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Multiple scattering effects due to a random array of obstacles are considered. Employing a ``configurational averaging'' procedure, a criterion is obtained for the validity of approximate integral equations describing the various field quantities of interest. The extinction theorem is obtained and shown to give rise to the forward‐amplitude theorem of multiple scattering. In the limit of vanishing correlations in position, the complex propagation constant κ of the scattering medium is obtained. Under appropriate restrictions, the expression for κ is shown to include both the square‐root law of isotropic scatterers and the additive rule for cross sections valid for sufficiently low densities of anisotropic obstacles. Some specific examples from acoustics and electromagnetic theory then indicate that at least in the simplest cases the results remain valid for physically allowable densities of obstacles.

Measurements of changes in ultrasonic attenuation together with changes in ultrasonic velocity have been made concurrently with load strain measurements in tensile tests on the same specimen of aluminum. The results of such measurements taken during loading of the specimen, during relaxation or recovery at constant strain, and during unloading show a number of interesting effects. These observed effects are interpreted in terms of dislocation behavior for the various stages of the experiment. The experimental results for attenuation α and velocity change Δv/v permit the calculation of changes in dislocation density and loop length based on a dislocation damping theory developed in this laboratory to include both megacycle and kilocycle frequencies. The strain due to dislocation motion was calculated with a simple model and with dislocation loop lengths and densities obtained from the data and the theory just mentioned. The comparison of the calculated strain and the measured strain lends support to the use of this model, and while this dislocation damping model may not be entirely correct it does seem to be possible to obtain information about stress‐strain behavior at low strains from sound propagation data alone.

An rf ion source has been constructed, which yields up to 15 ma of hydrogen ion current consisting of 80–90 percent protons. The ion current yield has been studied as a function of rf power and frequency, gas pressure, and dc (extracting) voltage. These relationships are correlated with internal phenomena of the gas discharge and of ion extraction therefrom.

A study is made of the ultrasonic field produced by a circular quartz crystal transducer and the integrated response of a quartz crystal receiver with the same dimensions as the transducer. The transducer and receiver are taken to be coaxial, and it is assumed that the transducer behaves as a piston source while the integrated response is proportional to the average pressure over the receiver area. Computations are made for cases of interest in the megacycle frequency range (ka=50 to 1000; a=piston radius; λ=wavelength; k=2π/λ). The results contain features of use in identifying and correcting for diffraction errors. These features which apparently have been missed in previous investigations are compared with available experimental data. Finally correction formulas to account for diffraction effects in the accurate measurement of attenuation are discussed. It is shown that the order of magnitude of the diffraction attenuation is given by one decibel per a2/λ.

The measurement of ultrasonic attenuation and velocity at megacycle frequencies demands that the sample faces be parallel within limits that require extreme care. The importance of maintaining the tolerances in question is shown in terms of the echo patterns for various cases in which the tolerances are held and in which they are not. In order to measure attenuation of crystalline quartz to within about 10% at 1000 Mc/sec, the required wedge angle θ should be about 5×10−8 rad or less. This turns out to be just barely possible. under rather special conditions. At the highest frequencies attained thus far (i.e., about 25 000 Mc/sec) with ultrasonic stress waves, the limits on parallelism are so severe that there are at present no instruments capable of making the necessary measurements to control the sample parallelism.

A brief review is given of the calculation of cross section expressions for the scattering of an elastic wave in an elastic medium. With scatterers that are elastic and spherical in shape, numerical calculations are carried out for non‐Rayleigh conditions of scattering. Comparisons of such cross sections are made with those obtained from the Rayleigh approximation.The deviation from Rayleigh behavior becomes pronounced, for the elastic scatterer, beyond ka≅0.5. In addition a ``peaking'' and ``fluctuating'' of the cross section occurs beyond this point.

Electrical charges developed during plastic deformation of sodium chloride were investigated. Experiments are chosen (orientation and shape of the crystals; direction of applied stress) to enable the edge and screw components of dislocation loops to emerge separately on two different pairs of the external surfaces of the specimen. The experimental results indicate that an electrical charge is associated only with the edge components of dislocation loops. It is also found that in tensile experiments of a bent specimen (accidentally or intentionally) negative charge appears on the concave side and positive charge appears on the convex side, and the negative charge is larger, in absolute value, than the positive charge, in the early stages of deformation. A model is proposed to explain these results.

Ultrasonic attenuation measurements in sodium chloride as a function of cobalt 60 gamma‐ray irradiation show no appreciable dependence on temperature in the temperature range from room temperature to liquid nitrogen temperature, hence no appreciable thermal activation energy for the process of immobilizing the dislocations. The influence of plastic deformation on the attenuation‐irradiation behavior is large; the greater the deformation the more rapid the attenuation change, at least in the deformation range from zero to one percent deformation. The consequences of these effects are discussed in terms of boundary conditions on possible mechanisms for the observed effects.

The method of Ying and Truell for studying the scattering of longitudinal elastic waves by spherical obstacles is applied to the particular case of scattering by a fluid‐filled cavity that is embedded in an isotropic elastic solid. Exact solutions for the expansion coefficients which describe the scattered elastic wave are obtained. The limiting case of ka≪1 (Rayleigh scattering) is considered in detail. A general expression for the scattering cross section is derived.

Aluminum single crystals oriented for single slip and for “polyslip” were deformed in tension, at room temperature. These tests were limited to a total tensile strain of up to 1 per cent. Simultaneous measurements of attenuation and velocity changes were made continuously during the tensile deformation. Both longitudinal and shear waves were used, at frequencies of 10 Mc/s and 13 Mc/s. The observed changes in attenuation in the polyslip orientations were consistent with an explanation based on the number of equally favored slip systems in each case. The results on crystals oriented for single slip indicate that in the easy glide range of strain hardening, dislocation multiplication is confined to the primary slip system only. The end of easy glide is associated with dislocations multiplying in other slip systems. An increase in attenuation is also observed prior to the onset of the macroscopic yield. This increase is presently attributed to an increase in dislocation loop length, caused by a breakaway mechanism. In the early stages of deformation, for all orientations, an increase of ultrasonic velocity, relative to the unstrained condition, is observed. The changes of attenuation and of velocity reported here, are consistent with theoretical results derived from the treatment of dislocation damping given by Granato and Lucke.

An ultrasonic interferometer technique for measuring the temperature dependence of the relative change in elastic constants is described in detail. Results are reported for measurements carried out as a function of frequency from 10 mc/sec to 170 mc/sec, for compressional wave propagation along the [100] axis of silicon over a temperature range of −60° to +15°C.The mean value of (1/c11)(dc11/dT) obtained is 55.59±0.81×10−6 (°C)−1, and no frequency dependence of (1/c11)(dc11/dT) was found.

The various mechanisms producing damping of ultrasonic elastic waves in solids are reviewed in this paper. Particular attention is given to the changes in attenuation produced by pinning of dislocation lines. Measurements of ultrasonic attenuation in irradiated single crystals of NaCl are presented and discussed, on the assumption that the dislocation density remains constant while the radiation displaced defects pin the dislocations. The effects on the velocity of propagation are discussed on the basis of the same model.

Pulse techniques for the measurement of attenuation in solids have been extended and refined sufficiently to obtain dependable measurements over a frequency range from 5 to 50 megacycles. Understanding of the relative importance of beam spreading, geometrical boundaries, and method of coupling has been improved. Coupling by means of water buffer and direct mounting is discussed in detail.Attenuation measurements in the frequency range from 5 to 50 megacycles have been made on chrome molybdenum steel specimens, and these measurements have shown large differences in ultrasonic attenuation for samples of the same chemical composition but different heat treatment. The resulting differences are connected with anisotropy which appears in the photomicrographs. The attenuation measurements are quite sensitive to heat treatment and other factors. The application of the methods to metallurgical problems and the physics of solids is suggested.

Scattering by a spherical obstacle of a plane longitudinal wave propagating in an isotropically elastic solid is computed. Expressions for the scattered wave and the total scattered energy are given. Three special types of obstacle—an isotropically elastic sphere, a spherical cavity, and a rigid sphere—are discussed in detail, especially for Rayleigh scattering. The result for the isotropically elastic sphere is compared with the well‐known result of scattering of a plane wave propagating in an ideal fluid by a sphere of another ideal fluid.

The problem of propagation of traveling waves in a right circular cylinder having hexagonal elastic symmetry is considered here. Exact expressions for the displacements produced by both the compressional and torsional modes are obtained. A conclusion of importance in ultrasonic methods is the result that the compressional and torsional modes propagate independently without coupling either through the equations of motion or the boundary conditions. A condition relating the frequency and wave number of the torsional wave is derived.

The temperature dependence of the attenuation of 30‐Mc/sec longitudinal ultrasonic waves has been measured from room temperature to 380°C in CdS. The observed attenuation is in good agreement with the theory of Hutson and White and with the temperature dependence of the resistance measured on the same specimen.

An ultrasonic interferometer technique for measuring the temperature dependence of relative changes in elastic constants has been utilized to study the variation of c11 for germanium. Results are reported for measurements carried out as a function of frequency from 30 Mc/sec to 170 Mc/sec for compressional wave propagation along the [100] direction in two sets of compatible germanium samples over the temperature range from −60°C to 0°C. The mean value of (1/c11) (dc11/dT) is 78.87×10−6(°C)−1, and no systematic frequency dependence of (1/c11) (dc11/dT) was found.

The relation Jn′(x)Nn′(kx)−Jn′(kx)Nn′(x)=0 arises in certain resonant cavity problems having cylindrical symmetry. The first roots of this relation are presented here as a function of k for n = 1, 2, 3, 4. The M'Mahon relation does not allow calculation of the first roots despite statements to the contrary in several places. It is shown that the functions Jn′(x)/Nn′(x) have relative maxima at x = n except for n = 0.

A method is described for determining the range of sizes that damaged regions created by fast neutrons may have in a silicon single crystal. In particular the use of precision ultrasonic velocity and attenuation measurements for compressional waves together with certain new results from scattering theory permit the determination of an upper and lower limit on the size of a darnaged region. The lower limit is determined from the fractional velocity change arising from the irradiation and the upper limit is found from the corresponding attenuation change-or in this case the lack of it. Under the assumption that such regions of damage are spherical the limits obtained lie in a range from 0.01 to 0.27 mu for the radius of the region in question. (auth)

The propagation behavior of high frequency stress waves in solids is determined by the measurement of the attenuation and velocity of the stress waves as a function of whatever variables are of interest. Such measurements permit one to study the influence, on the propagation behavior, of any property of a solid that is sufficiently well coupled to the lattice. Various types of defects in the lattice are, for example, closely coupled with the lattice and a change in the type or density of defects will, in such cases, change the propagation behavior of a stress wave. Altogether there are ten or more known “interactions” between stress waves and various properties of a solid. Electrons are coupled to the lattice and interact with stress waves in at least two distinct ways. Lattice waves interact with other stress waves, and nuclear interaction with stress waves has been studied. Radiation induced defects of several types have been investigated, and paramagnetic resonance effects can be influenced by stress waves of the proper frequency.

Evidence is presented showing that the interaction of high‐frequency stress waves with lattice vibration waves provides a means of studying crystal perfection. This discussion is concerned specifically with quartz single crystals, but the method has been found to apply to other materials as well.

Attention is called to the very rapid increase in acoustic attenuation at low strains, and it is pointed out that this effect has been observed only in the highpurity almost single-crystal sample. It is proposed that this rapid attenuation increase at very low strain is an example of dislocation''breakaway'' and that the effect is caused by an increase in dislocation loop length arising when the dislocations free themselves from relatively weak pinning,; at low strains. (A.C.)

Results of an ultrasonic measurement survey in which nine groups made compressional wave transit time measurements on a given sample of fused silica are presented. Seven of the nine reports agreed within 0.3%, and it is suggested that discrepancies in published values of elastic constants of a given material may be due to variations in the samples themselves.

The damage or irradiation effects arising from the reaction of slow neutrons with B/sup 10/ were detected and observed in a boron glass by means of high-frequency ultrasonic attenuation. The glass used (Corning 7070) contained 28% by weight of B2O3, or approximately 8% B. Irradiation for more than an hour in the Brookhaven Reactor caused the glass samples to crack. Reactor irradiation of 15, 30, 45, and 60 minutes produced both attenuation and velocity changes that could be measured in the frequency range from 10 to 50 Mc/sec. The results show an attenuation increase of approximately. 50% at a frequency of 50 Mc/sec in a sample irradiated for 30 min and somewhat less for an irradiation of 60 min. The ultrasonic velocity for iransverse waves at 10 Mc/sec decreases as a function of increasing irradiation. while at 50 Mc/sec ihe 7 -velocity increases as a function of increasing irradiation. (J.S.R.)

Ultrasonic propagation in Al2O3 at low temperatures and high frequencies (to 3000 Mc/sec) is discussed in terms of the relations given by Woodruff and Ehrenreich. Phonon relaxation times calculated from thermal conductivity measurements are used. The measured low‐temperature attenuation values at 1315 Mc/sec agree with the calculated values. A comparison of attenuation as a function of frequency is made among Al2O3, Ge, Si, and quartz. On comparing experimental values of attenuation for pure Al2O3 and impure Al2O3 (ruby) it is found that impurities lead to an increase in the attenuation as would be predicted from the corresponding effect of impurities on the thermal phonon relaxation time.

The dependence of ultrasonic attenuation and velocity on plastic deformation in 2S aluminum is compared at two frequencies, five and ten megacycles, and the comparison shows agreement as regards frequency dependence with what one can expect from dislocation damping theory.

This is a discussion of general methods of making ultrasonic attenuation measurements in the megacycle range (5–500 Mc/sec and higher). It includes some discussion of dislocation damping effects together with examples where ultrasonic attenuation measurements have led to new information. Work has been done on metals and alloys, semiconductors, and insulators; the work to be described will be mainly that of the Metals Research Laboratory at Brown University.

Some of the current methods of generating and using ultrasonic waves in the range of 1000 mc will be discussed together with speculation on some of the uses of such microwave ultrasonic energy.