Edmund G. Henneke

Reflection‐Refraction of a Stress Wave at a Plane Boundary between Anisotropic Media

A review is given of the pertinent equations necessary to describe the reflected and refracted waves at a plane boundary between anisotropic media and the utility of the wave surface in discussing this problem. The critical angle phenomenon in anisotropic media is discussed in terms of the energy flux vector associated with the reflected and refracted modes. The critical angle is shown to occur generally at that angle of incidence for which the energy flux vector of the reflected or refracted mode is parallel to the boundary and not when the wave vector is parallel to the boundary. The possibility of not needing a nonhomogeneous surface wave to satisfy the boundary conditions at angles of incidence greater than the critical angle is discussed for certain particular regions in some anisotropic materials.

Critical angle for reflection at a liquid–solid interface in single crystals

Recent investigations have utilized the measurement of the critical angle for reflection from a liquid‐solid interface for determination of the elastic constants of the solid. For anisotropic media, this technique is appropriate only for certain special cases of the incident plane and reflecting surface. We discuss here the general condition for the critical angel in anisotropic media and show that for some planes in quartz, major errors may arise if one employs the usual statement of Snell’s law for definition of the critical angle.Subject Classification: [43]20.30; [43]35.26.

Light‐Wave/Elastic‐Wave Analogies in Crystals

General consideration is given to the analogies that exist between the propagation of light waves and the propagation of ultrasonic waves in anisotropic media. A historical survey is given of prior work in this area, followed by a comparison of present views of the analogies with those of other investigators. The inability to obtain a direct one‐to‐one analogy for all phenomena is discussed.

Ultrasonic Orientation Determination of Single Crystals

A general method for determining crystallographic orientation by measurement of velocities of propagation of ultrasonic waves is given. The method supplements optical and x‐ray techniques and is primarily designed for orienting crystals to be used in ultrasonic pulse experiments. Aluminum and zinc single crystals are used as specific examples of the method. The required accuracy for such measurements is discussed.

Compilation of Elastic Wave Modes in Hexagonal Metals

Elastic wave velocities, particle displacement vectors, and directions of energy flux have been calculated and are given as a function of the polar angle θ in the five hexagonal metals Be, Cd, Mg, Ti, and Zn. Because of elastic transverse isotropy around the hexagonal axis, the wave characteristics are functions of the angle between the hexagonal axis and the direction of propagation in the crystal only. The results of these calculations are especially useful for experimental investigations utilizing the ultrasonic pulse‐echo technique.

Reflection of an elastic wave at a free boundary in hexagonal metals

Calculated reflection parameters for a quasi‐longitudinal wave incident upon a stress‐free boundary are presented for zinc, magnesium, and cadmium.

Orientation Dependence of Dislocation Damping in Hexagonal Metals

The orientation dependence of dislocation damping in the five hexagonal metals Mg, Cd, Be, Zn, and Ti has been predicted by calculation of the appropriate factors relating the stress field of the elastic waves to the slip systems in which the dislocations are moving. The four metals having basal slip systems have very similar qualitative behavior of the orientation factors. Titanium, with prismatic slip systems, has a quite different behavior and the orientation factors are smaller for all orientations than the values for the basal slip metals. For Zn, Cd, Mg, and Ti, the values of the orientation factors for standing waves (resonance technique) are larger than the corresponding values for the traveling waves (pulse‐echo technique). For Be, the orientation factors for both types of waves are nearly equal. Crystal orientations and wave types of particular interest for experimental study are delineated.

Experimental Determination of the Critical Angle for Reflection from a Water‐Quartz Interface

The critical angle of reflection for a wave incident from water upon the xy plane of quartz has been measured for planes of incidence about the z axis. These experimental measurements are compared with the theory that predicts that the critical angle for reflection from an interface of an anisotropic material occurs when the energy flux vector of the refracted wave becomes parallel to the interface. This differs from the usual definition of the critical angle as that angle for which the refracted wave normal becomes parallel to the boundary. The new definition of critical angle modifies the usual expression sinθc = VI/VR by the factor cosφ/cosβ as discussed in the previous abstract.

Critical Angle for Reflection from a Water‐Quartz Interface

A new, general definition is given for the critical angle of reflection in anisotropic materials. While the critical angle for reflection at a liquid‐isotropic‐solid interface can be written in the form sinθc = vI/vR, we show here that, in general, for an anisotropic solid, the right‐hand side of this expression must be modified by the factor cosφ/cosβ, where φ is the angle between the normal to the refracted wave in the solid and its energy flux vector and β is the angle between the energy flux vector and the component of the refracted wave normal on the interface. Numerical calculations have been made for this general definition for a water‐quartz interface for planes of incidence about the z axis (the xy Diane being the interface). These calculations are compared with the value of the critical angle as determined by the unmodified definition. It is shown that this earlier definition can lead to large errors when one uses the experimental technique of measuring the critical angle to evaluate the elastic...

Reflection of an Elastic Wave from a Free Surface in Quartz

The reflection of a plane, elastic wave in quartz is discussed. Numerical solution has been obtained for quasilongitudinal and quasitransverse waves incident upon the xy, xz, and yz crystal planes for angles less than critical. The angles of the reflected wave vectors, the associated reflection coefficients, and the directions of the energy flux vectors for the reflected waves have been calculated. The numerical results indicate that an incident critical angle occurs when a reflected wave has a Poynting vector parallel to the boundary as suggested in an earlier paper [E. G. Henneke II, J. Acoust. Soc. Amer. 51, 210 (1972)]. In some incidents, the wave vector of the reflected wave may be pointing in a direction in free space, but if its Poynting vector is directed internally, the reflection coefficient is nonzero. In addition, it has been found that for some incident angles greater than critical, three quasitransverse waves may be reflected, and the boundary conditions are satisfied without the necessity of assuming a surface wave. [This work was supported by the National Science Foundation].