Volume 46, Issue 3B, September 1969

Ultrasonic and Viscoelastic Relaxation in Bis[m‐(m‐phenoxy phenoxy) phenyl] Ether
View Description Hide DescriptionThe velocity of propagation and attenuation of longitudinal waves has been determined for a six‐ring polyphenyl ether, bis[m‐(m‐phenoxy phenoxy) phenyl] ether. The viscosity, density, and shear relaxation properties of this liquid are known from previous measurements. It is found that the relaxing part K _{2} of the compressional modulus is equal to times the limiting value of the shear modulusG _{∝}, but the volume viscosity η_{r} is approximately times the shear viscosity η. Within experimental error, the shape of the reduced relaxation curves for the components of the complex compressional modulus is the same as that found for the corresponding components of the complex shear modulus. This indicates that the shear and compressional relaxation processes have a common origin. These findings are compared with previously published results for associated liquids.

Effect of Pressure on Ultrasonic Absorption, Multistate Dissociation, and Chemical Equilibrium in Solutions
View Description Hide DescriptionMeasurements were made on the effect of pressure on ultrasonic absorption in a 0.5M aqueous solution of manganese sulfate at 25°C. A cylindrical resonant cavity was used for obtaining absorption in the frequency regions of 300 and 500 kHz, at pressures up to 20 000 psi. Results are similar to those obtained for , in that absorption decreases by a factor of about 3 for a pressure increase of 15 000 psi. Independent conductivity data show nearly the same behavior for as for solutions, namely, an increase of only about 14% in the concentration of dissociated ions for the same pressure increase. These results for appear to be inconsistent with the model used by Atkinson and Kor to analyze their data. The large effect of pressure on ultrasonic absorption is similar to that observed by Carnevale for 0.1M at 60°C from 9 to 75 MHz.

Thermal Relaxation in Humid Air
View Description Hide DescriptionA theoretical and experimental investigation of the molecular absorption process in air has been carried out with particular reference to the variation of Napier frequency (f _{max}) with water‐vapor concentration (h′). It turns out that the relationship between these two quantities is a quadratic only at very low humidities and that, over the rest of the humidity range, the general relationship derived here is best approximated by an equation of the form , where 1 < x < 2. Critical examination of previous theoretical relationships shows that the rate of resonant interchange of energy between oxygen and water vapor has been underestimated because of the quadratic approximation. Reassessment of this quantity on the basis of the analysis herein yields a more realistic value that is more consistent with previous experimental findings in oxygen and closer to what one would expect from theoretical calculations. Details are also given of a numerical method of calculating Napier frequency directly from measured values of molecular absorption.

Subharmonic and Other Low‐Frequency Emission from Bubbles in Sound‐Irradiated Liquids
View Description Hide DescriptionThis paper presents measurements of the acoustic emission from gas bubbles of controlled sizes “seeded” into water and glycerol‐water mixtures and subjected to sound fields over a wide range of frequencies and intensities up to, and beyond, the transient cavitation threshold. The emission from unprepared liquids was also recorded for comparison. A mechanical device has been developed for generating small bubbles of uniform sizes at controlled rates. The results suggest that there are at least two mechanisms for generating signals below the excitation frequency (f _{0}): (1) Subharmonics of f _{0} may be produced by forced oscillations of bubbles whose radial resonance frequencies are submultiples of f _{0}; (2) bubbles of any size may be shock excited to emit signals at their radial resonance frequencies. Large bubbles of certain preferred sizes are normally present, and their resonances probably account for many low‐frequency signals that are not integral submultiples of f _{0}. It also seems certain that some other mechanism unconnected with resonant bubbles contributes to the relatively strong signals observed at the first subharmonic (f _{0}/2).

Rôle of the Vibrational Relaxation of Nitrogen in the Absorption of Sound in Air
View Description Hide DescriptionThe vibrational relaxation of nitrogen is proposed for the mechanism of the anomalously large absorption of sound found at low frequency and high relative humidity in air.

Pressure Waves in an Accelerated Sphere Filled with a Compressible Liquid
View Description Hide DescriptionThe linearized equations for the flow of a compressible inviscid fluid in an accelerated sphere were solved for a constant step in the acceleration from a state of rest. Pressure waves whose maximum amplitude is nearly equal to the hydrostatic pressure from an equivalent steady gravitational field acting through the depth of the diameter are induced in the fluid. This maximum pressure occurs at the end of that diameter aligned in the direction of the acceleration. By means of the convolution integral, the pressure distribution on the inner surface of a sphere being accelerated sinusoidally was obtained and the reaction force resulting from the fluid motion was computed. The pressure oscillations build up in a resonant manner with amplitude linear with time when the driving‐reduced frequency is equal to an eigenvalue of the problem.

Propagation of Sinusoidal Small‐Amplitude Waves in a Deformed Viscoelastic Solid. I
View Description Hide DescriptionThe propagation of a sinusoidal wave of small amplitude in an initially isotropic viscoelastic solid, which is subjected to a static pure homogeneous deformation, is discussed. The secular equation for a plane wave propagating an arbitrary direction with respect to the principal directions of the static deformation is derived. The complex slownesses corresponding to propagation along the principal axes of the static strain are found. Relations are obtained between these and the principal stresses associated with the static deformation, which generalize earlier formulas obtained by Ericksen for elastic materials.

Analysis of Acoustic Wave Scattering from a Rough Layer
View Description Hide DescriptionThe theory of wavescattering from a rough surface has been used to calculate reflection from a rough layer. Mathematical expressions have been developed for the average value and the variance of the scattered power. These expressions take into account multiple reflections in the layer between the top and the bottom interfaces. The variance of the scattered power is expressed as a series. The terms of this series decrease rapidly.

Acoustic Field Produced by an Arbitrary Body in Good Vibration. I. Theory and Three‐Dimensional Synthesis
View Description Hide DescriptionArbitrary‐body acoustic radiation is simulated by an inside source distribution whose field satisfies the boundary vibration. Together with the expansion of Kirchoff's integrand in the ratio of source points and observation point locations, a proposed perturbation process is shown to converge for ratios that are less than unity. Within the frame of perturbation, the class of vibration (good vibration) was shown to bypass the eigenvalue difficulties by analytical inversion. The results are consistent with spherical pulsation, spherical oscillation, and Morse's solution for spherical‐cap vibration. Application of the method to transient analysis is also indicated.

Effective Dynamic Properties of a Fiber‐Reinforced Material and the Propagation of Sinusoidal Waves
View Description Hide DescriptionA theoretical study is presented on the propagation of a plane sinusoidal wave through a material that is reinforced with parallel fibers in one direction. The wave propagates in a direction normal to the fibers, and both fiber and matrix are made of linear elasticmaterials. An integral formulation for the scattered‐wave solution of an isolated fiber is used to study the multiple scattering in an infinite slab of the composite material. The transmitted and reflected waves from the composite and from a homogeneous slab are shown to be similar. By matching the two sets of results, formulas expressed in terms of the isolated‐fiber solution are derived for the wave speed, the effective density, and the modulus of the composite. In general, the effective density and modulus so defined are complex numbers and depend on the wave frequency. This fact indicates the possible existence of dissipation and dispersion in the composite under dynamic loadings. A series solution is presented for a composite containing circular fibers, and approximate closed form results are obtained for the limiting case of very long wavelength. As expected, the material does not exhibit dispersion and dissipation at this limit.

Wave Propagation in Anisotropic Bars of Rectangular Cross Section. II. Flexural Wave Propagation
View Description Hide DescriptionAn approximate method is employed to obtain the solution for the problem of steady‐state flexural wave propagation in anisotropic bars of infinite length and rectangular cross section. The first three dispersion curves are obtained for bars of orthotropic, tetragonal, hexagonal, cubic, and isotropic material with several different values of S (ratio of lateral dimensions). The work represents an extension of a previous study of the longitudinal wave‐propagation problem that is discussed by Nigro [J. Acoust. Soc. Amer. 43, 958–965 (1968)].

Dispersion of Flexural Waves in Circular Bimaterial Cylinders—Theoretical Treatment
View Description Hide DescriptionThe general form for displacements and the frequency equation are obtained for the first mode of flexural wave propagation in an infinitely long circular bimaterial cylinder. Based on the equations of linear elasticity, the theory follows the technique developed by Pochhammer and Chree for elastic bars. Several first‐branch dispersion curves for the first flexural mode are presented for various ratios of the constituent cylinder radii. Dispersion characteristics significantly different from those predicted by the theory for homogeneous cylinders are realized. Some implications in regard to equivalent properties are discussed.

Transmission of Plane Waves through Layered Linear Viscoelastic Media
View Description Hide DescriptionThe two‐dimensional propagation of time‐harmonic plane waves through a plane horizontally layered viscoelastic medium is discussed. The problem is formulated as the equivalent elastic plane‐strain case with modified Lamé constants, which are complex and frequency dependent, replacing the usual elastic Lamé constants. Rather than use potentials, incident angles, etc., we formulate the problem directly in terms of stresses and displacements and solve it by using matrix methods. This approach is felt to be more direct and leads to some interesting conclusions. If the incident wave is not attenuated in the direction parallel to the layering, interface waves can be generated only if one of the layers is “pseudoelastic,” i.e., has at least one real wave speed. In this case, the interface waves are generated in the same manner as in the purely elastic case. Such a physical problem would exist, for example, if the incident waves were to travel through a semi‐infinite elastic half‐space before striking the plane viscoelastic layers. If the incident wave is attenuated in the direction parallel to the layering, interface waves can be generated at specific angles of incidence and specific combinations of material parameters.

Decoupling of the Vector Wave Equation of Elasticity for Radially Heterogeneous Media
View Description Hide DescriptionThe potential representation of the vector fields is employed to effect the decoupling of the vector wave equation of elasticity for radially heterogeneous isotropic media. Two types of decoupling are considered: decoupling for all values of the frequency and decoupling for high frequencies, which is useful in the ray‐theory treatment. In the case of decoupling for all frequencies, the three constitutive parameters of the medium should satisfy a pair of nonlinear differential equations. For high‐frequency decoupling, one of these equations becomes unnecessary. A few useful examples of the media that satisfy the conditions of decoupling are given.

Time‐Domain Analysis of Broad‐Band Refraction and Diffraction
View Description Hide DescriptionThe emissions from a broad‐band impulsive point source are refracted by a spherically shaped interface between two fluids having different acoustic velocities. It is shown that, on axis, two disturbances are produced: The first is the waveform emitted by the source, and the second is its polarity inverse. As the observation point approaches the image point of the source, the waveform approaches the derivative of the waveform emitted by the source.Diffraction is a special case for which the acoustic velocities of the two fluids are equal. Two disturbances are produced: The first is the waveform emitted by the source, and the second is its inverse.

Stability of Nonlinear Oscillations of Stretched Strings
View Description Hide DescriptionThe stability of damped forced vibrations and undamped free vibrations of stretched strings is investigated. The regions of stability for the forced vibrations are plotted in the amplitude‐frequency plane. It is shown that undamped planar free vibrations are unstable.

Response of Linear Structures to Classes of Pressure Fields. I. Deterministic Nonconvecting Fields
View Description Hide DescriptionPressure fields are classified according to the analyticity of the input‐function wavenumber and frequency spectrum. This property is sufficiently general to allow for grouping of a host of different input functions into a class, and yet provides sufficient information for the calculation of the response. A methodology thus results for the theoretical prediction of the linear response of structures. Within the framework of this methodology, a detailed mathematical description of the excitation is not required; rather, knowledge of the nature of the singularities of the input‐function spectrum suffices. This paper considers the important class of input functions whose spectra are meromorphic functions of wavenumber and frequency. The methodology and computational procedure are exemplified by the calculation of the response of simply supported uniform beams and plates to deterministic excitation; the excitation is described by input functions of the foregoing class. With a single calculation, response solutions result that are applicable to a variety of input functions. It is shown that the dependence of the response on the physical parameters of interest is dictated by the location and order of the poles of the input‐function spectrum. The usefulness of the derived solutions in engineering applications is shown through a simple example.

Constants of a Trapped‐Energy Electromechanical Transducer Made by Evaporating a Thin Layer of a Piezoelectric Crystal on Each Side of a Quartz Plate
View Description Hide DescriptionOne of the problems in making overtone monolithic crystal filters is that the electrodes become so small and the electromechanical coupling so low that the impedance becomes so high and the bandwidth so low that useful filters are not obtained. It is suggested in this paper that both of these difficulties can be removed by evaporating thin films of a high‐electromechanical‐coupled crystal such as lithiumtantalate on both sides of a quartz‐crystal substrate. An alternate construction is to cement thin single crystals on each side and decrease their thickness by grinding or sputtering. The thickness of the thin transducers are in the order of R times the quartz‐crystal thicknesses where R is the plate‐back parameter for a monolithic filter. It is shown that such a structure has a resonant frequency determined by the sum of the thickness of the quartz crystal and evaporated piezoelectric layer and a coupling coefficient given by , where k ^{2} is the square of the coupling coefficient of the evaporated film,h is the thickness of one film, and b is half the thickness of the quartz crystal. For k = 0.6, as in lithiumtantalate, the composite coupling k_{c} is in the same order as for a quartz crystal. For an overtone mode, the composite coupling k_{c} does not vary, whereas it varies as 1/n for an overtone quartz crystal. Since the impedance is very low, the filter impedance can be matched by standard transformers.

Free Vibrations of Circular Membranes under Arbitrary Tension by the Finite‐Element Method
View Description Hide DescriptionThe method of finite elements is extended to problems of free vibrations of elastic homogeneous membranes of general shape under arbitrary tensile forces at the edge. Special consideration is given to drumhead‐type structures that are basically circular thin elastic sheets fixed at the edge and subjected to nonuniform tension. Six different loading conditions, including the uniform case, are considered. Particular attention is given to the influence of various force systems on natural frequencies and normal modes of vibration. It is found that the lowest natural frequencies are virtually identical for the various edge loading conditions studied, and the corresponding vibration modes vary little from one another. However, for the higher modes, not only do the natural frequencies of vibration differ much, but some of the nodal patterns are distorted in a very fascinating fashion. For the case of uniform tension, a comparison of natural frequencies with known exact solutions shows that the method gives very good results.

Vibration Characteristics of a Submerged Ring‐Stiffened Cylindrical Shell of Finite Length
View Description Hide DescriptionAn analytical procedure for determining the natural frequencies for a submerged cylinder is presented. The coupling between the shell and acoustic medium is taken into account. An experimental evaluation of this procedure for a particular cylindrical shell is presented. Good agreement is shown between measured and predicted natural frequencies of the cylindrical shell, both in air and submerged in water.