Characteristic Wavenumbers Research Articles

Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains

Potential non-uniqueness of boundary representations of the Helmholtz equation underscores the importance of investigating continuous boundary-based and domain-based formulations, which is the main purpose of this work. Uniqueness properties of the solutions of boundary integral equations are reviewed. We analyze formulations for domain-based computation that are derived by the DtN method, which imposes a relation between the function and its normal derivative on an artificial boundary. The DtN formulation is shown to possess non-reflective boundary conditions and to give rise to exact (and thereby unique) solutions. In practical implementation the DtN map is often truncated. The truncated DtN operator fails to completely inhibit reflection of higher modes, resulting in loss of uniqueness at characteristic wave numbers of higher harmonics. However, simple expressions that determine a sufficient number of terms in the operator for unique solutions at any given wave number are derived. We prove that a local approximation of the boundary conditions restores uniqueness for all wave numbers, and derive its three-dimensional version.

Acoustic radiation from a shell with internal structures

A method is developed to compute frequency response and acoustic radiation of a complex shell. The axisymmetric geometry of the shell includes cylindrical, conical, and spherical segments stiffened by discrete rings and bulkheads. The shell is coupled to internal masses and elastic frames. Shell segments are treated by transfer matrices. Rings, bulkheads, frames, and concentrated masses are treated by impedances at junctions of segments. The shell is coupled to an external acoustic fluid treated by Green’s function and curved surface elements. A major issue facing the method’s treatment of the fluid would be lack of existence or uniqueness encountered in the uncoupled, external acoustic problem at characteristic wavenumbers. By using a simple spherical shell, without internal structures, this potential hindrance is shown not to arise. A fuller application of the method awaits subsequent results.

Stochastic Modeling of Seafloor Morphology: Inversion of Sea Beam Data for Second‐Order Statistics

At scale lengths less than 100 km or so, statistical descriptions of seafloor morphology can be usefully employed to characterize ridge crest processes, off‐ridge tectonics and vulcanism, sedimentation, and postdepositional transport. We seek to develop methods for the estimation of seafloor statistics that take into account the finite precision, resolution, and sampling obtained by actual echo sounding systems. In this initial paper we restrict our attention to the problem of recovering second‐order statistics from data sets collected by multibeam devices such as Sea Beam. The seafloor is modeled as a stationary, zero‐mean, Gaussian random field completely specified by its two‐point covariance function. We introduce an anisotropic two‐point covariance function that has five free parameters describing the amplitude, orientation, characteristic wave numbers, and Hausdorff (fractal) dimension of seafloor topography. We formulate the general forward problem relating this model to the statistics of an ideal multibeam echo sounder, in particular the along‐track autocovariance functions of individual beams and the cross‐covariance functions between beams of arbitrary separation. Using these second moments as data functionals, we then pose the inverse problem of estimating the seafloor parameters from realistic, noisy data sets with finite sampling and beamwidth, and we solve this inverse problem by an iterative, linearized, least squares method. The inversion method is applied to Sea Beam transit data from both the Pacific and Atlantic oceans. Sea Beam system noise stands out as a sharp spike on the along‐track autocovariance function and can be modeled as a white noise process whose amplitude generally increases with beam angle. The five parameters in our second‐order model can be estimated from the inversion of data sets comprising ∼100–200 km of track length. In general, the cross‐track wave number is the most poorly determined, although uncertainties in the assumed Sea Beam response may bias the values of the fractal dimension. Using the assumed beamwidth, the measured noise values, and the seafloor parameters recovered from the inversion, we generate Sea Beam “synthetics” whose statistical character can be directly compared with raw Sea Beam data. For most of the track segments we have processed thus far the synthetics are similar to the data. In the case of one Atlantic profile, however, the comparison clearly indicates the necessity of incorporating higher‐order statistics. The space domain procedures described in this paper can be extended for this purpose.

Two-sensor methods for the measurement of sound intensity and acoustic properties in ducts

A unified theoretical approach to the development of two-sensor methods is presented. It is shown that various methods developed in the last 10 years for sound intensity measurement and for the measurement of acoustic properties in ducts may be systematically derived from a general decomposition theory. In the decomposition theory, the incident and reflected wave auto and cross spectra are obtained from a set of decomposition equations using the measurement of the total acoustic pressure at two arbitrary points in a one-dimensional steady, random sound field. The application of the wave spectra to the measurement of sound intensity and acoustic properties follows directly. It is further shown that the decomposition theory predicts a set of characteristic wavenumbers at which two-sensor methods fail to yield meaningful data. Experimental data are presented showing the application of the decomposition theory to acoustic property determination, sound intensity measurement, and the estimation of sound pressure and particle velocity in a duct.

Estimation of total aromatics in kerosene fractions by infrared spectroscopy—II

AbstractA general equation for the quick and accurate estimation of total aromatics in straight run kerosene fractions is derived using i.r. absorbances at characteristic wavenumbers from aromatic concentrates of kerosene of different crude oils. The accuracy of the method is found to be ±5%. When compared with the FIA method the deviation was found to be within 5.5%.

Intramolecular coupling of vibrational modes and the assignments of the partially-deuterated methyl, silyl and germyl halides

The fundamental vibrational wavenumbers of the mono- and dideuterated methyl, silyl and germyl halides have been determined by transferring the force fields established from the experimental wavenumbers of the normal and perdeuterated species. Reasonable agreement has been found between the calculated and the experimental wavenumbers of the methyl halides. The normalized potential energy distributions have been interpreted in terms of characteristic group vibrational wavenumbers and intramolecular coupling between some of the internal modes.

The use of CHIEF to obtain unique solutions for acoustic radiation using boundary‐integral equations

This report is concerned with the problem of obtaining a unique solution for radiation problems at characteristic wavenumbers when a boundary‐integral equation formulation is used. It is shown that the combined Helmholtz integral equation formulation (CHIEF) method works even when numerous interior points lie exactly on nodal surfaces of a mode corresponding to an eigenfrequency of the related interior problem, when at least one interior point does not lie on a nodal surface. The use of CHIEF is illustrated with numerical radiation experiments for the sphere and the finite cylinder. CHIEF is compared to the Helmholtz gradient formulation (HGF) for circumventing nonuniqueness and is found to yield a more accurate solution, in at least one published example. A procedure is used to indicate the solution error when using integral equation methods, with or without a technique to circumvent nonuniqueness. This procedure uses the velocity potential of an interior point as an indicator of solution error. In all cases considered, the interior potential correctly indicated a good or bad solution, whereas the matrix condition number falsely indicated a bad solution in several instances. [Work supported in part by ONR.]

The use of CHIEF to obtain unique solutions for acoustic radiation using boundary integral equations

This article is concerned with the problem of obtaining a unique solution for radiation and scattering problems at characteristic wavenumbers when a boundary integral equation formulation is used. It is shown that the combined Helmholtz integral equation formulation (CHIEF) method works even when numerous interior points lie exactly on nodal surfaces of a mode corresponding to an eigenfrequency of the related interior problem, when at least one interior point does not lie on a nodal surface. CHIEF is compared to the Helmholtz gradient formulation (HGF) for circumventing nonuniqueness and is found to yield a more accurate solution. A procedure is used to indicate the solution error when using integral equation methods, with or without a technique to circumvent nonuniqueness. This procedure uses the velocity potential of an interior point as an indicator of solution error. In all cases considered, the interior potential correctly indicated a good or bad solution; whereas the matrix condition number falsely indicated a bad solution in several instances.

Magnetic field generation in a relativistic plasma with high frequency waves

Magnetic field generation in a relativistic plasma in the presence of Langmuir or electromagnetic waves is studied theoretically. The generation is shown to occur in the form of the three simplest instabilities: anisotropic, magnetic-modulational and magnetic-transformational, or their combinations. Characteristic wave numbers and growth rates of the instabilities mentioned are found as function of the degree of plasma relativism. The growth rates are shown to attain their maxima at relativistic temperatures.

Reply to ’’Comments on ’Multiple‐obstacle acoustic scattering in the simple source formulation with accuracy checks’ ’’

Statements made regarding the failure of the simple source formulation at characteristic wavenumbers are clarified, and numerical results obtained from this particular implementation of the simple source formulation are presented. A possible explanation is offered regarding the discrepancies in results at high wavenumbers.

Comments on ’’Multiple-obstacle acoustic scattering in the simple source formulation with accuracy checks’’ [J. Acoust. Soc. Am. 66, 1536–1541 (1979)

Sorensen implies that in scattering from multiple objects, no failure of the simple source integral method occurs at the characteristic wavenumbers. Comparisons of the simple source integral method with a multiple body eigenfunction expansion solution indicates numerical difficulty with the integral solution near a characteristic wavenumber of one of the objects. Calculations which differ from the results in Sorensen’s paper at a high wavenumber are presented.

Characteristic wave numbers for simple source formulations of acoustic radiation and diffraction problems

It is shown that the characteristic wave numbers for which the simple source formulation of acoustic diffraction problems fails can be obtained directly by use of the model’s integral equation. Characteristic wave numbers for diffraction by a rigid cylinder and by a rigid sphere are determined.

Review of Integral Equation Methods in Sound Radiation and Scattering

Integral-equation methods are described for calculating the entire sound-pressure field when either the velocity distribution or, alternatively, the sound-pressure distribution is specified on an arbitrary closed surface. The theory is based on determining equivalent surface layers with either monopoles and/or dipoles. An indefinite number of appropriate integral equations are derived for the surface monopole and/or dipole density for each case and each boundary condition. Simple techniques are described for approximating each of the integral equations by a linear matrix equation with finite elements and for the numerical solution of the matrix equation. For every surface shape, and any specified ka, there exists an infinite sequence of complex characteristic numbers γi and associated sets of surface functions (sound pressure pi, normal gradient qi, dipole surface density μi, and monopole surface density σi) which are linear homogeneous functions of each other, and which provide a complete orthonormal basis for the solation to arbitrary Dirichlet or Neumann problems. In particular, for every closed surface, there exist two infinite sequences of characteristic wavenumbers such that, at one or the other of these wavenumbers, and for particular boundary conditions which are stated, each of the more common integral equations has an infinite or indeterminant solution. Special methods are described to eliminate the indeterminacy in such cases.

Variational Techniques in Acoustic Radiation Theory

The possibility of applying variational methods to the acoustic radiation problem for an arbitrary body with a given velocity distribution is investigated. It is found that since the stationary quantity involved is, in general, neither a maximum nor minimum, the variationally determined result may not always be accurate. Sufficient conditions for achieving accurate results are determined. Since there is no way of knowing a priori, whether for any given problem these conditions will be satisfied, methods must be devised which ensure and/or give a measure of the reliability of a given calculation. Several possible approaches are indicated, some which yield a quantitative measure of the accuracy of the results. It is found that these variational methods, although based on the surface Helmholtz integral formulation, are applicable even at characteristic wave numbers and, hence, may provide an attractive alternative to the CHIEF program. A simple example illustrating the use of one of these techniques is presented.

Improved Integral Formulation for Acoustic Radiation Problems

Three different integral formulations have been used as a basis for obtaining approximate solutions of the exterior steady-state acoustic radiation problem for an arbitrary surface whose normal velocity is specified: (1) the simple-source formulation, adapted from potential theory; (2) the surface Helmholtz integral formulation, based on the integral expression for pressure in the field in terms of surface pressure and normal velocity; and (3) the interior Helmholtz integral formulation, in which the surface pressure is determined by making a certain integral vanish for all points interior to the radiating surface. For certain characteristic wavenumbers, it is shown that no solution of the simple-source formulation exists in general and that there is no unique solution of the surface Helmholtz integral formulation. The interior Helmholtz integral formulation is subject to similar difficulties and has undesirable computational characteristics. A Combined Helmholtz Integral Equation Formulation (CHIEF) that overcomes the deficiencies of the first two methods and the undesirable computational characteristics of the third, is described. The significant improvement over the previous three methods, which is accomplished through the use of CHIEF, is illustrated by numerical examples involving spheres, finite cylinders, cubes, and a steerable array mounted in two different boxlike structures.