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SUMMARY This paper is in homage to Robert Stoneley for his many contributions to the analysis of acoustic wave propagation and, in particular, for his introduction of anisotropy into the computation of elastic surface-wave characteristics. The latter has proved to be fundamental to the industry which has grown up around the signal processing devices using surface acoustic waves. The problem reviewed here is that of the reflection of incident acoustic waves from planar interfaces of quite general geometries. The interface may consist of substrate and superstrate welded together, of periodic or non-periodic intervening layers between superstrate and substrate, of free surfaces or of layered free surfaces. Surface waves and layer waves are also included. The materials involved may be of arbitrary crystalline symmetry and may be piezoelectric. The goal is to summarize this range of reflection and transmission problems in a common transfer matrix formalism which is convenient for programming. The same format is used for all the problems and the size of the matrices depends not on the complexity of the interface geometry but only on the crystalline symmetry. The reflection results are expressed as scattering matrices linking the amplitudes of the modes propagating in the substrate and the superstrate.

The transmission and reflection of a Rayleigh wave through a step on a homogeneous half-space of Poisson's ratio σ = 0.253 have been investigated by a seismic models method. The experimental results show good agreement with numerical ones obtained by finite-difference calculations when the wave is incident onto a downward step. Phase shifts are given both for the upward and downward steps. The characteristics of the transmitted wave do not depend much on the direction of propagation but those of the reflected wave are clearly direction-dependent. In each case the transmission and reflection coefficients and phase shifts depend on the ratio of step height to wavelength. The group delays are calculated and they are positive or negative depending on the period. About 80 per cent of the incident energy is converted into body waves when the step is higher than 0.3 times the wavelength of the Rayleigh wave.

In non-isotropic single crystals the normals to the wavefronts of elastic waves are not colinear with the vectors representing either the energy flow or the particle displacement. Calculations have been carried out on the propagation characteristics of sound waves in two particular trigonal crystals, α-quartz and sapphire.The development of the eigenvalue equation for the velocity and the formulae for the components of the displacement and energy-flow vectors are summarized. The assumption that the wave has a plane wavefront normal to a given direction leads to three solutions, one representing a quasi-longitudinal wave and the other two representing quasi-transverse waves. The velocities of propagation, directions of displacement, and directions of energy flow for the three waves have been calculated for many orientations of the wave normal. Detailed results for propagation near one of the pure-mode axes are presented.

A homogeneous dielectric sphere illuminated by a small external source of electromagnetic radiation will be free, because of its special symmetry, from all off-axis aberrations such as coma and astigmatism. If the only monochromatic aberration, namely spherical aberration, is not too excessive or if it is partially corrected by a suitably designed source, a sphere could be used for purposes of rapid scanning of a microwave beam through an angle of 360° of arc without moving the lens. This possibility was investigated by computations based on geometrical optics and on the diffraction theory of optical aberrations. The results suggest that a sphere of fairly small refractive index and less than about 30 wavelengths in diameter will produce a good diffraction image of a point source. Methods are also discussed of designing appropriate sources for use with larger spheres.

A detailed study of elastic surface‐wave propagation on a variety of free surfaces on many anisotropic crystals has been made. Included in the search were cases of propagation on nonsymmetry as well as symmetry planes. To date, no combination of surface, direction of progagation, and crystal has been found for which an unattenuated surface wave satisfying the traction‐free boundary conditions cannot be propagated. Solutions are shown for examples which occur within previously reported ``forbidden'' ranges and for cases in which the free surface is a nonsymmetry plane. The ``forbidden'' ranges appear now to be those in which the penetration of the surface wave becomes relatively deep. A fast and general computer program has been developed for this study to allow arbitrary choice of propagation conditions. The direction of energy flow is computed in the general case by effectively constructing the normal to an appropriate segment of the slowness surface. Solutions have been found for pseudo‐surface waves in various ranges of directions. These are second surface‐type waves which can propagate almost unattenuated.

When the free surface is anisotropic, mode of elastic surface‐wave propagation can arise that has many of the properties of a normal surface wave but has a phase velocity higher than that of the transverse bulk waves in the corresponding direction. The pseudo surface wave is a coupled mode involving terms decaying rapidly beneath the free surface and a term representing a bulk wave radiating into the solid. For many choices of crystal and plane of propagation, the contribution of the bulk term over a range of directions is small enough that the energy of the wave is essentially concentrated within a few wavelengths of the free surface and flows parallel to the surface as with the normal elastic surface waves. Moreover, in certain specific directions, the bulk term disappears completely and the pseudo‐surface wave has all the properties of a normal surface wave. The method of computation of the characteristics of the pseudo surface waves is outlined here and typical results of velocity, displacements and e...

Detailed measurements at a wavelength of 3.2 cm have been made of the phase distribution in the diffraction image produced by an aberration-free lens illuminated from a point source. The behavior of the phase of the electric field is shown by contours of constant phase (wavefronts) in the plane of the image space containing the axis and the H-vector of the radiating source. The region of the image space considered extends from the solid dielectric lens to a surface somewhat beyond the focus. Results are also shown of the phase distribution measured within about one wavelength of certain phase singularities which occur at the minima of intensity on the focal arc and on the axis. In general, there is very good agreement between plots of measured phase contours and plots of phase contours previously calculated from scalar diffraction theory.

The finite difference iterative method yields the full wave solution to problems without exact solution that involve the scattering of elastic surface waves at vertical discontinuities in homogeneous media. The technique successfully predicts the results for a problem for which an analytical solution does exist, that of a Rayleigh wavelet propagating on a homogeneous, isotropic, semi-infinite half space. For a Rayleigh wave of unit amplitude incident normally at a 90° corner in a homogeneous medium of Poisson's ratio σ = 0.245, the amplitude coefficients for transmission and reflection were found to be 0.64 ± 0.02 and 0.36 ± 0.02, respectively, whereas the corresponding phase shifts were −79 ± 5° and 38 ± 5°. About 45% of the incident energy is converted into body waves at the corner, and more than 90% of this energy is radiated back into a sector of the plane included between lines making angles of 15° with the two free surfaces. All these results, which are independent of wavelength, agree well with other published data. In the related problem of Rayleigh wave scattering at a downward step discontinuity, the dependence on step height of the transmission and reflection coefficients and of the phase shifts for a given wavelength component has also been investigated. Results show good agreement with both experimental curves and earlier theoretical work. This type of numerical simulation may be applied to other two-dimensional geometries, including layered media problems.

The full‐wave solution to the problem of acoustic surface‐wave scattering on a homogeneous isotropic three‐quarter space of Poisson's ratio σ=0.245 has been obtained by a finite‐difference iterative method. The amplitude coefficients for Rayleigh wave transmission and reflection at the 270° corner were found to be 0.28±0.02 and 0.09±0.01, respectively, while the corresponding phase shifts were 140±10° and −125±10°. All of these results, which are independent of wavelength, agree well with previously published experimental and theoretical work. About 90% of the incident energy is lost to body waves and the scattered energy‐density pattern indicates that over half of this energy sweeps past the corner and propagates into the body of the medium in the form of bulk modes. Some of the incident energy tends to follow the free surface by swinging around the corner while the reflected wave appears to be launched mainly by the action of a virtual source located at the corner which is excited by the incoming wave and radiates into the three‐quarter space. The accuracy of the finite‐difference technique has also been verified by comparing the iterated results for Rayleigh wave propagation on a homogeneous isotropic semi‐infinite half‐space with the analytically exact results. This basic method may be extended to solve problems involving layered geometries.

The paramagnetism of the color centers in smoky quartz is caused by the electron which is missing from a nonbonding oxygen orbital near an aluminum impurity. The effects produced by externally introduced microwave phonons on the spin-resonance signal from these color centers have been studied in detail as a function of the relative phonon and spin-resonance frequencies, the phonon power, the spectrometer power, the magnetic field angle, and the phonon polarization. As has been noted previously, these interactions do not conform with those observed in other paramagnetic crystals. Effects of an applied electric field on the resonance lines and various further experiments on the color centers in smoky quartz are reported; two in particular suggest that the ''hole'', the missing electron, makes thermally excited transitions between two different sites adjacent to a given impurity atom. One of the experiments is a measurement of the cross relaxation which takes place between the various lines of the spectrum, while the other is a measurement of the low-frequency dielectric loss found at liquid helium temperatures. An attempt is made to discuss the spin-phonon results in terms of such transitions for the holes. (Author)

The usefulness of the immersion–reflection technique for the measurement of the angular dependence of the dispersion curves for elastic surface waves in anisotropic thin-film, substrate combinations is demonstrated. The example chosen consists of gold films on a substrate of Y-cut lithium niobate, the latter being both highly anisotropic and piezoelectric. The velocity accuracy of the measurements is about ±0.3% and the frequency of operation is limited by the attenuation in the liquid medium in which the test specimen is immersed.

Converging and diverging surface acoustic waves (SAW) have been generated and detected using a noncontacting method. A yttrium aluminum garnet laser together with an axicon lens produces a focused optical ring which excites both converging and diverging SAW on the test sample. An optical interferometer is used to measure the vertical displacement of the SAW. Variations of the SAW pulse waveforms in different positions along the radial direction of the ring are given. It is found that about 10 SAW wavelengths away from the focus, a 90° phase difference exists between the Fourier components of the SAW going toward the focus and of that having passed through and moving away.

The concept of a planar acoustic microscope lens to be used in the standard geometry of a scanning acoustic microscope (I) has been discussed previously (2,3). In this paper we wish to look a little more closely at the focussing mechanism of this lens and to consider some further extensions of the concept., Twelfth International Symposium on Acoustic Imaging, July 1982, London, UK, Series: Acoustical Imaging: Recent Advances in Ultrasonic Visualization: Proceedings of the International Symposium

The dark rings of the Airy pattern in the focal region of a microwave lens form phase singularities. In an error-free lens these dark rings lie on the surface of a sphere. It is shown that in the presence of primary aberrations the line singularities remain closed loops and the phase behavior near them is similar to that in the ideal lens. Third-order astigmatism of relatively small amounts produces a marked distortion of the shape of the line singularity.

The behavior of the phase along the axis of aberration-free microwave lenses is shown by curves of the axial phase anomaly calculated from scalar Kirchhoff diffraction theory. These curves are drawn for three different wavelengths and two typical focal lengths. Also illustrated in the calculated results are the errors introduced by the use of certain geometrical approximations commonly employed in light optics.Detailed experimental measurements have been made of the electric field along the axis in the point-source image produced by solid dielectric lenses at a wavelength of 3.2 cm. Despite the limited size of the exit pupil (radius approximately 8 wavelengths) the agreement between the measured values of the axial phase anomaly and those calculated from scalar theory is very good except within about one lens radius of the lens surface.