give rise to resonances in the scattering amplitude. We here show, using a method devised by Werby for an analogous acoustic scattering problem, that the set of resonances can be mathematically related to the appearance of pulsed multiple returns, hereby providing an alternate, simplified approach to the study of traveling wave pulses and their generating of return pulses in radar scattering from wires.

Water-filled elastic shells present more complicated backscattered echoes when compared to evacuated shells and are less well understood. The objective here is to determine and explain features peculiar to such objects and to use these features to aid in object characterization. Towards that end, an acoustic background suitable for such targets in conjunction with "residual" partial wave analysis obtained from subtracting the background from the elastic response is used. This analysis is aided by the determination and comparison of plate modes and resonance locations of evacuated unloaded, evacuated loaded, and loaded water-filled shells. Analysis of water-filled shells suggests that there exist isolated narrow, uniformly spaced eigenmodes in addition to the elastic modes. The broader elastic modes appear split in comparison to their unloaded counterparts. Otherwise, the elastic modes are not greatly affected by the included eigenmodes. The isolated (included) eigenmodes are extremely well modeled by a water inclusion in an infinite elastic matrix composed of the elastic shell material. The splitting of elastic modes proves to be due to interference of the included eigenmodes and the elastic body resonances. This leads to what appear to be separate branches of dispersion curves related to the elastic modes. Because of the numerous and usually evenly spaced included eigenmodes, dispersion curves are dominated by the abundance of these modes. A clear picture of the physical processes emerges from this analysis that explains all features of this event. Simple sets of rules that lead to tractable calculations are introduced that facilitate analysis of this interesting physical process.

Exact equations exist for the determination of the backscattered signals excited by acoustic signals on submerged elastic shells. In the absence of resonances these signals may be cast in terms of conservation principles of inertial components of the interaction of the signal with the object. This indeed leads to an adequate description of the acoustic background of such targets. What is determined to be critical is the inclusion of an entrained mass related to the displacement of the mass equivalent of the fluid due to the movement of the object. But what happens to such inertial effects in the presence of elastic resonances? We explore this issue and demonstrate that the effective or inertial component ranges from zero to infinity over a small frequency range that characterizes the resonance width. Over this range there is also a phase shift of 1800. We may as well examine the effect of radiation loading in this range. It is also possible to investigate a phase plot of the real and imaginary components of the effective mass factor which leads to closed trajectories for certain classes of resonances and to hyperbolic trajectories for other types. The meaning of these observations is discussed. [Work supported by NRL and ONR.]

A comparative study is presented of the acoustical excitation of circumferential (surface) waves on fluid-immersed cylindrical or spherical metal shells, which may be either evacuated, or filled with the same or a different fluid. The excited surface waves can manifest themselves by the resonances apparent in the sound scattering amplitude, which they cause upon phase matching following repeated circumnavigations of the target object, or by their re-radiation into the external fluid in the manner of head waves. We plot dispersion curves versus frequency of the surface waves, which for evacuated shells have a generally rising character, while the fluid filling adds an additional set of circumferential waves that descend with frequency. The resonances of these latter waves may also be interpreted as being due to phase matching, but they may alternately be interpreted as constituting the eigenfrequencies of the internal fluid contained in an elastic enclosure.

Microwaves can exhibit Bragg diffraction effects when their wavelength becomes shorter than the spacing of grooves in a macroscopic grating of parallel grooves or ridges. For spacings of the order of a few centimeters, these effects will appear at about 10 GHz or above, or at lower frequencies if the incidence is closer to grazing. We study these diffraction phenomena for a general (3D) incidence of the microwave beam, both regarding their geometry, and regarding their intensity, using certain simple scattering approximations for the latter purpose.

This paper considers acoustic scattering from submersed, elongated elastic objects, i.e., solid spheroids and finite-length cylinders. Surface waves are generated on the objects by incident acoustic waves; they can cause resonances in the objects’ vibrations which become evident in the scattered acoustic amplitudes. As it has been shown previously, the resonance frequencies can be obtained by a phase matching condition for multiply circumnavigating surface waves as they propagate on the object over closed paths. Comparing with calculated resonances in the scattering amplitude, it is found that the phase matching condition furnishes correct values for the calculated resonance frequencies for the case of Rayleigh-type surface waves. For spheroids and cylinders of larger aspect ratio, it is shown that bar-wave induced eigenfrequencies may also be used to approximate the calculated resonance frequencies.

Acoustic signals scatter from an elastic shell and excite elastic shell resonances. For evacuated shells resonances due to proper Lamb waves, A0 and S0 waves and the pseudo-Stoneley resonances, are the only ones allowed. When the shell is fluid filled then the eigenfrequencies of the included fluid may be excited. Further, the presence of the fluid may alter the elastic body resonances. When the impedance of the entrained fluid is not much smaller than the mechanical impedances of the elastic material then the scattered signal is greatly influenced by the entrained fluid, but only at their allowed eigenfrequencies. Since the fluid is entrained this leads to a discrete spectrum and an eigenvalue problem which is quite manageable. In this work we outline a method for determining the eigenfrequencies as well as their nature and use their values to isolate the actual body resonances due to the elastic material. We also illustrate the influence of the presence of the entrained fluid on the elastic resonance frequencies. [Work supported by NRL and ONR.]

Received 14 January 2002DOI:https://doi.org/10.1103/PhysRevB.66.149901©2002 American Physical Society

Negative group velocities have been reported in calculations for waves generated on elastic objects. It is usual to interpret a group velocity with the travel time of a wave front or the time that the energy of a wave generated travels along some trajectory. One may look at the group velocity as the rate at which an envelope of a wave train travels. More generally, one may exploit the principle of stationary phase along with the Fourier transform in frequency and derive the concept of group velocity from the first order asymptotic term. This derivation requires that the modulus of the integral is changing slowly relative to the phase term. In addition it is required that the group velocity is not near a stationary point in frequency. In that case first order asymptotics is no longer valid and one must employ higher asymptotics. Here, the standard concept of group velocity is no longer meaningful. Further, the signal becomes amplitude modulated rather than frequency modulated. We examine this for some examples, assuming losslessness. In the event of loss the concept of group velocity has already been reported to be of dubious value.

The onset of the excitation of flexural resonances for fluid loaded evacuated elastic shells produces a striking event. This issue has been interpreted theoretically in 1988 by means of a partial wave decomposition which showed a very narrow-peaked subsonic wave and a broad-peaked weak wave that begins at the speed of sound of the entraining fluid and increases. The narrow peaks are identified with subsonic water-borne waves that resonate in the fluid along the surface of an elastic object, and the broad peaks correspond to the inception of flexural waves. They exist over a small interval in wave number at the point when the flexural wave begins to couple with the fluid. Here we examine the pulse solution. With increasing frequency the partial waves change phase (a 90 degree phase change) and the overlapping flexural waves transition from a partially coherent constructive signal to one that is partially destructive. This leads to an envelope or hump effect, also called mid-frequency enhancement; it is a function of shell thickness as well as material property. We demonstrate how this effect may be employed to identify a submerged elastic shell in either the pulse or frequency solution.