Pulsating Sphere Research Articles

Wave motion in a random hydroacoustic medium using boundary integrals/elements methods

Two algorithms for solving the general problem of wave propagation, radiation and/or scattering in a random hydroacoustic medium using the boundary element method are presented here. The formulation is based on an observation about the predominant type of randomness in a hydroacoustic medium, i.e. the dominance of sound velocity fluctuations over density fluctuations, that allows employment of a single perturbation scheme. Perturbations may be applied at either the differential or the integral operator level. In the former case, each of the resulting nth order governing differential equations, n = 1,2…, preserves the deterministic nature of the original wave operator with the randomness appearing as a forcing function (inhomogeneity). The differential equations are subsequently recast as integral equations with both surface and volume type integrals that are solved using routine boundary element techniques. In the latter case, a random integral equation statement is written and the random kernels are expanded using perturbations. Only surface type integrals result, but different expressions must be evaluated at each perturbation order level n. Second-order moment statistics are finally computed once the total solution has been reconstituted from the perturbed solutions. Random forcing functions and /or boundary conditions are included in the development of these approaches but these are taken as deterministic in the applications. The methodology is for steady-state waves. Finally, a numerical example involving a pulsating sphere in a random fluid under stationary conditions serves to illustrate the method.

Internal wave generation in uniformly stratified fluids. Part 1. Green's function and point sources

In both Boussinesq and non-Boussinesq cases the Green's function of internal gravity waves is calculated, exactly for monochromatic waves and asymptotically for impulsive waves. From its differentiation the pressure and velocity fields generated by a point source are deduced. by the same method the Boussinesq monochromatic and impulsive waves radiated by a pulsating sphere are investigated.Boussinesq monochromatic waves of frequency ω < N are confined between characteristic cones θ = arccos(ω/N) tangent to the source region (N being the buoyancy frequency and θ the observation angle from the vertical). In that zone the point source model is inadequate. For the sphere an explicit form is given for the waves, which describes their conical 1/r½ radial decay and their transverse phase variations.Impulsive waves comprise gravity and buoyancy waves, whose separation process is non-Boussinesq and follows the arrival of an Airy wave. As time t elapses, inside the torus of vertical axis and horizontal radius 2Nt/β for gravity waves and inside the circumscribing cylinder for buoyancy waves, both components become Boussinesq and have wavelengths negligible compared with the scale height 2/β of the stratification. Then, gravity waves are plane propagating waves of frequency N cos θ, and buoyancy waves are radial oscillations of the fluid at frequency N; for the latter, initially propagating waves comparable with gravity waves, the horizontal phase variations have vanished and the amplitude has become insignificant as the Boussinesq zone has been entered. In this zone, outside the torus of vertical axis and horizontal radius Nta, a sphere of radius a [Lt ] 2/β is compact compared with the wavelength of the dominant gravity waves. Inside the torus gravity waves vanish by destructive interference. For the remaining buoyancy oscillations the sphere is compact outside the vertical cylinder circumscribing it, whereas the fluid is quiescent inside this cylinder.

Interaction of two pulsating spheres in a cylindrical cavity filled with an incompressible fluid

Interaction of two pulsating spheres in a cylindrical cavity filled with an incompressible fluid

Studies on surface discretization in boundary element method applications

The implementation of the boundary element method (BEM) to sound radiation problems consists of discretizing the vibrating surface into a number of elemental radiators and then applying the Helmholtz integral equation for predicting the acoustic field. As with other numerical techniques such as the finite element method, the accuracy of the model is dependent upon, among other factors, the discretization scheme. Creating an efficient mesh is important in order to insure accurate results with a reasonable amount of computational effort. Results of work conducted comparing the use of planar as well as linear and quadratic isoparametric elements are shown. The case studies involve modeling the acoustic field and radiation efficiency of a pulsating sphere at various wavenumbers and modeling the directivity pattern of a vibrating finite cylinder. Procedures for checking accuracy of solution are compared such as matrix condition number and velocity potential of interior points.

The ?-field theory of gravitation and cosmology

The present paper presents, within the Einstein conceptual framework, a scalar field theory of gravitation. The Ω-field, however, is not superimposed on the space-time manifold from outside the given geometry but is derived from the space-time torsion which is shown to be the generator of gravitation and inertia as well as spin. The theory predicts the three crucial tests of gravitation to the same degree of accuracy as does Einstein's theory. But it also predicts gravitational radiation emitted by a pulsating sphere and a singularity free cosmology in contradiction to Einstein's results. Above all, the scalar gravitational field can be easily quantized in the present context.

Internal gravity waves generated by oscillations of a sphere

We consider the radiation of internal gravity waves from a spherical body oscillating vertically in a stratified incompressible fluid. A near-field solution (under the Boussinesq approximation) is obtained by separation of variables in an elliptic problem, followed by analytic continuation to the frequencies ω < N of internal wave radiation. Matched expansions are used to relate this solution to a far-field solution in which non-Boussinesq terms are retained. In the outer near field there are parallel conical wavefronts between characteristic cones tangent to the body, but with a wavelength found to be shorter than that for oscillations of a circular cylinder. It is also found that there are caustic pressure singularities above and below the body where the characteristics intersect. Far from the source, non-Boussinesq effects cause a diffraction of energy out of the cones. The far-field wave-fronts are hyperboloidal, with horizontal axes. The case of horizontal oscillations of the sphere is also examined and is shown to give rise to the same basic wave structure.The related problem of a pulsating sphere is then considered, and it is concluded that certain features of the wave pattern, including the caustic singularities near the source, are common to a more general class of oscillating sources.

The sound field of a pulsating sphere in unsteady rectilinear motion

A sphere, whose radius changes in time, moves with non-uniform speed along a straight line through an inviscid compressible fluid. The acoustic velocity potential is determined asymptotically when the characteristic speed U 0 of the sphere is small compared with the mean sound speed c 0 in the fluid. It is assumed that both the sphere radius and the translational speed vary slowly, with appreciable change on a time scale t 0 = a 0 / U 0 , where a 0 is a characteristic radius; furthermore, the sphere is assumed to be compact, in the sense that the ratio between its radius and a typical wavelength is small. The sound field is expressed by means of moving source and dipole potentials in a fashion that appears simpler and more general than previously given.

Oscillation of a sphere in a saturated poroelastic medium using singular solutions

The Biot equations [J. Acoust. Soc. Am. 28, 168–178, 179–191 (1956)] for a saturated poroelastic medium, modified to include the mass and momentum sources, are solved for the fundamental monochromatic singularities. The oscillatory point source (sourcelet) and force (forcelet) and their higher‐order poles are directly applied to solve the problems of a rigid sphere embedded in an infinite medium in various modes of vibration. These closed‐form solutions include the torqued and pulsating sphere, as well as the sphere in rectilinear oscillation. The sourcelet, being radially symmetric, emits only dilatational waves and represents the total volume of solid and fluid supplied. The forcelet, in addition to the two compressional waves, induces a rotational wave and is applied to the solid and fluid constituents in proportion to the porosity.

Propagation of finite amplitude sound waves radiated from a pulsating sphere

The propagation of weakly non-linear acoustic waves is studied, which are radiated from a harmonically pulsating sphere in an inviscid perfect gas. A representation of the solution is presented for a far field equation of the first order, which is closely related to the solution obtained by the method of renormalization. The applicability of the method to the present problem is proved within the first order approximation. The asymptotic solution in the far field is obtained up to the second order by using the method of renormalization. A uniformly valid solution is, to the second approximation, constructed by matching the near field solution with the far field solution. Also investigated is the effect of dimensionless parameters and the second order correction on the acoustic shock formation distances and the non-linear distortion of waveforms.

Models for representing the acoustic radiation impedance of the mouth

The acoustic radiation impedance of the mouth is important in the production of speech in that it affects the amount of acoustic power radiated away from the mouth and it influences the acoustic response of the excited vocal tract. Five existing methods of modeling the acoustic radiation impedance of the mouth are: 1) the pulsating sphere, 2) the circular piston in an infinite baffle, 3) the circular piston in a spherical baffle, 4) an approximation by Fant, and 5) an approximation by Flanagan. A comparison has been carried out between the acoustic impedances as a function of frequency predicted by these models for mouth aperture areas of 0.9 and 5.0 cm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . The basis for comparison was the piston-in-a-spherical-baffle model. This model was chosen as being the most physically representative one which is computationally tractable, although it is by far the most computationally difficult of the models considered. Special precautions were needed to prevent errors due to arithmetic overflow. On the basis of the comparison, a new approximation is proposed. This new model is based on the piston-in-an-infinite-baffle model, but includes more terms than the Flanagan model, resulting in increased accuracy for higher frequencies and larger mouth apertures.

Dynamics of a seismoemitter in the form of a pulsating cylinder buried in a horizontal half-space

Four theoretical models are examined for their use as vibratory sources acting in an unlimmited elastic medium and at depth in an elastic half-space. The spherical models of type I and II are utilized extensively to estimate the wave fields being excited and to describe the dynamics of vibratory sources of the oscillating and thrust type, respectively, interacting with a medium. Model III is of the pulsating sphere type and is simpler and preferable over models I and II. Model IV of the pulsating cylinder type can be of interest in the development of an underground seismic source and in an estimate of efficiency. The expediency of its practical realizaatin is dictated by technical factors that govern the efficiency of this model as compared with the others. These factors are presented.

An Integral Equation Method for Predicting Acoustic Emission within Enclosures

A method is presented for calculating the acoustic emission of a vibrating body within an enclosure whose surface has known absorption characteristics. It is based on a numerical solution of the Helmholtz integral equation. Solutions are given for the case of a pulsating sphere within a sphere, and good agreement with the exact analytical solution is reported. The method is of value for small and medium scale problems at lower frequencies, where traditional techniques are less reliable. It is also potentially less demanding computationally than finite element methods.

Electrodynamic loudspeaker for low and medium sound frequencies

An electrodynamic loudspeaker radiates sound in all directions and with a high output for a low and medium sound frequencies. It comprises two hemispherical movable diaphragms which are arranged on either side of a disc-shaped carrier part. The diaphragms are connected with the carrier part by flanges and constitute a pulsating sphere. In this sphere and connected in each case with one of the diaphragms are actuating units with permanent magnets and in each case, an oscillatory coil. Each oscillatory coil is connected to its respective diaphragm through stiff transition parts, preferably in the form of spherical shells. Openings are provided for an atmospheric pressure equalization between the interior and exterior of the breathing sphere.

Subwavelength monopole underwater sound radiator

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Active suspension piezoelectric polymer transducer

An electromechanical transducer comprising a radiating structure whose active element is formed by a polymer film placed between two electrodes. The invention provides a transducer in which a closure element having the exact shape of a spherical surface portion is connected to at least one active peripheral suspension which simulates the movements of a pulsating sphere portion completing the closure element.

Holographic reconstruction of odd‐shaped 3‐D sources

As previously described [J. D. Maynard, J. Acoust. Soc. Am. Suppl. 174, S37 (1983)], nearfield acoustical holography can reconstruct the surface velocity and pressure over a level surface in a separable coordinate system from measusrements of the radiated pressure field over a parallel level surface. This study investigates the feasability of reconstructing the surface pressure and normal velocity over surfaces which do not conform to a level surface. Such a reconstruction involves numerically determining a distribution of surface pressure and velocity which satisfies the surface Helmholtz integral equation and which is consistent with the Helmholtz integral equation for the pressure field measured on the level surface. Details of the algorithm and results of computer simulations for a uniformly pulsating sphere and a piston set in a sphere (measurements simulated over a planar surface) are presented. [Work supported by ONR and NASA.]

Acoustic intensity measurement error for complex sources

There has been much discussion in recent literature of the accuracy of the two-microphone method of acoustic intensity measurement. The errors introduced by the finite difference formulation for intensity have been a frequent topic of investigation. However, these treatments have been limited to ideal plane wave and point sources. This paper describes a study of finite difference approximation errors for more realistic, finite sized sources. Error expressions are developed for a pulsating sphere, an oscillating sphere, and a piston in an infinite baffle. For the spherical sources, the error expressions derived are identical to those of their ideal equivalents, the point monopole and dipole. In the case of the piston source, large errors are predicted for particular conditions. For nearfield measurements along the piston axis, intensity level measurement errors can approach infinity.

Motion of two pulsating spheres in an ideal incompressible fluid

The problem of the interaction of two pulsating spheres in an ideal incompressible fluid was first investigated in detail by Bjerknes [1]. However, his and subsequent studies on this subject [2–5] were restricted to the interaction forces between the spheres, whereas the law of their motion was not considered because of the much greater complexity of the corresponding problem. The aim of the present paper is to find an approximate analytic solution to the problem of the motion of two pulsating spheres in an ideal incompressible fluid filling the entire space exterior to the spheres under the assumption that the flow of the fluid is irrotational.

Response of viscoelastic fluids in extensional flow generated by a pulsating sphere

A simple analysis of the periodic extensional flow generated by a pulsating sphere in an infinite sea of viscoelastic fluid has been carried out. The general procedure is illustrated by two specific constitutive equations: the corotational Jeffreys fluid and the Oldroyd fluid model B. The response of these fluids is reflected in the temporal variation of the pressure on the surface of the sphere, with Reynolds and Deborah numbers and parameters of the constitutive equations as independent variables. For the case of pulsation with infinitesimal amplitude the fluid response is summarised in the form of pressure amplitude and phase lag versus Deborah number plots. The role of the pulsating flow in the characterisation of viscoelastic fluids and the extension of the procedure to other constitutive equations are briefly discussed.

The entrapped elasticity of a fluid‐loaded plate

It is basic knowledge that the wave equation for an acoustic fluid represents a force balance between an inertia and an elastic‐restoring component in much the same way a balance exists between the mass and spring forces of a simple oscillator. It is also commonly known that only the mass component contributes to sound radiation from a simple pulsating convex source like a pulsating sphere, cylinder, or circular piston in a baffle. (By simple source one usually means a source whose dimensions a, are small compared to the acoustic wavelength, that ka < 1.) This paper will explore the situations under which the elastic component of an acoustic field arise, and specifically, consider how both the mass and elastic components of the acoustic field contributes to the vibratory response of a fluid‐loaded plate.