The diffraction field of a Gaussian planar velocity distribution is a Gaussian beam function under the condition (ka } 2•, 1. This property makes a series of Gaussian functions attractive as a possible base function set. The new approach presented enables one to express any axisymmetric beam field in a simple analytical form-the superposition of Gaussian beams about the same axis but with beam waists of different sizes located at different positions along the axis. A computer optimization is used to evaluate the coefficients, as well as the beam waists and their positions. The extreme case of a piston radiator is used to test the approach. Good agreement between a ten-term Gaussian beam solution and the results of numerical integration (or analytical solution on axis) is obtained throughout the beam field: in the farfield, the transition region, and the nearfield. Discrepancies exist only in the extreme nearfield (< 0.1 times the Fresnel distance). For surface velocity distributions that are less discontinuous (smoother), the number of terms in the Gaussian beam solution is reduced. In the extreme case of a Gaussian radiator, only one term is needed. The approach, then, reduces the study of any axisymmetric beam field to the study of the much simpler Gaussian beam.

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The non-Hookeian behavior of a solid is studied using the distortion of an initially sinusoidal ultrasonic wave. The nonlinear differential equation describing the propagation of a finite amplitude ultrasonic wave in a cubic crystal is shown to take the same form as that describing a fluid. The propagation distance required for discontinuity in the particle velocity is calculated for various directions in the crystal. It is shown that in a typical experiment the path length can be appreciably less than the discontinuity distance, so that the distortion can be represented as a linear increase of second harmonic content with path length. Experimental results are given for copper single crystals which have been annealed or neutron irradiated.

Axisymmetric flow equations for a viscous incompressible fluid are transformed into the vorticity transport and Poisson’s equations. They are numerically solved via a finite difference method imposing appropriate initial and boundary conditions. A model source of 1-cm radius and 5-cm focal length with Gaussian amplitude distribution radiates 5-MHz ultrasound beams in water. Numerical examples are shown for buildup of acoustic streaming along and across the acoustic axis. Evidently, hydrodynamic nonlinearity has an essential effect on the streaming generation in comparison with a linear flow case; the nonlinearity reduces the streaming velocity in the focal and prefocal region, whereas it tends to accelerate the flow in the postfocal region.

Elsevier, Amsterdam, 1987. 229pp. Price $48.00.The most significant contribution to the recorded ground displacement during an earthquake comes from surface waves propagating along the ground. A consistent mathematical theory describing such surface wave propagation, then, is at the heart of seismology. Since the Earth can be approximated by a layered, elastic half-space that is infinitely deep, this problem can be handled as a special waveguide problem. The theory of open waveguides is of exceptional importance to this effort. Thus a book describing waveguide theory applied to geology really is quite useful and quite logical.The Earth is a highly dispersive system. A temporally short impulse in the source region reaches a distant observer as a disturbance of long duration. The dispersion, which affects surface waves, also is a typical waveguide phenomenon. For this reason, it is not surprising that surface waves are suited to treatment using the theory of waveguides. Such a treatment is presented in great mathematical detail in the book under review. Some general remarks are made about waveguides with discontinuities in the Introduction. In Chap. 2, Malischewsky considers boundary conditions and their seismological interpretation, and concludes with differential operators in closed domains and in open domains.The real mathematical ingenuity begins with Chap. 3, "The Undisturbed Waveguide." After demonstrating that waveguides exhibit an eigenspectrum, Malischewsky shows that the formalism of quantum mechanics is appropriate for their description. The list of topics covered is suggestive of a quantum mechanics text: 3.1. The mode conception .... 3.2.2. The eigenspectrum .... 3.2.3. The eigenfunctions .... 3.2.4. Orthogonality and normalization .... and 3.3.6. The radiation conditions.The mathematics does not stop at this point because Malischewsky selects his models by considering the principle "As simple as possible, as complicated as necessary." Thus Chap. 4, "The Disturbed Waveguide," continues the discussion of the interrelations with quantum theory. The surface and body waves, respectively, of the ideal waveguide can be compared with the eigenstates of an atom. Whereas the eigenstates of an atom are characterized by their energy, for surface waves each "state" is characterized by its phase velocity (eigenvalue). The ionization energy at which the continuum of the atom begins is analogous to the shear velocity of the layered half-space that determines the cutoff of a given Love mode. In one respect, the analogy breaks down: The bound states of an atom are a denumerable infinity, whereas the surface wave modes are finite in number at any given period. However, since one most often concentrates on the lowest level modes in any mathematical analysis, this shortcoming of the analogy is not of great importance. Since the other analogies hold completely, it is useful to introduce the bra-ket formalism in order to apply some methods of quantum theory to surface wave calculations. To illustrate its usefulness, Mali...

The behavior of a Gaussian ultrasonic beam incident on a liquid-solid interface at the Rayleigh angle, the angle at which surface waves are excited on the interface, has been studied in some detail. The reflected beam is displaced in the manner predicted by Schoch; however, the ’’Schoch displacement’’ in general is too large. Good agreement is obtained between experimental results and the theory of Bertoni and Tamir, which assumes that the incident beam couples resonantly into a leaky surface wave at the Rayleigh angle and that the energy reradiated from this leaky surface wave interferes with specularly reflected energy. The propagation distance of the ultrasonic beam is explicitly included in describing the ultrasonic wave reflection at the Rayleigh angle.

Results of ultrasonic harmonic generation are combined with those of ultrasonic beam mixing [R. W. Dunham and H. B. Huntington], to obtain a complete set of truly adiabatic third-order elastic constants of a strain-free sample of G.E. type 151 fused silica: C111+64.8±0.5×1011 dyn/cm2; C144=+5.4±0.3×1011 dyn/cm2; and C456=−1.32±0.08×1011 dyn/cm2. Differences among values of the third-order elastic constants for different samples are great enough that care in sample selection is necessary for consistent results.