Volume 50, Issue 3B, September 1971

Absorption Coefficients of Surfaces Calculated from Decaying Sound Fields
View Description Hide DescriptionA modified form of the Eyring equation is derived which explicitly takes account of the anisotropy of a sound field due to the deviation of the energy reflection (absorption) coefficient of individual surfaces from the mean value for all surfaces. When all surfaces have the same energy absorption coefficient, the equation simplifies to that of Eyring; when in addition these absorption coefficients are small, the usual simplification to the Sabine equation is valid. Using a simplified version of this modified Eyring equation, one can show that the effective absorption coefficient of a single absorptive surface in a hypothetical reverberation room is a complicated function of the location and total area of randomly oriented scattering panels. Apart from any correction for diffraction, the effective absorption coefficient is also a function of the area of the absorptive surface. In typical situations, this theory predicts that the effective coefficient will be greater (0 to 40%) than the energy absorption coefficient. In general terms, this agrees with measured values. In extreme situations not usually found in reverberation rooms, effective absorption coefficients are predicted to vary more widely and to be either greater or less than the energy absorption coefficient.

Enhancement of Heat Pulses in Crystals due to Elastic Anisotropy
View Description Hide DescriptionThe generation of thermal acoustic waves by a heated metallic film on the surface of a dielectric crystal at low temperatures is discussed. It is shown that even if the waves produced correspond to a uniform distribution of directions of wave vector, there may be a large enhancement of energy flow in some crystallographic directions compared to the average. This effect arises because of elasticanisotropy. Explicit expressions are given for the enhancement expected in the principal directions of cubic crystals.

Reflection and Refraction Coefficients at a Fluid‐Solid Interface
View Description Hide DescriptionFor an infinite plane elastic wave which strikes the plane interface separating two semiinfinite isotropic media, the calculation of the complex reflection coefficient,R*, for varying angles of incidence, θ, is not difficult. In the past, calculations have been made for a number of lossless media and, consequently, an important facet of the R*−θ curve for real materials has been overlooked. The total reflection, R*=1, which occurs at the longitudinal‐ and shear‐wave critical angles is well known, but the appearance of a minimum (sometimes a zero) in R* is not and its existence defines a third critical angle, sometimes inappropriately called the Rayleigh‐wave angle, at which a wave with large surface components is generated. During experiments with beams of acoustic waves, there is an apparent lateral displacement of the reflected beam at the third critical angle which manifests itself markedly only when there is a near‐zero in R*. Notwithstanding the calculations of Schoch [Ergeb. Exakt. Naturw. 23, 127–234 (1950)] and the experimental measurements of others apparently to the contrary, it is strongly suggested that no actual lateral displacement occurs and that re‐radiation from the region outside that ensonified by the incident beam gives rise to the apparent lateral displacement. Some outcomes of the investigation are discussed and of these the more important are: (1) Rayleigh and other interface and surface waves in real media are degenerate; (2) no interface wave can exist independently at the junction of two nonideal media; (3) Huyghens's principle in its elementary secondary‐wavelet form does not apply to lossy media; (4) the usefulness of the sensitivity of R* to minute changes in elastic parameters at the Rayleigh angle as a measuring tool has a wide application; (5) elliptical polarization of both shear and longitudinal waves can occur, owing to boundary influences in lossy refracting media. The real part of the propagation vector lies in the plane of the ellipse.

Acoustic Horns with Spatially Varying Density or Elasticity
View Description Hide DescriptionThe transformation of acoustic impedance can be accomplished either by a controlled variation of wavefront area as the wave progresses, or by transmission of the waves through a medium in which there is a gradient of density or elasticity. The second concept does not seem to have been exploited despite the fact that the effective density of a fluid medium can be increased by introducing suitable arrays of obstacles, and the bulk modulus of a liquid decreased by compliant inclusions. The present analysis shows that for plane waves a medium in which density varies as (B + mx)^{−2} or elastic modulus as (B − mx)^{2}, where x is distance and B and m are constants, is analogous to the familiar exponential horn. Simple analytical solutions are also found for radial variations of density or elasticity which transform the behavior of sectoral and conical horns into the acoustic equivalents of exponential horns. Since the wave equation is separable in both cylindrical and spherical coordinates, these solutions are exact. Three examples of artificial fluid structure are given and the relationships between array size, bandwidth, and damping are discussed.

Velocity Dispersion of Rayleigh Waves Propagating along Rough Surfaces
View Description Hide DescriptionThe frequency dependence of the velocity of Rayleigh waves propagating along rough surfaces is examined from the viewpoint of the mass loading of a smooth surface. Predicted values are given for examples of idealized surfaces on a solid of Poisson's ratio 0.29.

Forced Vibration of Internally Damped Circular and Annular Plates with Clamped Boundaries
View Description Hide DescriptionClosed‐form expressions are presented for the driving‐point impedance, transfer impedance, and force transmissibility of circular and annular plates with clamped boundaries. The plates are internally damped and are driven either by a sinusoidally varying central point force or by one or more concentric ring forces of variable radii. The circular plates are mass loaded or elastically restrained at their midpoints, or are loaded by an ideally rigid annular mass (rib) of arbitrary radius. Computed results illustrating the frequency dependence of impedance and transmissibility are described and their physical significance is discussed. Added mass is very effective in increasing plate impedance levels and in blocking force transmission to the plate boundaries, particularly when the mass is positioned directly beneath the driving force.

Equations of Elastic Motion of an Isotropic Medium in the Presence of Body Forces and Static Stresses
View Description Hide DescriptionBy taking account of second‐order coupling between stress and strain, modified equations of elastic motion have been derived which include the effects of static surface forces and body forces upon the propagation of elastic waves in isotropic materials.Anisotropy of the kind found in hexagonal crystals is shown to occur when an isotropic substance is acted upon by a uniaxial static stress. A brief discussion of some inferences to be drawn from these results is given.

Longitudinal and Shear Magnetoelastic Behavior of Metals
View Description Hide DescriptionThe magnetic‐field dependence of the complex Young's and shear moduli are determined for a variety of nonferromagnetic metals. An experimental technique using capacitive excitation of torsional and longitudinal modes in low‐loss systems is described and the criterion for frequency resolution is discussed. Contributions to the magnetic‐field dependence of the elasticity due to the charge carriers and dislocations are considered.

Surface‐Wave Propagation over an Elastic Cosserat Half‐Space
View Description Hide DescriptionThe equations describing an elastic isotropic Cosserat continuum are presented. Solutions of these equations for the various modes of plane harmonic waves in an infinite medium are briefly discussed. The analysis then concentrates on the development and interpretation of the surface‐wave solution for straight crested waves on a Cosserat half‐space. It is found that a wave analogous to the Rayleigh wave of the classical elasticity theory exists, except that it is a dispersive wave in this theory. The phase velocity of the surface wave may either increase or decrease with frequency, depending on the relative magnitude of the micromaterial moduli.

Early‐Time Interaction of Spherical Acoustic Waves and a Cylindrical Elastic Shell
View Description Hide DescriptionThe three‐dimensional transient interaction of spherical acoustic waves with a cylindrical elastic shell is investigated using the concept of fictitious Riemann surface. Integral transform techniques are used. The transform double integrals with respect to the two shell coordinates are evaluated asymptotically by the method of “critical points.” Simple expressions for the early‐time responses of the shell and the scattered wave are obtained. The results of this investigation show that the effects of the curvature of the incident wave are quite significant with respect to the bending moments of the shell.

Scattering by a Spherical Cap
View Description Hide DescriptionThe boundary‐value problem for plane‐wave scattering by a spherical cap is formulated in terms of either the radial velocity in the aperture or the equivalent‐vortex strength of the shell. The resulting complementary integral equations are used to construct complementary variational expressions for the scattering cross section and also are solved approximately by Galerkin's method. Numerical results are given for the scattering cross section, say σ, with special reference to the Helmholtz resonator (small aperture) and the hemispherical shell. The function σ(k), where k is the dimensionless wavenumber based on the radius of the sphere, rises initially as k ^{4} (Rayleigh‐scattering regime) to a first peak, σ(k _{0}), and then executes a decaying oscillation about the asymptotic value, σ_{∞} (twice the transverse area intercepted by the incident wave); as the polar angle of the aperture, β, tends to zero, whilst for the hemispherical shell.

Waveforms and Frequency Spectra of Acoustic Emissions
View Description Hide DescriptionIllustrations are given of information carried in the waveforms and frequency spectra of acoustic emissions. The effects of multiple reflections and resonances are discussed. A model of the acoustic‐emission sourcewave is developed, and arguments are given why this wave should be a pulselike function, rather than an oscillatory function of stress. Further use of this model may allow more quantitative treatment of emission amplitudes, energies, and spectra. Experimental results show the use of two instruments for the evaluation of emission spectra. The feasibility of using frequency analysis to obtain information about source events is demonstrated.

Propagation of Acoustic Waves in a Turbulent Medium
View Description Hide DescriptionA theoreticalanalysis of the propagation of the coherent, or average, acoustic wave in a turbulent medium is presented. The analysis is based on the smoothing method developed by Keller and others. The dispersion equation is derived for a statistically homogeneous medium for three special cases: waves in a fluid at rest with density and temperature fluctuations, high‐frequency waves in a fluid in turbulent motion, and waves in an isentropic irrotational turbulent flow. This equation is solved approximately for the propagation constant of high‐ and low‐frequency waves by assuming that the medium is statistically isotropic. From the propagation constant, the propagation speed and attenuation coefficient are obtained. The results indicate that generally the propagation speed of the coherent wave is less than the average sound speed of the medium. The results also indicate a similarity between the effect of turbulent velocity fluctuations and the effect of inhomogeneities the thermodynamic properties of the medium on the attenuation of high‐frequency waves, whereas there is apparently a fundamental difference between the effect of velocity fluctuations and the effect of thermodynamic inhomogeneities on the attenuation of low‐frequency waves.

Acoustic Waves Transmitted through Solid Elastic Cylinders
View Description Hide DescriptionThe acoustic field of a plane wave incident on an elastic cylinder may be decomposed into two parts, one representing circumferential waves that propagate around the cylinder, the other part obeying in the high‐frequency limit the laws of geometrical optics. The latter part is studied in this paper, and is decomposed into the specularly reflected wave and into a series of transmitted waves which traverse the cylinder either directly along a secant, or undergo additional internal reflections with and without mode convention (compressional to shear type, or vice versa). Wavefront loci and amplitudes of these waves are calculated using the saddle‐point method, and the corresponding reflection and transmission coefficients are shown to reduce, for the case of a large cylinder, to the known expressions for a flat elastic half‐space in contact with a liquid.

Volume Backscattering in the South China Sea and the Indian Ocean
View Description Hide DescriptionVolume backscattering strengths averaged over four depth intervals and at the frequencies 3.5, 6, 8, and 12 kHz were measured at a number of stations in the South China Sea and the eastern region of the equatorial Indian Ocean. Measured values ranged from −91 dB re m^{−1}, which occurred at 3.5 kHz, to −61 dB re m^{−1}, which occurred at 8 kHz. On the average, the scattering was a maximum in the depth interval 0<z<500 m at 3.5, 6, and 8 kHz; but at 12 kHz, the maximum was usually in the interval 350<z<500 m. At one station, a deep scattering layer with a resonant frequency of 3.5 kHz was observed at a depth of 1700 m. In the Indian Ocean, negative correlations existed between the scattering at 8 kHz and the water temperature at a depth of 100 m. It is shown that the conventional sonar equation for volume reverberation is applicable to resonantscattering as well as nonresonant scattering and that it is necessary for the acoustic source to be at a finite depth.

The Acoustical and Mechanical Properties of Sonite
View Description Hide DescriptionSonite is a material used for acoustic decoupling in deep‐ocean sonartransducers. Its density, bulk modulus, soundvelocity, and attenuation constant have been measured as a function of precompression and ambient pressure. The soundvelocity and attenuation constant were measured at 40 kHz and 12°C, utilizing a water‐filled impedance tube. By using a delay line technique at 1 MHz, the material was found to exhibit transverse isotropy. Sample preparation and measurement techniques are described.

Wood's Refraction Correction Equations for a Constant‐Gradient Medium: An Alternate Derivation
View Description Hide DescriptionClosed‐form solutions have been developed by Wood [J. Acoust. Soc. Amer. 47, 1448–1452 (1970)] for the exact computation of the true distance d and direction θ_{ i } of a sound source relative to the observer in an unbounded constant‐gradient medium. Wood's solutions are given in terms of the sound speedc at the receiver depth, the apparent direction θ_{0}, the one‐way travel time t, and the positive gradient g. All of these quantities are measurable at the receiving point. Wood makes use of the Pekeris triangle formulas to arrive at these solutions; however, it is of interest to know whether and how the refraction correction equations can be derived from the conventional ray propagation and time equations in a constant‐gradient medium. Such an alternate derivation is presented in this article. Also, a few applications illustrating the utility of Wood's solutions are presented.

An Intensity Differential Equation in Ray Acoustics
View Description Hide DescriptionA desired goal of either ray acoustics or normal‐mode analysis in underwater sound propagation problems is the prediction of the intensity at some field point. The ray‐bundle approach in ray acoustics is a technique which has been employed to predict intensity. By formulating the ray‐path problem in terms of a differential equation, rather than in terms of a range integral, a second differential equation can be obtained which describes the continuous variation of intensity along the ray. An extension of the calculation of intensity to allow for variation of the speed of sound with range as well as depth is easily accomplished. It is shown that, for the intensity function to be continuous, a sufficient condition on the velocity function is that it be twice continuously differentiable except at caustics where the theory breaks down.

The Transient Response of Arrays of Transducers
View Description Hide DescriptionA general approach is developed to compute the time‐dependent velocities of transducers within an array resulting from a set of prescribed electrical inputs. It is noted that the effect of the dynamic response of a transducer of any complexity and the effect of the time‐dependent mutual interactions are included within the approach which is based on a Green's function solution to an initial boundary value problem. The solution of the initial boundary value problem is reduced to a set of time‐domain‐coupled convolution integral equations which can be easily solved on a digital computer. In addition to presenting a clear physical understanding of the transient behavior of arrays, the approach leads to a more basic understanding of array operation and limitations caused by its transient response. Specific application of the approach to the analysis of planar arrays mounted in rigid baffles is discussed and related to earlier work. Finally, a simple numerical example is presented to verify the approach and offer insight into the behavior of coupled systems exhibiting array‐type characteristics.

A Continuous‐Gradient Curve‐Fitting Technique for Acoustic‐Ray Analysis
View Description Hide DescriptionA method of computing acoustic‐ray paths is developed. The velocity of sound is approximated by a function of depth that has a continuous gradient and allows one to evaluate the range and travel time integrals in closed form. The advantages of this technique over others are discussed, and some interesting phase effects are illustrated.