Volume 48, Issue 1B, July 1970

Transient Fields of Acoustic Radiators
View Description Hide DescriptionA set of short sketches is presented, giving an overview, unobscured by mathematical detail, of the literature on transient fields of acoustic radiators. Related papers are grouped together and interrelationships between results emphasized.

History of Standardization in Mechanical Vibration and Shock
View Description Hide DescriptionThe history of standardization of mechanical vibration in the United States is traced from its inception in 1932 to date, from its beginning in the Z‐24 Sectional Committee to the present S2 Standards Committee. In 1964, International Standards Organization Technical Committee 108, “Mechanical Vibration and Shock,” was organized. The current activities and future plans of ISO/TC 108 are discussed in terms of the tasks to which its seven working groups are committed.

Statistics of Combined Sine Waves
View Description Hide DescriptionThe first‐order probability density of sums of sine waves having random phases and amplitudes is approximated using a method developed by H. E. Daniels [Ann. Math. Stat. 25, No. 4, 631–650 (Dec. 1954)]. Three cases are discussed: (1) random phase and equal amplitude, (2) random phase and sine‐distributed amplitudes, and (3) random phase and rectangularly distributed amplitudes. Detailed results are presented for the first two cases. It is shown that extremal statistics may be quite poorly estimated by an assumption of normality.

Length Optimization for Constrained Viscoelastic Layer Damping
View Description Hide DescriptionViscoelastic materials are used extensively to damp flexural vibrations of metallic structures; it has been known for some time that the energy dissipation due to shear strain in the viscoelastic layer can be increased by constraining it with a stiffer covering layer. In this paper, we will discuss a method for increasing this damping by cutting the constraining layer into appropriate lengths. The analysis for a single layer of this treatment is relatively straightforward. The damping can be increased still further by using several layers; in this case, the analysis is based upon effective complex elastic moduli of an equivalent homogeneous medium. One result found from this analysis is that, if the element length of the constraining layer is optimum, the damping depends primarily upon the stiffness of the constraining layer and the loss coefficient of the viscoelastic material, and only indirectly on the shear modulus of the viscoelastic layer. Experimental data is presented for comparison with the theoretical predictions.

Potential of High‐Intensity Noise Testing
View Description Hide DescriptionVery nearly all vibration of consequence in missiles and space vehicles is excited by aerodynamic turbulence or noise propagating from turbulence. The presence of noise began to stimulate interest, a decade or more ago, in supplementing the traditional vibration tests of the flight equipment by some form of acoustic noise test. More recently, interest and activity have focused on the application of the technique to major sections of space vehicles. This application is attractive because the noise environment is in principle easier to predict than the vibration environment; the noise test permits simulation of three‐dimensional vibration with realistic equipment‐to‐structure mechanical‐impedance relationships, equalization of the excitation signal for armature and fixture resonances is avoided, and the observation of functional interactions between different equipments of the same system during environmental testing becomes possible. Such system environmental testing is made possible by the development of high‐intensity noise generators, especially in the form of electropneumatic transducers. The high‐intensity noise test has its own peculiar limitations, which must be considered by the shock and vibration engineer if he is to use it to best advantage. As with vibration testing, much of the new technology must be learned while it is being used. For the immediate future, the test will be developed as a means of simulating, by fixed sound‐pressure levels and point‐to‐point correlations, a composite condition of flight. In time, it may become possible to introduce variable time delays in the electrical random noise signals to make the point‐to‐point correlations of the test sound field correspond more closely to those of successive instants along the trajectory.

Transient Flexural Vibrations of Ship‐Like Structures Exposed to Underwater Explosions
View Description Hide DescriptionThe flexural vibrations of ship‐like structures are easily excited by underwater explosions, particularly if the explosion bubble period approximately synchronizes with a period of flexural vibration. A summary of the theoretical analysis shows how the transient motions are related to the principal vibration modes of the structure, and to the distribution and history of the applied pressures. The latter depend on the size, depth, and relative position of the explosive. Experiments on small idealized ship and submarine models are described which vary the weight, depth, and position of the explosive, and which measure the instantaneous deformations of the structure with strain gauges connected into special bridge circuits to yield the normal coordinates associated with the principal flexural vibration modes. Comparison of the theory and experiments verify the theory as applied to the fundamental flexural mode, and with less precision, as applied to the higher flexural modes. The principal mode patterns and the history of the explosion bubble are verified by independent experiments.

Motion of Bubbles in a Stationary Sound Field
View Description Hide DescriptionSmall air bubbles can be propelled through a liquid by a sound field. This paper presents observed and calculated values of the translational velocities at which bubbles smaller than resonance size are propelled through a standing acoustic wave in water and isopropyl alcohol. Bubble radii ranged from 29 to 149 μ, acoustic‐pressure amplitudes ranged up to 1.1 bar, and the frequencies ranged from 23.6 to 28.3 kHz. The resulting bubblevelocities, ranging to 23 cm/sec, were measured by photographing a moving bubble under stroboscopic illumination. Experiment and theory are in agreement as long as the bubble translation is rectilinear. Experimental results also indicate that the bubble translation becomes erratic when the pressure amplitude exceeds a threshold value. This threshold appears to be identical to the threshold for a similar onset of erratic dancing by bubbles that are trapped without translational motion in the sound field.

Third‐Order Elastic Constants of
View Description Hide DescriptionBy use of the pulse‐echo technique, the hydrostaticpressure and uniaxial stress dependence of the ultrasonicwavevelocities in single‐crystal have been measured, and the values of the 14 independent third‐order elastic stiffnesses were determined using a least‐squares fit to 31 different experimental measurements. A generalized Gruneisen theory was used in the quasiharmonic approximation to calculate the high‐ and low‐temperature limits for the two coefficients of linear thermal expansion. A numerical integration over 171 directions in the crystal was used for these calculations. The low‐temperature limit does not compare well with the available thermal data; however, measurements of thermal expansion below 100°K, which are needed for a realistic comparison, are not presently available.

Nonlinear Effects in Microwave Acoustic Surface‐Wave Delay Lines
View Description Hide DescriptionNonlinear effects have been observed in acoustic surface‐wave delay lines operating at a fundamental frequency of 905 MHz. Second‐, third‐, and fourth‐harmonic generation, and their buildup and subsequent decay with distance, have been measured using laser‐light deflection. The acoustic power entering these harmonics can be made comparable to that in the fundamental, since surface‐wave attenuation has been seen to be dependent on input power. In fact, under certain conditions harmonic power is returned to the fundamental causing a region of negative attenuation or “gain.” Similar effects have been observed for the sum and difference frequencies (mixing) when two input frequencies are used. For linear operation at 905 MHz, surface‐wave devices using Y‐cut Z‐propagating should be limited to power densities of 10 mW/mm.

Effects of Winds on the Dispersion of Acoustic‐Gravity Waves
View Description Hide DescriptionThe influence of winds at various levels in the atmosphere on the propagation of acoustic‐gravity waves is studied theoretically using dispersion curves for various atmospheric models. It is found that short‐period waves (periods less than about 400 sec) are influenced mainly by winds close to the ground, whereas long‐period waves (periods more than about 400 sec) are influenced by high‐altitude as well as low‐altitude winds. Strong winds at altitudes near 100 km in the direction of propagation of the waves and winds near the ground blowing in a direction opposite to that of the waves are found to be favorable for inverse dispersion in group velocities at long periods. Exactly opposite wind conditions favor normal dispersion at all periods.

Sound Field of Plane or Gently Curved Pulsed Radiators
View Description Hide DescriptionWhen a single pulse is applied to a plane radiator in a large rigid baffle or to a convexly curved baffled radiator having dimensions and radii of curvature large compared with the relevant wavelengths, the pressure at a field point is shown theoretically to consist, generally, of a sequence of pulses, each of which is, approximately, a scaled replica of the applied pulse. The number of pulses and their relative size and spacing are functions of position of the field point. In the direction of the main beam, if the radiating surface is plane, these pulses are not resolved and a single nearly undistorted pulse is obtained. A form of reciprocity is shown to exist between the structure of the acoustic signal at a point in the field of a pulsed transducer when transmitting and the structure of the electrical signal when the same transducer receives an acoustic pulse. Simple relationships are presented between the formulas for pulsed radiation, reception, and backscattering from a plane surface.

Transient Interactions of Spherical Acoustic Waves and a Cylindrical Elastic Shell
View Description Hide DescriptionThe three‐dimensional transient interaction between spherical acoustic waves with infinitely steep wavefronts and a circular cylindrical elastic shell of infinite length is investigated. The incident spherical wave is transformed into cylindrical partial waves by using the addition theorem for the modified Bessel function. The governing wave equation and equations of motion of the shell are solved by a series expansion‐Laplace‐Fourier transform technique. The transformed solution of the problem is obtained in closed form exact within the limit of series solution imposed by the Gibb's phenomenon. The physical solution is obtained by an accurate numerical scheme for the two‐fold inverse Laplace‐Fourier transforms. Detailed numerical results are obtained for the transient response of the shell and some quantitative effects of the sphericality of the incident waves on the response of the shell are also revealed.

Interaction of Acoustic Waves with Moving Media
View Description Hide DescriptionThe problem of interaction of a plane incident acoustic wave with n‐layer stratified moving media has been solved. The differential equations governing wave behavior in a moving medium are derived by employing first‐order Lorentz transformation instead of the Galileian transformation that has been used in previous works. Solutions are facilitated by the introduction of propagation matrices. Reflection and transmission coefficients are then calculated by making use of these matrices.

Lateral Waves on Diffuse Interfaces of Finite Thickness
View Description Hide DescriptionAn investigation of the lateral wave excited on a transition layer of finite thickness is made. The layer is located between two homogeneous half‐spaces of different refractive index. A line source is placed in the medium with the lower refractive index, and an exact integral representation is obtained for the scattered field. This integral representation is asymptotically evaluated for large k _{0} in two parameter ranges; first, when the layer is thick compared with wavelength; second, when the distance between the source and the observation point is large compared with wavelength. The results of the asymptotic analysis show that for thick layers there exists a close connection between layer continuity and lateral‐wave strength.

Scattering of Waves by a Cylindrical Piezoelectric Inclusion
View Description Hide DescriptionThe general problem of the scattering of waves from a transversely isotropic piezoelectric cylinder is examined. The solution for the interaction of an electromagnetic wave with a circular piezoelectric cylinder or antenna is presented. Solutions for the scattering of axial shear and compressional elastic waves by a piezo‐electric circular cylinder embedded in an elastic medium are also presented. A few numerical results are presented for bariumtitanate.

Effect of Diffraction on Stress‐Wave Measurement and a Concept for an Omnidirectional‐Dynamic‐Stress Gauge
View Description Hide DescriptionThis paper presents an analysis of the transient response of the pressure in an embedded elastic inclusion due to an incident compressional wave. It is found that the pressure or mean stress at the center of the inclusion is insensitive to the curvature of the incident wave. An early time analysis reveals that internal reflections in the inclusion can be minimized by matching the acoustic impedance of the inclusion with that of the matrix. An estimate is made of the time taken for these reflections to decay due to radiation damping. The method makes use of the calculus of residues and the high‐frequency response to sum the resulting infinite series. The inverse problem is solved exactly; i.e., the incident‐wave pressure is found in terms of the inclusion pressure. A design for an omnidirectional‐pressure transducer is also discussed.

Physical Interpretation of the WKB or Eikonal Approximation for Waves and Vibrations in Inhomogeneous Beams and Plates
View Description Hide DescriptionThe WKB or eikonal approximation is derived for flexural waves of constant frequency in inhomogeneous Euler‐Bernoulli beams and plates and for waves in inhomogeneous Timoshenko beams. The derivation is based on the assumption of negligible partial reflection of waves and holds when the relative variation of parameters characterizing the medium is small over distances comparable to one wavelength. The approximate relations governing the variation of wave amplitude from point to point are derived from the requirement that no energy be lost from the wave. The extension of this technique to include evanescent waves as well as propagating waves is also discussed. As an application of the method, some simple results are given for the modal frequencies and normal modes of a cantilevered inhomogeneous Euler‐Bernoulli beam. A general development of the eikonal method applicable to a wide class of mechanical systems is also given, which establishes the consistency of the energy‐conservation technique with the results obtained by taking the eikonal approximation as the zeroth‐order approximation to a series.

Integrated Signal on Circular Piston Receiver Centered in a Piston Beam
View Description Hide DescriptionBass's method for computing the integrated signal at a circular receiver, equal in size to the source, parallel to it, and centered on the same axis, is generalized. Expressions are derived for (1) the velocity potential (and, therefore, the acoustic pressure) at a distance from the axis equal to the source radius; (2) the radial component of particle velocity; and (3) the integrated signal at a circular piston receiver, located as specified above but with radius differing from that of the source. Approximate limits of validity are given. Case 3 is tested against three different sets of experimental data, with good agreement.

Free Vibration of the Rectangular Parallelepiped
View Description Hide DescriptionThe “associated‐periodicity” extension of Fourier analysis is used to obtain an exact solution of the classical, three‐dimensional elasticity problem of free vibration of the rectangular parallelepiped. This problem has been completely stated for more than a century and has been solved for only a very few special cases. The characteristic determinant yielding the eigenvalues is formulated for the completely free, rectangular parallelepiped, although the method of associated periodicity can be straightforwardly applied to arbitrary boundary conditions. Modes are classified into eight mutually exclusive and collectively exhaustive symmetry classes. Numerical results are presented for the frequency spectrum of plane‐strain vibrations of completely free rectangles according to two‐dimensional elasticity and are compared with classical Bernoulli‐Euler beam theory and Timoshenko beam theory (including the effects of shear deformation and rotary inertia).

Approximate Technique for Treating Random Vibration of Hysteretic Systems
View Description Hide DescriptionA technique is presented for predicting response statistics of a nonlinear hysteretic system by seeking an approximate equivalence between the hysteretic system and a nonlinear nonhysteretic system for which certain statistics of stationary response can be computed by an existing analytical method. The response statistics that can be computed include stationary displacement and velocity levels, and probability distributions. The technique is applied to the bilinear hysteretic oscillator, and the results are compared with experimental results and with the results of the equivalent linearization approximate technique of Krylov and Bogoliubov.