Volume 40, Issue 4, October 1966

Inverse Design for Flexural Vibrators
View Description Hide DescriptionIt is shown how flexural vibrators with prescribed vibrational characteristics can be designed by an inverse method. An illustration of the algebraic procedure is given.

Vibration of a Thick Cable
View Description Hide DescriptionThe undamped free vibration of a thick cable is studied by taking into consideration the effects of its flexural stiffness, shear rigidity, and rotational inertia. These effects can be expressed in terms of a dimensionlesss parameter ε. When the end of a cable of span L is clamped, the influence of flexural stiffness is mainly confined to a distance εL at the end. As a contrast to a beam, a cable is defined as one with ε small as compared with unity, and, therefore, can be analyzed with ε as a perturbation quantity. The cable with clamped ends is analyzed with matched asymptotic expansions, and a simple equation is obtained for the natural frequencies of vibration. Experimental verification has been performed with ε up to 0.1 for the fundamental mode, and with ε up to 0.05 for the second mode.

Axially Symmetric Waves in Hollow, Elastic Rods: Part I
View Description Hide DescriptionA system of approximate, one‐dimensional equations is derived for axially symmetric motions of hollow, elastic rods of circular cross section. The theory is valid for a range of wall thicknesses from the very thin to thick walls and, in fact, is valid in the limit for the solid cylinder. The theory takes into account the coupling between the longitudinal, radial, and axial shear modes. The theory is based on expansions of the displacements in a series of orthogonal polynomials in the radial coordinate, retaining only the earliest terms representing the longitudinal, radial, and axial shear deformations. To offset the error introduced by omission of the terms of higher order, four adjustment factors are introduced and chosen in such a way that the behavior of the first three branches of the exact frequency spectrum is reproduced at long wavelengths. Comparison is made between the three spectral lines developed from this theory when the propagation constant is real and the comparable spectral lines from the exact three‐dimensional theory.

Response of Viscoelastic Cylindrical Shells to Moving Loads
View Description Hide DescriptionThis paper presents the steady‐state response of infinite‐length shells of linear viscoelasticmaterials to moving axisymmetric ring loads based on the shell theory with corrections for the shear‐deformation and rotatory inertia effects. The complex Fourier transform is used to obtain the exact solution. Numerical results for shells of Maxwellmaterials with various viscous damping coefficients under various velocities of moving loads are given. The special case, response for elastic shells, is also discussed.

Natural Frequencies of Closed Spherical Sandwich Shells
View Description Hide DescriptionSolutions for the axially symmetric motion of an elastic spherical sandwich shell are obtained from a theory of shells that includes the effects of transverse shear deformation, transverse normal stress, and rotatory inertia. Frequency equations and mode shapes are derived for the free vibrations of a closed spherical shell. It is shown that three branches appear in the spectrum of torsionless modes, and two branches appear in the spectrum of torsional modes.

Sloshing of a Liquid in Connected Cylindrical Tanks Owing to U‐Tube Free Oscillations
View Description Hide DescriptionWhen two parallel cylindrical tanks are connected by a pipe joining the tank bottoms, the liquid can oscillate back and forth between tanks, just as in a U shaped tube. This paper deals with the axisymmetric sloshing oscillations excited in each tank because of the flow back and forth between the tanks. By applying the results from a recent paper [P. G. Bhuta and G. C. K. Yeh, “Liquid Sloshing Due to a Time Dependent Discontinuous Boundary,” Intern. J. Mech. Sci. 7, 475–488 (1965)], the expressions of velocity, pressure, and waveheight are obtained in terms of the liquid density, tank dimensions, pipe radius, and U‐tube natural frequency, which, in turn, is a function of the tank dimensions, pipe dimensions, and the magnitude of acceleration vector. Resonance will occur whenever the U‐tube natural frequency is equal to any one value of the natural frequencies for the axisymmetric sloshing modes of the liquid in each cylindrical tank. Pertinent dimensions to be avoided are pointed out. Extension of the results to asymmetric sloshing based on a paper by Yeh [“Liquid Sloshing in a Moving Tank with a Time Dependent Discontinuous Boundary,” TRW Space Technol. Labs. Rept. EM 14‐12 (6438‐6002 RU‐000) (June 1964)] is also indicated.

Analog‐Computer Program for Studying Underwater Sound Rays and Refraction Effects
View Description Hide DescriptionAn analog‐computer program has been developed that traces sound rays and gives a continuous value for the refractive anomaly. The program uses the second derivative of the sound velocity with respect to depth as a description of the medium through which sound waves travel. Tests have been made using constant values for the second derivative. Excellent agreement has been obtained between computer results and mathematical results predicted by theory.

Transmission of Random Sound and Vibration through a Rectangular Double Wall
View Description Hide DescriptionThe existence of many modes of vibration in a complex system makes a detailed classical analysis almost intractable. However, when excited by broad‐band random noise, the detailed response characteristics may be overlooked and statistical properties such as mean‐square values and power spectra provide a measure of the vibration. The existence of classes of similarly excited modes allows theories of room acoustics and thermodynamics to be applied successfully. For sound transmission through a rectangular double wall, this technique gives results that agree within experimental error of measured transmission loss, much better than do conventional loss estimates. Both theory and experiment confirm that the product of modal density, average joint acceptance of the first panel, and the ratio of radiation to total resistance of the second panel are the most important characteristics in determining the transmission characteristics of the wall.

Signal‐Detection Analysis of Equalization and Cancellation Model
View Description Hide DescriptionAn analysis of the equalization and cancellation model from a signal‐detection standpoint is presented. At low frequencies, say 250 cps, one can neglect the time‐error parameter because any slight phase error is small as compared with the period of a sinusoid and all the error in the binaural mechanism can be treated as small‐amplitude error. Using this assumption, it is possible to derive the equations presented in the text. Three main conclusions can be drawn from this analysis. (1) The signal‐detection analysisgenerates the same equations as Durlach's signal‐to‐noise‐ratio approach, except for a multiplicative factor that reflects the accuracy of the monaural energy measurement. (2) The parameter associated with the amplitude error, because of the multiplicative factor, is different from Durlach's by a factor of 5. (3) The new analysis predicts that shortening the signal duration or decreasing the noise bandwidth will increase the size of the masking‐level difference. Both effects have been observed.

Reliability of TTS from Impulse‐Noise Exposure
View Description Hide DescriptionA comprehensive damage‐risk criterion (DRC) for impulse‐noise exposure is needed, and it is desirable to state the DRC in terms of allowable TTS (temporary threshold shift), since TTS is both a valid and convenient measure of noise effects on hearing. This is possible only if TTS is also a reliable measure. Four TTS‐reliability studies are reported. The following conclusions are reached. (1) Individual subject's TTS's are not sufficiently reliable to permit generalization of impulse‐noise effects. (2) Group mean TTS varies only slightly across a series of exposures and is considered to be a reliable (consistent, repeatable) measure. This is true for the exposure of normal‐hearing subjects to different impulse‐noise conditions, for the TTS's of subnormal‐hearing subjects, and for frequencies representative of the whole range of human hearing. (3) The formulation of an impulse‐noise DRC should be based on group data (means, quartiles, etc.). Samples should be as large as possible and should be representative of the population to which generalization of results is desired.

Experimental Study of the Fundamental‐Frequency Component of a Plane, Finite‐Amplitude Wave
View Description Hide DescriptionThe fundamental‐frequency component of a plane, finite‐amplitude wave is measured in water at 5.65 Mcps by a fixed‐distance pulse method giving variable‐distance results. The absorption coefficient of the fundamental‐frequency component of the wave is examined from the standpoint of its maximum, average, and instantaneous values. It is found that there is a maximum amount of sound‐pressure amplitude that can be transmitted over a fixed path, varying as the −0.80 power of the distance. A value for the nonlinearity parameter of water is obtained, giving B/A = 6.3±0.8.

Relation between Third‐Order Elastic Moduli and the Thermal Attenuation of Ultrasonic Waves in Nonconducting and Metallic Crystals
View Description Hide DescriptionThe background loss of all materials, and sometimes the principal loss in single crystals, is connected with the conversion of acoustic waves into thermal phonons. One source of conversion is the thermoelastic loss resulting from the flow of heat from the compressed (hotter) part of the wave to the expanded (cooler) part. This source accounts for about half the thermal attenuation in a metal, but only about 4% of that in an insulator. Another source of thermal attenuation—first suggested by Akhieser—is connected with the separation of phonon‐mode temperatures by a suddenly applied stress, followed by an equilibration of these temperatures with a relaxation time τ. This source has been called “phonon viscosity.” A method for evaluating this source of attenuation was proposed in previous papers [W. P. Mason and T. B. Bateman, J. Acoust. Soc. Am. 36, 644–652 (1964); W. P. Mason, Physical Acoustics (Academic Press Inc., New York, 1965), Vol. 3B, Chap. VI, pp. 235–286]. It involves the use of the third‐order elastic moduli to determine the energy stored by the phonon‐mode temperature separations, together with a relaxation time τ to equilibrate this energy. Recently, new determinations of these third‐order moduli have been made for NaCl, KCl, MgO, and (YIG). The present paper shows that the formulas predict the measuredattenuations (longitudinal and shear waves) within about 25%. Silicon, without oxygen, has a higher thermal conductivity than silicon with dissolved oxygen. This is reflected in the higher attenuation of oxygen‐free material, in agreement with theory. A theoretical evaluation of the attenuation is made for a longitudinal wave propagated along a (111) axis. For metals, it is shown that the “phonon‐viscosity” attenuation is in the same order as thermoelastic attenuation. A method for evaluating this loss in single‐crystal lead is devised. The attenuationmeasured over a temperature range is consistent with the sum of the thermoelastic attenuation and the “phonon‐viscosity” attenuation.

Ultrasonic Diffraction Loss and Phase Change in Anisotropic Materials
View Description Hide DescriptionThe loss and phase change in progressive longitudinal waves from a finite circular piston source radiating into certain anisotropic media are calculated in this paper. Three‐dimensional calculations are presented for propagation along 3‐, 4‐, and 6‐fold symmetry axes. Relationships sufficient to permit the performance of the calculation for 2‐fold axes are derived. The anisotropy is introduced as a term in the spatial part of the phase in the integral for the pressure in the field of the piston. Geometry proper for pulse‐echo measurements was assumed in the calculation. The diffraction loss fluctuates with (z is the distance into crystal, λ, is the wavelength, a is the transducer radius) before becoming monotonic increasing as the logarithm of S. The positions of the peaks in the loss are functions of the anisotropy of the medium. The phase change, on the other hand, increases monotonically to a limit of π/2 from S‐0 to infinity. However, the phase has plateaus where the loss has peaks. Both new and previously published experimental data on diffraction loss along 3‐, 4‐, and 6‐fold axes in crystals agree quantitatively with theory. Some data on 2‐fold axes are also presented. The methods for correcting attenuation and velocity measurements for diffraction are given.

Propagation of Waves from a Spherical Surface of Time‐Dependent Radius
View Description Hide DescriptionThe dynamic response of an infinite elastic medium to a time‐dependent pressure in a spherical cavity is investigated. For a finite length of time, the wall of the cavity is subjected to an ablation process and the radius of the cavity increases monotonically from the initial value a _{0} to the permanent value b. Polar symmetry is assumed throughout the analysis. Explicit expressions are derived for the radial and the tangential stresses. The analysis is valid for arbitrary ablation rates and cavity pressures. For a constant ablation rate and a constant cavity pressure, the solutions are compared with the corresponding quasistatic results.

Scattering and Reflection by Elliptically Striated Surfaces
View Description Hide DescriptionClosed‐form approximations for the power‐reflection coefficient, for the phase of the coherently reflected wave, and for the differential‐scattering cross section per unit area, are given for random distributions of semielliptic protuberances on free or rigid base planes. For low frequencies, the real and imaginary parts of the required scattering amplitude (f) of an isolated protuberance are given explicitly to the eighth and sixth power of the frequency, respectively; for high frequencies, we use an approximation for f consisting of the geometrically reflected term plus a Fraunhofer diffracted term. Illustrative numerical examples are given for semiellipses ranging from perpendicular strips, through semicircles, to the near‐limiting case of nearly flat strips.

Acoustic Experiment to Determine the Composition of an Unknown Planetary Atmosphere
View Description Hide DescriptionMeasurement of the speed of sound and the acoustic impedance, combined with a knowledge of ambient pressure and temperature, provide sufficient data for determining the relative abundance of constituents in a mixture of nitrogen, argon, and carbon dioxide.Sound absorption and dispersion yield additional information for checking the consistency of the results. An instrument has been developed to measure the acoustic properties of an unknown gas mixture. Laboratory results demonstrate the feasibility of determining the gas composition within a few percent. The technique of velocity measurement by use of phase comparison over several wavelengths is quite sensitive and may be useful in other applications.

Energy Density of Sound in a Dispersive Medium
View Description Hide DescriptionThe linearized field equations are written with phenomenological frequency‐dependent parameters to describe dispersion, in a manner analogous to the use of frequency dependent permittivities and permeabilities in electromagnetism. It is then shown that the energy density for such a medium should include contributions from the frequency derivatives of these parameters.
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Acoustical Hazards of Children's “Toys”
View Description Hide DescriptionMeasurements of impulse noises produced by four toy firearms are presented. It is suggested that these “toys” constitute a potential hazard to children's hearing.

Pitch Shifts of Tones in Wide‐Band Noise
View Description Hide DescriptionPitch shifts of pure tones occuring in wide‐band noise were measured for five frequencies (250, 500, 1000, 3000, and 6000 Hz) at four levels (5, 15, 25, and 35 Db] above quiet and masked thresholds for each of 11 subjects. Amount of noise employed was sufficient to shift threshold of a 1000‐Hz tone by 35 dB. Alternate monaural presentation of 500 msec standard and variable signals was employed. The amount of pitch shift expressed in Hertz was found to increase with frequency. Relative pitch shift (absolute shift/standard frequency) increased with frequency between 250 and 3000 Hz but was less for 6000 than for 3000 Hz. The data suggested that the relative amount of pitch shift may decrease with signal level for frequencies less than 3600 Hz, increase with signal level for 6000 Hz, and be independent of signal level for 3000 Hz. All observed pitch shifts were in all upward direction, except for the two lowest frequencies.

Objective Audiometry
View Description Hide DescriptionIt has been suggested [W. D. Keidel and M. Spreng, “Neurophysiological Evidence for the Stevens Power Function in Man,” J. Acoust. Soc. Am. 38, 191–195 (1965)] that human evoked cortical responses to acoustic stimuli have properties that can be related directly to the sensation of loudness. Evidence is quoted to support the argument that, although the evoked responses are certainly related to loudness, the direct equation of sensation magnitude to potential is not entirely justifiable.
