Volume 47, Issue 3B, March 1970

Free‐Field Technique for Measuring Ultrasonic Dispersion and Absorption in Gases
View Description Hide DescriptionThis is a method for determining the phase velocity and attenuation of a progressive sonic wave by means of electronic interferometry. The wavelength λ and the frequency f are very accurately measured and then combined in the well‐known relationship, V = λf, to give the velocityV. To eliminate the effects of echoes, sine bursts are produced by gating an amplifier driven by a CW reference oscillator. Utilizing a calibrated attenuator, the received burst is added to the CW reference to show a null each time the sound receiver is moved through a wavelength. This addition and null are displayed on a cathode‐ray oscilloscope. Wide‐band solid dielectric transducers are employed to give a frequency span of 1 kHz to 1 MHz. The precision and reproducibility of the velocity measurements are within 0.01%. Sound transmission can be measured over a range of 110 dB to within 0.1 dB. Temperature is controlled from −20° to +60°C to within 0.1°C.

Instability of the Motion of a Pulsating Bubble in a Sound Field
View Description Hide DescriptionThis paper describes two related instabilities of spherical bubbles that are set into pulsation by a sound field. One instability is the observed onset of erratic dancing by bubbles that are trapped in a standing wave. This instability occurs when the sound‐pressure amplitude exceeds a threshold value, and we measured the threshold for bubbles driven below resonance in water and in isopropyl alcohol. The other instability, which also requires that the sound‐pressure amplitude exceed a threshold value, is the theoretically predicted onset of oscillation of the bubble shape. This threshold was calculated for the conditions of the previous experiments by a theory of parametric excitation based on Hill's equation. All results refer to pressure amplitudes less than 0.7 bar and frequencies from 23.6 to 28.3 kHz. From the close agreement of the measured dancing thresholds and the calculated shape‐oscillation thresholds, we conclude that the erratic dancing of pulsating bubbles in a sound field is caused by shape oscillations that are parametrically excited by the bubble pulsations.

Absorption and Dispersion of Ultrasonic Waves in Mixtures Containing Volatile Particles
View Description Hide DescriptionThe absorption and dispersion of ultrasonicwaves in a mixture of an inert gas and a vapor with volatile suspended particles of the condensed phase is studied. An equilibrium distribution of particles is found by performing a numerical experiment using equations describing condensation in an initially condensate‐free vapor at a given supersaturation. Hydrodynamic equations are developed that include the effects of the interaction of the mixture components with each other and with a polydispersion of suspended particles and the effects of evaporation and condensation. Results are presented for mixtures of air and water vapor containing waterdroplets.

Interfacial and Love‐Type Waves in Materials with Monoclinic Elastic Symmetry
View Description Hide DescriptionThe existence of interfacial and Love‐type waves in materials with monoclinic elastic symmetry is shown, if the displacement is in the direction of the digonal axis of a monoclinic crystal.

Stability of Acoustic Waves within a Viscous Compressible Heat‐Conducting Fluid
View Description Hide DescriptionThe question of stability of propagation of acoustic waves within a continuum of compressible viscous heat‐conducting fluid has been investigated. A bounded solution (i.e., a stable solution) is assured if either of the following two conditions is satified: (i) or , where P is the Prandtl number, γ the ratio of specific heats, and υ and η the bulk and shear viscosities, respectively. According to the theory, the above inequality must be obeyed for all passive fluids with γ> 1. Using experimental values for many different fluids, it appears to be generally obeyed except for the molten metals including mercury and, possibly, the superfluids.

Hypersonic Velocity and Absorption in Aqueous Electrolytic Solutions
View Description Hide DescriptionHypersonic velocity and absorption measurements were made in a number of aqueous solutions, using the technique of Brillouin scattering. These light‐scattering experiments represent the first direct measurement of the solvent absorption in the presence of salt ions. The results indicate that in dilute solutions this is about equal to the absorption in pure water, being the principal exception. Concentrated (4M) NaCl and NaBr solutions also showed an absorption value different from that of water. The differing absorption values were accompanied by substantially larger values of the speed of hypersound.

Piezoelectric and Photoelastic Properties of Lithium Iodate
View Description Hide DescriptionConsiderable interest has recently been drawn to the iodates owing to the large nonlinear optical interactions (second harmonic and photoelastic) reported in . In the present work, many of the photoelastic and piezoelectric constants and related properties of single‐crystal lithium iodate have been determined. The acousto‐optic figure of merit for has been found to be only about 15% of that for , making the material unattractive as an acousto‐optic medium. However, has been found to have a rare combination of a large piezoelectric effect (k _{ t }=0.51, k _{15}=0.60) and low dielectric constant, which makes it particularly attractive for broad‐band high‐frequency transducer applications.

Determination of the Elastic Constants of a Unidirectional Fiber Composite Using Ultrasonic Velocity Measurements
View Description Hide DescriptionUltrasonic velocitymeasurements have been made to obtain the dynamic elastic stiffnesses necessary to determine fully the elastic properties of a unidirectional glass‐reinforced epoxy‐fiber composite. In units of 10^{6} psi, these stiffnesses are C _{11}=6.01, C _{22}=C _{33}=2.58, C _{12}=C _{13}=0.70, C _{23}=1.42, and C _{44}=0.49, where the subscript 1 refers to the fiber direction. Since more velocities were measured than were necessary to obtain the five constants required by the symmetry of this composite, the extra measurements were used to check on the experimental method. Analysis shows the ultrasonic technique to be satisfactory for measurement of the elastic stiffnesses of a fiber composite. The experimental results are compared with the elastic constants predicted for this composite from expressions based upon several theoreticalmodels. Good agreement is obtained when the theoretical calculations are made using the dynamic (as opposed to the static) modulus of the epoxy matrix.

Curvature Corrections to Rough‐Surface Scattering at High Frequencies
View Description Hide DescriptionA variational principle is constructed for the problem of acoustic scatter from a rough surface. The use of the physical‐optics approximation as trial function leads to an improved expression for the surface field on the illuminated surface, although the form of the curvature dependence at least implies the “penumbra” or transition width near grazing incidence. Its application to the theory of scattering from very rough random surfaces yields an angular distribution proportional to the probability density of slopes, but for a fictitious surface with rms slope dependent on the wavelength. This scattering distribution is corrected for a shadowed surface.

Energy Conservation for Rough‐Surface Scattering
View Description Hide DescriptionThe neglect of multiple‐scatter and shadowing effects in the theory of high‐frequency scatter from random rough surfaces is manifested as a nonphysical energy loss or gain. This is demonstrated most clearly when the scattered intensity is cast in the form of the probability density of specular slopes. Shadowing is accounted for by introduction of the probability of illumination, conditional on the slope. Energy is then conserved for near‐grazing incidence.

Propagation of Harmonic Waves in Composite Circular‐Cylindrical Rods
View Description Hide DescriptionIn this investigation, the general frequency equation for harmonic waves having an arbitrary number of circumferential nodes, traveling in composite traction‐free, circular‐cylindrical rods is established on the basis of the linear three‐dimensional theory of elasticity. The composite rods consist of a circular core made of one material, bounded by and bonded to a circular casing of another material. Simpler degenerate cases of the frequency equation are reduced and discussed. A numerical evaluation of the frequency equation is presented. The results are obtained by programming an iteration procedure on a digital computer. The effect of the variation of the physical and geometric parameters of the rod on the frequencies and mode shapes of the first few modes is illustrated and discussed. Moreover, the feasibility of utilizing composite rods as delay media in guided‐wave ultrasonic delay lines is considered briefly.

Distribution of Maxima in the Response of an Oscillator to Random Excitation
View Description Hide DescriptionWhen a lightly damped linear oscillator responds to stationary wide‐band Gaussian excitation, the root‐mean‐square level and average frequency of the response are relatively insensitive to increases in the bandwidth of the excitation, but the average rate of maxima increases without limit. This phenomenon is interpreted by reexamining the exact distribution of the maxima and by giving a heuristic explanation based on decomposing the response into macroscopic and microscopic components. It is shown that the maxima of the combined response occur in clusters of micromaxima. The distribution of all maxima is estimated by accounting for the number of micromaxima introduced at each macromaximum.

Optimum Damping and Stiffness in Nonlinear Single‐Degree‐of‐Freedom Systems. I. Ground Acceleration Shock
View Description Hide DescriptionThe protection of a single‐degree‐of‐freedom system from ground shock is examined; the shock is an acceleration pulse with a vertical front. The item to be protected is mounted on an isolator containing both nonlinear stiffness and nonlinear damping (exponentially “hardening” or exponentially “softening”). A large range of the system parameters and several values of pulse duration are examined, using a digital computer; a total of nearly 700 time‐dependent solutions are involved. The results show that components possessing nonlinear stiffness and nonlinear damping can be used to significant advantage. If an allowable relative displacement specified, it appears that the peak absolute acceleration can be reduced as much as 40%; if an allowable absolute acceleration is specified, it appears that the peak relative displacement can be reduced as much as 30% to 40%. However, for small allowable displacements or for moderately large allowable accelerations, the results from nonlinear and linear systems are not significantly different. The smallest peak acceleration achieved by the nonlinear systems is 10% to 30% below the smallest peak acceleration of the linear systems, depending upon pulse duration. “Softening” damping functions proved to offer the greatest advantages, but the range of favorable stiffness functions included both “softening” and “hardening” types.

Optimum Damping and Stiffness in Nonlinear Single‐Degree‐of‐Freedom Systems. II. Velocity Shock
View Description Hide DescriptionCases of landing impact (velocity shock), such as the dropping of a packaged item or the landing of an airplane or spacecraft, are investigated. The packaging material, or the shock‐absorbing landing gear, is assumed to be nonlinear (exponentially “hardening” or exponentially “softening”) in both stiffness and damping. A large range of the system parameters is investigated, using a digital computer to solve the nonlinear differential equations, involving a total of about 500 individual time‐dependent solutions. It is found that, for the range of parameters investigated, significant advantages are possible by using nonlinear systems, as compared to what can be achieved with purely linear systems. If an allowable displacement is specified, it appears that the peak acceleration can be reduced as much as 35%; with a given allowable acceleration, the peak displacement can be reduced as much as 25% to 30%. However, for moderately large allowable displacements or for small allowable accelerations, the results from nonlinear and linear systems are not significantly different.

Asymptotic Nature of Extensional Waves in an Infinite Elastic Plate
View Description Hide DescriptionA theory based on a perturbation method is constructed to describe extensional waves and stress distributions in an infinite elastic plate of finite thickness. The plate is subjected to arbitrary axisymmetric disturbances, which are loadings located symmetrically at the surfaces of the plate, and in‐plate disturbances distributed symmetrically with respect to the plate's midplane. The displacements and stresses are expanded by infinite series of a small parameter and are required to satisfy the three‐dimensional equations of motion. The equation of motion for the first‐order terms is derived, and the disturbance is shown to propagate at the plane‐stress wave speed. For a given order higher than 1, correction terms can be obtained to modify the first‐order equation of motion and field variables. An equation describing a type of dispersive wave motion in any number of dimensions is also considered. A self‐similar solution of this equation is presented that represents the farfield nature of the waveform. The solutions are products of two parts. One gives the decay law and the other is in terms of Airy function. The latter shows the modification of the sharp wavefront by dispersion.

Vibrations of Closed and Open Sandwich Cylindrical Shells Using Refined Theory
View Description Hide DescriptionThe governing dynamic equations are derived for sandwich cylindrical shells including thickness‐shear deformation in both the core and face layers. From these equations,solutions are obtained for the free vibrations of both the closed and open configurations for infinite as well as freely supported cylinders. Particular emphasis is placed on the comparison of these results with those based on the face‐membrane theory in which the thickness‐shear deformation and flexural rigidity of the face layers is suppressed. The importance of including these higher‐order effects in the face layers is clearly demonstrated for the lower natural frequencies, even for relatively small ratios of face layer to core thickness. In addition, the effect on the eigenvalue spectrum of varying the opening angle of split sandwich cylindrical shells is investigated.

Oscillatory Impact of an Inhomogeneous Viscoelastic Rod
View Description Hide DescriptionA stress is applied to the end of a semi‐infinite, inhomogeneous, “standard” viscoelastic rod. The rod is initially unstressed and at rest. The applied stress is uniformly distributed over the end, and it is oscillatory in time. The propagation of small‐amplitude, one‐dimensional, longitudinal waves is studied. A formal asymptotic expansion of the solution is obtained. A partial justification of the method for homogeneous rods is given. The leading term in the expansion represents a modulated, oscillating, progressive wave that propagates with variable velocity. The modulation depends on the elastic and viscousmoduli. The velocity depends only on the elastic moduli. The Maxwell rod is studied as a limiting case of the standard rod. Application of the method to finite rods is discussed.

Transient Analysis of Stress Waves around Cracks under Antiplane Strain
View Description Hide DescriptionThis paper carries out a mathematical formulation of the antiplane strain problem of a crack with finite width subjected to any time‐dependent loadings. It is based on integral tranforms and a technique employed originally by Cagniard and simplified subsequently by De Hoop in geophysical‐layer problems. The procedure obviates the contour‐integration difficulties and permits the explicit recovery of the transient result by insepction. An illustrative example is given for the case in which a finite crack suddenly appears in an elastic solid under anitplane shear. A Fredholm integral equation of the second kind is obtained in the Laplace transform domain and solved numerically on the computer. Asymptotic forms for the transient stresses near the crack tip are found and a dynamic stress‐intensity factor, which is held important in the current theory of crack propagation, is defined. Numerical results showing that the transient stress reaches a peak very rapidly and then oscillates about the static value are displayed graphically.

Forced Vibration of Internally Damped Circular Plates with Supported and Free Boundaries
View Description Hide DescriptionThis paper considers the vibration response of circular plates that are driven at their midpoints by a sinusoidally varying point force. The plates have simply supported or free boundaries and are assumed to be internally damped. The corresponding solutions to the thin‐plate wave equation are described. Because the solutions involve ordinary and modified Bessel functions of complex argument, methods for obtaining the real and the imaginary parts of these functions are described and their results compared. Expressions are derived for the driving‐point and transfer impedances and for the force and displacement transmissibilities of the plates. Representative calculations showing the frequency dependence of these quantities for internal plate damping of the solid type are presented, and their physical significance is discussed. The effect of attaching lumped masses to the plates at their midpoints is also examined.

Normal‐Mode Vibrations of Systems of Elastically Connected Concentric Rings
View Description Hide DescriptionThis paper presents the development and solution of the differential equations of flexural vibrations of a system of n elastically connected concentric rings. By using these equations, the particular case of a two‐ring system is analyzed in detail. The frequency equation and displacement functions are then presented. The analysis includes a consideration of the effect of rotatory inertia. Vibration experiments were made on physical model of the two‐ring system. The results are compared with results of calculations based on the theory. An interesting phenomenon encountered during the investigation suggests the possibility of exciting, at a definite frequency, patterns of displacement that probably represent the combination of two normal modes.