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The existing non-paraxial expression of audio sounds generated by a parametric array loudspeaker (pal) is hard to calculate due to the fivefold integral in it. A rigorous solution of the Westervelt equation under the quasilinear approximation is developed in this paper for circular PALs by using the spherical harmonics expansion, which simplifies the expression into a series of threefold summations with uncoupled angular and radial components. The angular component is determined by Legendre polynomials and the radial one is an integral involving spherical Bessel functions, which converge rapidly. Compared to the direct integration over the whole space, the spherical expansion is rigorous, exact, and can be calculated efficiently. The simulations show the proposed expression can obtain the same accurate results with a speed of at least 15 times faster than the existing one.

Previous work on scattering by Bessel beams shows that expansion of incident sound fields in term of these beams has application to scattering [P. L. Marston, J. Acoust. Soc. Am. 122, 247–252 (2007)]. In this work, an expression for the expansion coefficients of propagating, axisymmetric, sound fields are derived. In this paper, this expression is applied to a linear focused axisymmetric sound field and is expanded in terms of Bessel beam components. This is done for focused beams radiated from a spherical cap source. A physical optics model is applied to sound propagation close to the source to facilitate the calculation of the Bessel beam expansion coefficients. This type of model is useful for focused scattering [P. L. Marston and D. S. Langley, J. Acoust. Soc. Am. 73, 1464–1475 (1983)]. Once the expansion coefficients are found, the sound field can be evaluated by superposition. The model agrees approximately with Chen, Schwarz, and Parker [J. Acoust. Soc. Am. 94, 2979–2991 (1993)] and O'Neil [J. Acoust. Soc. Am. 21, 516–526 (1949)] on axis and with direct integration of a Kirchhoff integral both on and off axis. This type of expansion will have applications to scattering problems.Previous work on scattering by Bessel beams shows that expansion of incident sound fields in term of these beams has application to scattering [P. L. Marston, J. Acoust. Soc. Am. 122, 247–252 (2007)]. In this work, an expression for the expansion coefficients of propagating, axisymmetric, sound fields are derived. In this paper, this expression is applied to a linear focused axisymmetric sound field and is expanded in terms of Bessel beam components. This is done for focused beams radiated from a spherical cap source. A physical optics model is applied to sound propagation close to the source to facilitate the calculation of the Bessel beam expansion coefficients. This type of model is useful for focused scattering [P. L. Marston and D. S. Langley, J. Acoust. Soc. Am. 73, 1464–1475 (1983)]. Once the expansion coefficients are found, the sound field can be evaluated by superposition. The model agrees approximately with Chen, Schwarz, and Parker [J. Acoust. Soc. Am. 94, 2979–2991 (1993)] and O'Neil [J. Acou...

A scheme for evaluating the effective quasistatic speed of sound c in two- and three-dimensional periodic materials is reported. The approach uses a monodromy-matrix operator to enable direct integration in one of the coordinates and exponentially fast convergence in others. As a result, the solution for c has a more closed form than previous formulas. It significantly improves the efficiency and accuracy of evaluating c for high-contrast composites as demonstrated by a two-dimensional scalar-wave example with extreme behavior.

Previous work on scattering by Bessel beams shows that expansion of incident sound fields in term of these beams has application to scattering [P. L. Marston, J. Acoust. Soc. Am. 122, 247–252 (2007)]. In this work, we expand a focused cylindrically symmetric sound field in terms of Bessel beam components. In particular, the authors are interested in a focused beam radiated from a spherical cap source. A physical optics model is applied to sound propagation close to the source to facilitate the calculation of the Bessel beam expansion coefficients. This type of model is useful for focused scattering [P. L. Marston and D. S. Langley, J. Acoust. Soc. Am. 73, 1464–1475 (1983)]. Once the expansion coefficients are found the sound field can be evaluated by the appropriate superposition. The model gives results in agreement with O’Neil [H. T. O’Neil, J. Acoust. Soc. Am. 21, 516–526 (1949)] and Chen [X. Chen, K. Q. Schwarz and K. J. Parker, J. Acoust. Soc. Am. 94, 2979–2991 (1993)]. Comparison is also made with direct integration of a Kirchhoff integral for both on and off axis pressures with good agreement. [Work supported by ONR.]

A theory for acoustic radiation force on a sphere in soft tissue was developed for arbitrary incident compressional wave fields [Ilinskii et al., Proc. Meet. Acoust. 19, 045004 (2013)]. This theory includes two contributions to the radiation force. The first depends only on the incident and scattered compressional waves, whereas the second depends on the scattered shear waves as well. Each contribution in turn has two parts, one due to direct integration of the time-averaged Piola-Kirchhoff stress tensor over the surface of the sphere, and the other due to the irrotational component of the body force on the sphere. While both parts are known analytically for the compressional waves, only the first part has been obtained analytically for the contribution involving shear waves. The irrotational portion associated with shear waves and its effect on the total radiation force is the subject of this presentation. The analysis is conducted via Helmholtz decomposition of the body force associated with shear waves...

A fundamental problem in structural acoustics is the scattering or radiation of sound from a fluid‐loaded plate with discontinuities. In calculating the response of the plate, as well as the radiated pressure field, the traditional approach has been to evaluate the transform representation of the response or radiated pressure solution using a contour integral. The contour integral evaluation of these representations is numerically inefficient when compared to an approach that utilizes an FFT algorithm to compute the inverse transform by a direct integration along the real wave‐number axis. The latter approach however can run into problems because of the presence of singularities along the real wave‐number axis. In previous work by other investigators, this problem has been overcome by artificially including structural damping in their formulations, which in some instances may obscure the true effects of the radiation damping. In the present work, a direct integration approach is used after first analytica...

Experimental modal analysis can be used to measure the modal parameters and vibrational energy in structures. This presentation will compare biases in the total vibration energy at high frequencies for a circular and rectangular aluminum plate that are freely supported. Both rectangular and cylindrical coordinates were used to create the set of excitation points for the circular plate, while only rectangular coordinates were used for the rectangular plate. Singular value decomposition was then used to obtain the natural frequencies and mode shapes in each of these plates. The obtained modes were used to synthesize the vibration response and to compute the vibration energy. Biases were found at high frequency using a direct integration method to compute the vibration energy. The dependencies of the bias on the plate geometry and set of excitation points will be discussed.

The present formulation differs from those appearing in earlier literature in that the boundary conditions are explicitly incorporated, so that not all admissible trial functions need to satisfy them. If the class of considered trial functions is restricted so that each is a finite sum of simple functions with arbitrary coefficients, then the answer selected by the variational principle attempts to satisfy the boundary conditions plus the partial differential equations imposed in the interior simultaneously, but only achieves a trade‐off. An example is a waveguide of variable cross section, with one end open and the other closed. The Rayleigh integral solution for sound radiated by a baffled aperture with an arbitrary distribution of velocity amplitude on the aperture yields the open‐end boundary conditions. Individual trial functions are combinations of cross‐sectional mode‐shape functions with axially dependent coefficients. Numerical instabilities encountered in previous direct integration of the coupl...

Many structures of interest contain regularly spaced elements that leave their imprint on the acoustic scattered field. For periodic arrangements, even a few elements may imprint on the scattered field the signature of the periodic structure Floquet modes that characterize an infinite array [L. Carin and L. B. Felsen, J. Acoust. Soc. Am. 90, 2355(A) (1991)]. Explored here, for various collections of flat, soft, and rigid strip scatterers, is how departures from periodicity due to array truncation, as well as displacement, change in size, etc., of individual and groups of elements affects the time‐harmonic and short pulse plane wave scattered response. Reference solutions obtained numerically by direct integration of a (spectral domain)–(moment method) procedure are interpreted phenomenologically through asymptotic reductions to yield truncation edge diffracted contributions, local Floquet modes from the overall bulk of the array, and individual scatterings from displaced elements. The contracting behavior...

This paper concerns the circumferential propagation of shear horizontally polarized waves in hollow, infinite, elastic cylinders. The exact solution of this classical problem expressed in terms of Bessel and Hankel functions gives rise to serious numerical difficulties when the circumferential wavenumber takes large values. A different method is developed to treat such cases. The method is based on a direct integration of the problem cast in the Sturm–Liouville form. Calculations performed with the Sturm–Liouville solver of Bailey et al. [ACM Trans. Math. Software 4, 193–208 (1978)] allow accurate determinations of SH modes for a wide range of geometrical parameters, reduced frequencies, and circumferential wavenumbers. This direct approach may be extended to other modal problems which are amenable to a standard Sturm–Liouville formulation.

The driving‐point impedance and the force and moment transmission to the terminations of a simply clamped beam have been determined when the beam is excited at its midpoint by an alternating force, by a moment, and by a force and moment of like phase. The influence of rotary inertia, shear displacement, and internal damping upon the response of the beam has been considered in detail. When the beam is excited by an applied moment, the mean level of the force exerted upon the beam terminations is found to increase essentially in proportion to the square root of frequency—at frequencies above the first resonant frequency of the beam. In consequence, the transmitted force attains values at high frequencies that are significantly greater than the constant value observed at low frequencies. When the beam is excited simultaneously by an in‐phase force and moment, the force exerted upon the beam terminations can likewise take much larger values at high frequencies than at low; moreover, this transmitted force can greatly exceed in magnitude the force that is exerted in the absence of the applied moment.

In this study, we first present a two‐dimensional theory of high‐frequency vibrations of piezoelectric crystal surfaces, coated with thin electrodes. The theory is systematically and consistently extracted using a method of direct integration [cf. M. C. Dokmeci, J. Acoust. Soc. Am. 54, 292 (1973) and L. Al, Nuovo Cimento 7, 449–454 (1973)] appropriate to the linear expansions of the mechanical and electric displacement fields, from the three‐dimensional field equations. The interactions between crystal plate and electrodes are taken into account through the usual continuity conditions. Next, we use logarithmic convexity arguments to enumerate a set of sufficient conditions for uniqueness in the theory [cf. M. C. Dokmeci, “On the Higher‐order Theories of Piezoelectric Crystal Surfaces,” to appear in J. Math. Phys. 15, (1974)]. Further, we exhibit some applications of the theory, and, in particular, study wave propagations in an infinite circular cylindrical anisotropic thin shell of uniform thickness, with its inner and outer faces being both traction free and completely coated with electrodes. The transcendental equations of resonant frequencies are obtained and the dispersion curves are calculated for a rotated Y‐cut quartz shell. The results are compared with those of unplated shell, and considerable deviations in the dispersion spectrum are found. [Work supported by the Turkish Office of Naval Research.]

A survey is made of the methods available for the calculation of axial wavenumbers in lined ducts. Rectangular, circular and annular ducts with both uniform and nonuniform flow are considered as are ducts with peripherally varying liners. A historical perspective is provided by a discussion of the classical methods for computing attenuation when no mean flow is present. When flow is present these techniques become either impractical or impossible. A number of direct eigenvalue determination schemes which have been used when flow is present are discussed. Methods described are extensions of the classical no‐flow technique, perturbation methods based on the no‐flow technique, direct integration methods for solution of the eigenvalue equation, an integration‐iteration method based on the governing differential equation for acoustic transmission, Galerkin methods, finite difference methods, and finite element methods. Special emphasis is given to the case of uniform flow and an approach based on converting th...

An analytical simulation and experimental measurement of stress wave propagation through a guided projectile structure was made and the results correlated through the use of fast Fourier transform (FFT) methods. Two guided projectile structures were simulated; one with and one without bolted joints such that signal attenuation across a joint could be evaluated. The analytical simulation consisted of a linearly elastic finite element structural model subjected to a time‐history force input and solved using SAP IV via direct integration of the coupled differential equations. The experimental measurements were made using a pulse‐testing technique wherein a structure was instrumented with accelerometers and impacted with a force input and the accelerometer response and force input measured. The resulting time history response data, whether analytical or experimental, was then analyzed using FFT methods to generate a frequency domain representation of the response signal for comparison. Results indicate finite element structural simulations of elastic stress wave propagation are a valid design tool for studying the noise characteristics of a structure being impacted with low order force inputs. [Work sponsored by Honeywell Independent Development Funding.]An analytical simulation and experimental measurement of stress wave propagation through a guided projectile structure was made and the results correlated through the use of fast Fourier transform (FFT) methods. Two guided projectile structures were simulated; one with and one without bolted joints such that signal attenuation across a joint could be evaluated. The analytical simulation consisted of a linearly elastic finite element structural model subjected to a time‐history force input and solved using SAP IV via direct integration of the coupled differential equations. The experimental measurements were made using a pulse‐testing technique wherein a structure was instrumented with accelerometers and impacted with a force input and the accelerometer response and force input measured. The resulting time history response data, whether analytical or experimental, was then analyzed using FFT methods to generate a frequency domain representation of the response signal for comparison. Results indicate finite...

Undulatory motions of a separated shear layer in a rectangular slot are found to give rise to three forces that are believed responsible for jet‐edge and jet‐edge‐resonator oscillation at low Mach number. The three are (1) a transverse force on the downstream edge which gives rise to the transverse dipole feedback radiation postulated by Powell, (2) a longitudinal force on the downstream wall, which results in longitudinal dipole radiation believed responsible for oscillation in shallow cavities, and (3) a longitudinal force in the body of the fluid which is associated with pipetone drive. By direct integration of the Lighthill tensor over the entire region of fluid motion between the edges, the magnitude of each force is estimated. In every case the results are directly proportional to shear layer width, and cannot therefore be derived from theories using the vortex sheet approximation.

The hydrodynamical field equations yield to three methods of solution. Two of these, the method of characteristics and that of separation of variables, have been widely used for both analytical and numerical investigation, but usually with either physical or mathematical approximations. The third, direct integration of the partial differential equations has always existed in principle but not until the advent of the fast Fourier transform could this approach be exploited in a systematic and economical manner. H. W. Marsh recognized this fact and introduced the concept of the fast field program (FFP). Several examples are presented and discussed which demonstrate that the FFP is not only feasible but a highly efficient algorithm for providing propagation estimations rapidly without undue approximations to the basic field equations. The results are compared with those predicted by both ray theory and normal‐mode theory. [This research was supported by the U. S. Underwater Sound Laboratory.]

A series solution has been obtained for the mutual acoustic impedance between two identical, circular disks vibrating in an infinite plane. The problem was formulated in terms of Bouwkamp's method of integrating over real and complex angles the square of the directional characteristic (relative sound pressure at a large, fixed distance) to yield the total radiation impedance. Integrals involved here are similar to a type previously evaluated by Stenzel and may be expressed in terms of a double series containing Bessel functions of integral and half‐integral order. Numerical values of the mutual acoustic impedance obtained by this method for two rigid disks are in good agreement with values obtained by Klapman by direct integration of the pressure at the surfaces of the disks. The acoustic self‐impedance and mutual impedance may also be calculated by the same methods for a more general type of circular disk having a prescribed radially‐symmetric velocity distribution.