A. L. Van Buren

An exact expression for the Lommel‐diffraction correction integral

A number of authors have obtained diffraction corrections for a circular piston source by numerical or graphical integration of an approximate expression for the piston field attributable to E. Lommel [Abh. Bayer. Akad. Wiss. Math.‐Naturwiss. Kl. 15, 233 (1886)]. Lommels expression gives the piston field in terms of trigonometric functions and Lommel functions of two variables. It is shown here that the required integral of Lommels expression can be evaluated analytically to obtain a simple closed‐form expression for the diffraction correction. The extrema of this expression are obtained as roots of simple transcendental equations, and approximation formulas for these roots are given. It is also shown that the same expression can be obtained by taking the limit as ka → ∞ (k is the wavenumber and a is the piston radius) of Williamss exact integral expression [J. Acoust. Soc. Am. 23, 1–6 (1951)] for the diffraction correction. Finally, it is shown both analytically and by comparison with numerical values...

Reflection of Finite‐Amplitude Ultrasonic Waves. I. Phase Shift

When an ultrasonic wave is reflected from an interface, a phase shift that is dependent on the angle of incidence may occur. If the wave is nonsinusoidal, this phase shift causes relative spatial shifting of the Fourier harmonic components. The assumption of independent reflection of the Fourier harmonics allows the use of linear theory to calculate the change in the phase angle between the fundamental and the second‐harmonic components in the distorted wave. A pulse technique is used to measure this change in phase angle. Agreement between theory and experiment is good for the phase shift upon reflection from water‐copper, from water‐duralumin, and from water‐mica‐glycerin interfaces when correction is made for finite‐amplitude effects between the interface and the receiving transducer.

Phase calibration of hydrophones

Determination of the phase information in a received signal requires knowledge of the phase angle of the receiving sensitivity of the measuring hydrophone. Except for frequencies well below the lowest hydrophone resonance, this phase angle varies considerably with frequency. This paper describes the extension of conventional reciprocity calibration to include phase. It also describes a unique measurement procedure that eliminates phase errors resulting from uncertainties in element location and sound speed. Several hydrophones are calibrated using the new procedure. Measured results are in good agreement with theoretical results based on diffraction constant calculations.

Nonlinear scattering of acoustic waves by vibrating obstacles

The problem of scattering of an acoustic wave (at angular frequency ω) by an obstacle whose surface vibrates harmonically (at angular frequency Ω) was studied both theoretically and experimentally. The theoretical approach involved solving the nonlinear wave equation, subject to appropriate boundary conditions, by use of a perturbation expansion of the fields and a Greens function method. This problem was previously studied theoretically by D. Censor [(J. Sound Vib. 25, 101–110 (1972)], who used the linear wave equation together with nonlinear boundary conditions to obtain his solution. In addition to ordinary rigid‐body scattering, Censor predicted nongrowing waves at the sum and difference frequencies ω± = ω ± Ω. The solution to the nonlinear wave equation also yields scattered waves at frequencies ω±. However, the amplitudes of these waves tend to grow with increasing distance from the scatterers surface and after a very small distance dominate those predicted by Censor. Preliminary experimental resu...

Censor's acoustical Doppler effect analysis--Is it a valid method?

In the article, ‘‘Acoustical Doppler effect analysis—Is it a valid method?’’ [J. Acoust. Soc. Am. 83, XXX–XXX (1988)], Censor considers the problem of scattering in the presence of moving objects under conditions of space‐ and time‐dependent moving media. He considers this situation in the context of a generalized linear wave equation. This equation predicts the generation of Doppler‐type spectral components at frequencies that are equal to (i) the sum of the frequencies of the primary waves, (ii) the difference of the frequencies of the primary waves, and (iii) harmonics of these sum‐and‐difference frequencies. However, since the nonlinear wave equation also predicts scattered spectral components at these same frequencies, and since those predicted by the nonlinear theory are usually much stronger than those predicted by Censor’s theory, the generalized linear wave equation used by Censor is generally inadequate for accurately predicting the amplitudes of the spectral components of interest. However, a l...

Acoustic radiation impedance of rectangular pistons on prolate spheroids

The self and mutual radiation impedances for rectangular piston(s) arbitrarily located on a rigid prolate spheroidal baffle are formulated. The pistons are assumed to vibrate with uniform normal velocity and the solution is expressed in terms of a modal series representation in spheroidal eigenfunctions. The prolate spheroidal wave functions are obtained using computer programs that have been recently developed to provide accurate values of the wave functions at high frequencies. Results for the normalized self and mutual radiation resistance and reactance are presented over a wide frequency range for different piston sizes and spheroid shapes.

A planar array for the generation of evanescent waves

Wavenumber‐frequency calibration of underwater, planar, receiving arrays requires the ability to generate single‐wavenumber pressure fields over the surface of the array. When the wavenumber‐frequency region of interest is evanescent, transmitting arrays previously constructed have been found to generate fields contaminated with harmonics, acoustic wavenumbers, and nonacoustic wavenumbers from the excitation of antisymmetric Lamb waves. An array that greatly reduces contamination has recently been constructed using a sheet of polyvinylidene fluoride (PVDF) with independent rectangular electrode stripes. The array operates in the frequency range of 500 Hz to 2 kHz and generates evanescent waves with phase speeds between 30 and 150 m/s. Contamination due to the excitation of antisymmetric Lamb waves is eliminated by shifting the phase speed of the Lamb wave out of the region of interest. This is accomplished by bonding the thin sheet of PVDF directly to a thick plate of LEXAN. Contamination from harmonics a...

Reflection of Finite‐Amplitude Ultrasonic Waves. II. Propagation

The reflection of a finite‐amplitude ultrasonic wave from an interface was described by assuming that the Fourier Components in the wave are shifted in phase as though they were independent waves (Part I). In Part II, the behavior of the distorted wave after reflections that produce nonzero component phase shifts is predicted using a computer model. The progress of these “unstable” waves toward the “stable” waveform is described. Experimental results obtained using a pulse technique are in good agreement with the computer results.

Technique for Evaluating Indefinite Integrals Involving Products of Certain Special Functions

A new technique is described for evaluating a general class of indefinite integrals involving products of many of the special functions of physics such as Bessel functions, Legendre functions, Hermite functions, etc. The technique is a generalization of the method used by Sonine to evaluate certain indefinite integrals of Bessel functions. It involves replacing the integral to be evaluated by a coupled set of linear, inhomogeneous differential equations. A particular solution of the set of differential equations is then sufficient to express the result of integration. Several examples are given to illustrate the technique.

Comments on ‘‘Distortion of finite amplitude ultrasound in lossy media,’’ by M. E. Haran and B. D. Cook [J. Acoust. Soc. Am. 73, 774–779 (1983)]

Readers of the subject paper might conclude that Haran and Cook are the first to derive an alogorithm based on Burgers’ equation which calculates the nonlinear propagation of a plane wave in a lossy medium with nonquadratic frequency‐dependent absorption coefficients. In fact, we presented such an algorithm in an earlier paper [D. H. Trivett and A. L. Van Buren, ‘‘Propagation of plane, cylindrical, and spherical finite amplitude waves,’’ J. Acoust. Soc. Am. 69, 943–949 (1981)] and we feel that the Haran and Cook’s reference to our work was insufficient. Although the derivation given by Haran and Cook is correct, it contains an inconsistency in notation. Also, their numerical results for propagation in fresh water correspond to different initial conditions than stated in the paper.