Index of content:
Volume 97, Issue 1, January 1995

Elastic wave propagation in transversely isotropic media. II. The generalized Rayleigh function and an integral representation for the transducer field. Theory
View Description Hide DescriptionThe basis for the derivation of elastodynamic holography for arbitrarily oriented transversely isotropic materials, given in Part I of this presentation [M. Spies, J. Acoust. Soc. Am. 96, 1144–1157 (1994)], is Huygens’ principle and a resulting relationship which links the spatial spectra of surface traction and displacement distribution. Similar to deriving the plane‐wave spectral decomposition of elastic wavefields for given displacement, this relationship yields a corresponding decomposition for the case of given surface traction, which can be applied to model the problem of transducer radiation as significant to nondestructive testing. For a physically reasonable distribution of surface traction within the transducer aperture, an integral representation for the resulting transducer field is obtained. The main problem in this approach is the inversion of the 2‐D space‐time spectral representation of Green’s triadic function. A specifically interesting result of this inversion is the Rayleigh function for arbitrarily oriented transversely isotropic media, which characterizes the propagation of the respective Rayleigh wavefronts. Since the resulting expressions are explicitly dependent on the orientation of the material’s axis of rotational symmetry, their numerical evaluation will be much more complicated than in the isotropic case.

Causal theories and data for acoustic attenuation obeying a frequency power law
View Description Hide DescriptionThis study compares causal theories, based on Kramers–Kronig relations, fractional calculus, and on those derived from new time domain causal relationships, to diverse data. All these theories are based on the assumptions that the functional form of the attenuation persists beyond the measurement range and that attenuation is much smaller than the wave number. The data, for lossy media with attenuation having a power‐law frequency dependence with an exponent y, include cases for both liquids and solids, ranging from acoustic to ultrasound frequencies. Data are in closer correspondence with the new theory which predicts decreasing dispersion as the power exponent y approaches zero or an even integer. Experimental results and supporting evidence show that the classical case of frequency‐squared attenuation is dispersionless. An approximate nearly local Kramers–Kronig theory is in agreement with the time causal theory when the exponent is close to one, but deviates for other values. The comprehensive time causal theory is shown to be equivalent to two other theories derived from exact Kramers–Kronig relations and from fractional calculus and it covers the y odd integer cases which are missing or incomplete in these approaches. Attenuation–dispersion relations are presented in two forms: one for a general frequency range and another for a finite range. It is demonstrated that complete velocity dispersion (within a signal bandwidth) can be predicted from knowledge of the attenuation data and velocity at a single frequency including the velocity at either zero frequency (y≳1) and or at a high‐frequency limit (0<y<1). In addition, attenuation data can be predicted from velocity data.

Decomposition theorem of sound intensity with application
View Description Hide DescriptionThe sound intensity I is rotational and may be decomposed into an irrotational component expressed by a gradient of potential function and a rotational component expressed by a curl of potential vector. It is proven that the decomposition is unique. It is shown that the rotational and irrotational components are orthogonal in the form of the inner product in the sound field and that in the far field the constant potential surfaces of the potential function are spherical surfaces. It is the irrotational component of sound intensity that transfers sound energy into the far field. The sound energies transferred by the rotational components cancel out each other. The effect of the rotational component on the results measured by the two‐microphone technique is discussed.

Leaky waves on weakly curved scatterers. II. Convolution formulation for two‐dimensional high‐frequency scattering
View Description Hide DescriptionA simple high‐frequency approximation is developed for leaky wave contributions to two‐dimensional scattering by curved elasticsurfaces. Following Bertoni and Tamir [Appl. Phys. 2, 157–172 (1973)] the method relies on general features of the Laurent expansion of the plane surfacereflection coefficientR(k _{ x }) about the leaky wave pole of interest at the complex surface wave number k _{ x }=k _{ l }+iα. The formulation uses the real part k _{ l } and radiation damping rate α for leaky waves on the curved elasticsurface of interest rather than an analysis of the response of any specific class of structures such as thin shells. The high frequency limit of the complex coupling coefficient G _{ l } [see, e.g., P. L. Marston, J. Acoust. Soc. Am. 83, 25–37 (1988)] is recovered for right circular cylinders and the physical origin of the π/4 phase shift is discussed. An O(kh)^{−1} phase correction important for empty thin shells of thickness h is obtained in agreement with results from other approaches. The importance of the Fresnel width of the coupling region is illustrated by consideration of a cylinder with an ideal coating having an abrupt edge. The leaky wave contribution becomes proportional to a Fresnel integral having a complex argument. The integral manifests the degree to which launching of a leaky wave can be considered to be a local process. The product of α and the Fresnel width is an important parameter. The detachment of the ray to the far field is taken to be separated from its launching by more than the Fresnel width. Leaky wave contributions to scattering by surfaces of variable curvature are approximated and applications for ultrasonic beams are noted.

Modal structures for axial wave propagation in a continuously twisted structurally chiral medium
View Description Hide DescriptionThe stiffness of a continuously twisted structurally chiral medium (CTSCM) varies helicoidally about the axis of spirality. The structures of axial propagation modes in a CTSCM having a tetragonal reference symmetry have been investigated in this paper. Salient features relating to the management of the phase and the vibration ellipse have been brought out, and may be of assistance in conceptualizing novel devices with plane‐stratified geometries.

An approximate integral method for calculating the diffracted component from multiple two‐dimensional objects
View Description Hide DescriptionA method for determining the diffracted or scattered components due to the interaction of a wave with multiple two‐dimensional objects is given. The method presented uses approximate boundary conditions to simplify the numerically cumbersome set of integral equations associated with nonapproximate analytic solutions of the Helmholtz integral. The approximate boundary conditions, defined here, are shown to decouple the N by N set of simultaneous equations. The approximate boundary conditions are used to calculate the insertion loss for various sizes of two‐dimensional objects. The predictions are compared to measured results and limitations are discussed.

The iterative time reversal process: Analysis of the convergence
View Description Hide DescriptionThe efficiency of a time reversal acoustic mirror to focus on a reflective target through an inhomogeneous media has been demonstrated. In a multitarget media, the ability of such a mirror to work in an iterative mode in order to focus selectively on the strongest target was shown [C. Prada, F. Wu, and M. Fink, J. Acoust. Soc. Am. 90, 1119 (1991)]. The theory of how the iterative time reversal process is built is based on a matrix formalism and treats the array of Ltransducers in a given medium as a linear system of L inputs/L outputs. The system is characterized at each frequency by its transfer matrix K and the time reversal iterative process is then described by a time reversal operator K*K. Because of the reciprocity principle, this operator is Hermitian. The following result is shown: If the scattering medium is a set of well resolved targets of different reflectivities then each eigenvector of the operator K*K with nonzero eigenvalue corresponds to one of the targets in the set and provides the optimum phase law to focus on it. Furthermore, the eigenvalue is proportional to the reflectivity of the target. In particular, the ‘‘brightest’’ target is associated to the eigenvector of greatest eigenvalue so that the iterative time reversal process leads to a wave focusing on this target. This analysis is illustrated by numerical and experimental results.

Wave scattering in a rough elastic layer adjoining a fluid half‐space
View Description Hide DescriptionWave propagation and scattering are considered in a medium consisting of a slightly rough elastic layer adjoining a fluid half‐space. The solution is obtained for the mean wave potentials in both media, due to multiple scattering within and reradiation from the layer. This is found by deriving effective transmission and reflection coefficients for each irregular boundary, and then showing that the problem is equivalent to the deterministic one for a plane‐sided layer with these effective coefficients. The solution exhibits the dependence upon the variance and correlation length of each boundary explicitly.

A direct boundary element method for acoustic radiation and scattering from mixed regular and thin bodies
View Description Hide DescriptionIn this paper, the direct boundary element method is applied to acoustic radiation and scattering from mixed regular and thin bodies. A typical application problem involves a regular body with thin fins either attached to or in the neighborhood of the body. It is shown that the mixed‐body integral formulation requires interactions between the conventional Helmholtz integral equation and the hypersingular thin‐body integral equation. If at least one regular body is involved in a mixed‐body environment, the direct integral formulation possesses the well‐known nonuniqueness difficulty. This nonuniqueness difficulty is overcome by either the CHIEF method or the Burton and Miller method. Since both the thin‐body integral equation and the Burton and Miller equation contain a hypersingular integral, a regularized form recently derived by Krishnasamy et al. [ASME J. Appl. Mech. 57, 404–414 (1990)] is adopted in this paper. Numerical results show excellent agreement with a multidomain boundary element solution and a point‐source solution.

Modal superposition in the time domain: Theory and experimental results
View Description Hide DescriptionTime domain expressions of the modal superposition are derived. It is shown that modal superposition in the time domain is a valuable tool to describe the stationary and nonstationary behavior of acoustic systems, especially in active noise control applications. The mathematical expressions show physical insight in the fundamental differences between stationary and decaying sound fields. Simulations and measurements based on the time domain expressions have been carried out for the special case of a lightly damped rectangular cavity. These simulations and measurements show a good agreement.

Phase and energy velocities of cylindrically crested guided waves
View Description Hide DescriptionThe excitation and propagation of axisymmetric, cylindrically crested, azimuthally polarized elastic waves in a homogeneous isotropic plate is analyzed. As is well known, the frequency spectrum of the azimuthally polarized waves is identical to the analogous (straight crested) horizontal shear (SH) modes. It is shown, however, that the phase and energyvelocitydispersion curves of the azimuthally polarized waves can differ significantly from their SH mode analogs. In addition, it is shown that both the phase and energyvelocitydispersion curves of the azimuthally polarized waves are functions of radial position. Finally, it is demonstrated analytically that at observation distances large in comparison to the ‘‘far‐field wavelength’’ the phase velocity curves of the azimuthally polarized waves converge to the corresponding SH curves and that the energyvelocity curves of the cylindrically crested azimuthally polarized waves converge to the group velocitydispersion curves of the SH modes. The convergence is, in both cases, nonuniform with frequency and is shown to depend on the ratio of observation distance to the far‐field wavelength of the mode.

A fast Fourier transformation algorithm for the Kirchhoff integral formulation
View Description Hide DescriptionA fast Fourier transformation‐based Kirchhoff integral formulation is developed for calculating sound radiation from finite objects. This formulation is applicable to both transient and steady‐state radiation in both near and far fields. The radiated acoustic pressure is solved in the frequency domain first and then converted to the time domain by taking a discrete inverse Fourier transformation. Since the spatial domain is decoupled from the temporal domain in numerical computations, and since commercially available fast Fourier transformationsoftware packages are utilized directly, numerical computations are highly efficient and accurate. Another advantage of using this formulation is that it always yields unique solutions for sound radiation from an object vibrating at a constant frequency.

Measuring the sound power of a moving source
View Description Hide DescriptionThe sound power of a moving source was measured with a vector sound‐intensity probe, for the simple case of an axisymmetric source moving along a known straight path. The probe tracks the position of the source and, at the same time, measures the component of intensity normal to a hypothetical measurement cylinder enclosing the source, the axis of the cylinder lying along the path of the source. The sound power of the axisymmetric source is determined by integrating sound intensity over the surface of the cylinder. The contributions of the ends of the cylinder can be neglected if the cylinder is long enough. The measuredsound power of the moving source was compared with the sound power of the source, (a) estimated from the applied voltage and (b) measured with the source stationary, and it was found to be greater than (a) by 14% and greater than (b) by 18% or 0.7 dB. The difference is believed to occur because the source is not completely axisymmetric due to reflections in the tank. The method can be applied to a nonaxisymmetric source using a circular array of sound‐intensity probes instead of a single probe.

Supersonic acoustic intensity
View Description Hide DescriptionA new acoustic quantity called the supersonic acoustic intensity vector is defined in this paper for application to measurements on plate and cylinderlike structures. As the name implies, the supersonic intensity is composed only of wave components which radiate to the far field (supersonic), with the nonradiating (subsonic) components eliminated. The normal component of this vector, measured in the extreme near field or on the surface of the structure, provides an accurate tool for locating regions (‘‘hot spots’’) on the structure which radiate to the farfield. Furthermore, the supersonic intensity provides an accurate quantification of these source regions, providing a ranking of the strength of the identified source regions as a function of frequency. This identification and ranking provides a powerful new tool in the understanding and control of radiated noise.

Response on a cylindrical surface due to an incident plane wave. I. Pressure‐release surface
View Description Hide DescriptionBecause the Debye approximation to the Hankel function is not applicable at points in or near the shadow zone of a cylinder insonified by an incident plane wave, it cannot be used to obtain asymptotic representations of the surface response at arbitrary locations. This paper proposes the use of an Airy function approximation to the Hankel function, as is done in electromagnetic problems. Solutions to the latter are sometimes approximated in terms of an integral of the Airy function called the Fock function, which is calculated by series expansions. The departure here is the evaluation of the integral by a saddle point approximation, and it is shown that this approximation to the Fock function agrees well with tabulated values. The method is applied to calculating the radial velocity on the surface of a pressure‐release cylinder, and the approximate solution is shown to agree well with the modal solution, for ka above 2. The relevance to structural acoustics is that it may be possible to estimate the surface response of a structural shell at arbitrary locations by using the saddle point of the Airy function integral to describe the admittance properties of a shell and the acoustic medium.

Response on a cylindrical surface due to an incident plane wave. II. Uniformly point‐reacting surface
View Description Hide DescriptionAsymptotic estimates of the response in the radial direction, at arbitrary locations in the shadow and transition zones, of a cylindrical surface due to an incident plane wave can be obtained through an integral that includes an Airy function representation of the Hankel function. A previous paper [J. Acoust. Soc. Am. 97, 128–133 (1995)] showed that, for a pressure‐release surface, the relevant integral could be estimated through a saddle point evaluation. The motivation was the hope that, when applied to the response of an elastic shell, the saddle point would have significance in quantifying the shell admittance and the surface acoustic admittance. The present paper applies the method to the intermediate case of a point‐reacting surface, which introduces the complication of a creeping wave that must be accounted for in the saddle point evaluation. Results are presented for the radial velocity of a surface having a masslike surface admittance equivalent to the mass per unit wall area of a steel shell with a wall thickness that is 1% of the radius. The ambient acoustic medium is water. Good agreement with the modal solution is obtained for ka greater than 2.

Sound propagation through atmospheric turbulence: Multifractal phase fluctuations
View Description Hide DescriptionPhase fluctuations of sound propagating through turbulent air show frequency spectra following an F ^{−β} power law (β=const.) over several decades. This is consistent with a simple fractal description involving the dimension D=(5−β)/2. However, many similar natural phenomena have been shown to be multifractal. Universal multifractals are characterized by two additional parameters: The Lévy index 0≤α≤2 for the type of multifractal and the codimension 0≤C _{1}≤1 for the intermittence. These parameters are a statistical measure of the nonlinear dynamics.Analysis of experimental data obtained at 5 kHz, range ≊4 m and wind speed ≊12 m/s is reported here and results indicate that phase fluctuations are ‘‘hard multifractal’’ (α≳1). The actual estimate is near the limiting value α=2, which is consistent with Kolmogorov’s lognormal model for turbulence.

Calculation of average turbulence effects on sound propagation based on the fast field program formulation
View Description Hide DescriptionDaigle has published a series of papers in which he has applied the turbulentscattering theories of Chernov and Karavainikov to sound propagation over hard and finite impedance grounds. In these papers, Daigle has introduced the decorrelation in phase and amplitude due to turbulence along the direct and reflected path into the spherical wave reflection analysis for a nonrefracting atmosphere. Here the phase and amplitude decorrelation terms have been incorporated into the evaluation of the spectral integral of a fast field program for propagation in a refracting atmosphere. Although the calculation involves two significant approximations, it reproduces Daigle’s results for homogenous atmospheres and compares well with the upward refraction measurement of Parkin and Scholes and with measurements taken under a variety of refractive conditions at Bondville, Illinois by the U.S. Army Construction Engineering Research Laboratory.

The effect of realistic ground impedance on the accuracy of ray tracing
View Description Hide DescriptionThe heuristic model of sound propagation in the atmosphere has been developed by L’Espérance et al. [Appl. Acoust. 37, 111–139 (1992)] to provide rapid calculation of sound‐pressure levels. This model uses an analytic ray trace method to calculate ray paths from the source to the receiver. The effects of finite ground impedance are treated by calculating the spherical wave reflection factor as if the atmosphere was homogeneous. The actual reflection angles and times of flight are used in this formulation. Since the speed of sound varies linearly with height, no focusing factors are calculated. The heuristic model has been compared to full wave calculations and is found to agree well at long ranges. In this paper the agreement between the heuristic model and the fast field program is investigated in detail. Criteria for the accuracy of ray tracing are discussed. It is demonstrated that the heuristic model produces realistic predictions even though certain criteria are violated. The finite ground impedance is shown to remove those rays for which the ray tracing criteria do not hold.

The effect of supersonic aircraft speed on the penetration of sonic boom noise into the ocean
View Description Hide DescriptionIn 1968 Sawyers presented a theory which predicts the acoustic pressure underwater due to a supersonic aircraft’s sonic boom in the air. The theory has since been validated in laboratory experiments. In the present paper Sawyers’ theory is utilized to predict the effect of a supersonic aircraft’s speed on the penetration of the sonic boom into the water. By taking into account the variation in a sonic boom’s duration as a function of the aircraftMach number, it is shown that higher aircraft speeds are associated with higher acoustic pressures in the water. For fixed depths of 10 m or less the peak SPL varies less than 6 dB over a wide range of Mach numbers. For greater depths, 100 m for example, increased Mach numbers may increase the SPL by 15 dB or more. The actual levels are always diminished for deeper depths. These observations may be important for evaluating the possible effects of sonic boomnoise on marine mammals.