Diffraction of a spherical acoustic wave on a cone

Diffraction of a spherical acoustic wave due to the coupling of pressure release and absorbing half planes in a moving fluid

Scattering of acoustic waves from a point source over an impedance wedge

This chapter considers the diffraction of a plane acoustic wave illuminating completely the surface of a right-circular impedance cone. Formulation of the problem is also given for an arbitrary convex cone and uniqueness of the solution is discussed. By separating the radial variable the problem at hand is then reduced to that for a spectral function. Further separation of the angular variables for the circular cone leads to integral equations for the Fourier coefficients. The use of the Sommerfeld integral enables one to describe the far-field asymptotics for, in particular, the reflected wave, the spherical wave from the vertex of the cone, and the surface waves.

Acoustic transients generated by high voltage pulses in longitudinal KDP modulator crystals were investigated. Photographic method has been developed to visualize the acoustic waves. The experiments directly demonstrate the piezoelectric acoustic wave generation, the acoustic wave propagation, the reflexion, diffusion and diffraction of acoustic waves and the interaction of acoustic waves with crystal imperfections. From the experimental results the velocity of shear waves and the related elastic constant can be evaluated.

The diffraction of a spherical acoustic wave by a half plane in an infinite plane, porous barrier is considered. The barrier is modeled as a rigid material filled with narrow pores, more or less normal to the plane, that provide sound damping. However, the compressibility of the gas in the pores is also considered so that some sound transmission takes place. An approximate boundary condition is derived that models both these effects. The principal purpose of the barrier is to reduce the transmitted sound so that the flow resistance of the pores is large. This consideration is important in the sense that point sources are also regarded as reasonable substitutes for real sources. The far field is calculated using integral transforms, the Wiener-Hopf technique and asymptotic methods.

An acoustic or electromagnetic wave propagating in a material medium experiences fluctuations in amplitude and phase that can be attributed in part to scattering from inhomogeneities due to variations in the refractive index of the transmission medium. Since such “patches of inhomogeneity” are often regarded as lenticular in shape, it is the purpose of this paper to extend, for both plane and spherical sound waves, contemporary wave-fluctuation theory (based on a Gaussian correlation function for refractive index) to include the circumstance of spheroidal inhomogeneities. Mean-square amplitude and phase fluctuations are considered for acoustic waves propagating in a statistically isotropic medium in which random deviations of the refractive index are small. Large-scale inhomogeneities of arbitrary spheroidal shape are treated, and the influence of transmitter-to-receiver range, acoustic wavelength, and “patch” dimensions on variations in wave amplitude and phase is derived for this general case. Particular attention is given the high-frequency ray limit (Bergmann region) and the low-frequency (Fraunhofer diffraction) region. The significant effects on wave fluctuations of both spherical and nonspherical inhomogeneities are compared for plane and spherical acoustic waves.

С помощью непрерывно-неоднородных упругих покрытий можно эффективно изменять характеристики рассеяния тел в определенных направлениях, если подобрать соответствующие законы неоднородности для механических параметров покрытия. В статье рассматривается задача дифракции сферической звуковой волны на однородном изотропном упругом цилиндре с радиально-неоднородным упругим покрытием. Полагается, что бесконечный круговой цилиндр с покрытием помещен в идеальную безграничную жидкость, законы неоднородности материала покрытия описываются дифференцируемыми функциями, на тело падает гармоническая сферическая звуковая волна, излучаемая точечным источником. В случае установившихся колебаний распространение малых возмущений в идеальной жидкости описывается скалярным уравнением Гельмгольца, а в упругом однородном изотропном цилиндре — скалярным и векторным уравнениями Гельмгольца. Колебания неоднородного изотропного упругого цилиндрического слоя описываются общими уравнениями движения сплошной среды. Аналитическое решение рассматриваемой задачи получено на основе известного решения аналогичной задачи дифракции плоской волны. Потенциал скорости сферической волны представляется в интегральной форме в виде разложения по цилиндрическим волновым функциям. При этом подынтегральное выражение оказывается аналогичным по форме выражению потенциала скорости плоской волны. Поэтому потенциал скорости рассеянной волны в случае падения сферической волны на цилиндр с покрытием записывается в виде интеграла, подынтегральное выражение которого аналогично по форме выражению потенциала скорости рассеянной волны при падении плоской волны на тело. Для вычисления подынтегральной функции необходимо определить поле смещения в неоднородном покрытии, решая построенную краевую задачу для системы обыкновенных дифференциальных уравнений второго порядка. Рассматриваются вычислительные аспекты оценки интеграла.

Fresnel diffraction is a fundamental wave phenomenon. This article explains its physical nature using the examples of the diffraction of acoustic waves at soft and hard half-planes and at large apertures on a black screen. It is shown that the shadow radiation by opaque screens plays a central role in these diffraction phenomena. Fresnel-Kirchhoff diffraction at large apertures is presented as an asymptotic form of the shadow radiation. Fresnel and Grimaldi-type diffraction at the soft and hard half-planes is revealed as interference of the shadow radiation and the incident wave.

A new asymptotic high-frequency solution is presented for the problem of diffraction of acoustic waves emanating from a ring source by a cylindrical impedance rod of finite length. This problem is described by a modified Wiener-Hopf equo tion of the third kind which can be solved approximately. The solution involves two infinite sets of constants satisfying two infinite systems of linear algebraic equations.

Application of zero-radius potentials to problems of diffraction by small inhomogeneities in elastic plates

For a long time, numerous problems in various applied fields [1-3] have been investigated with the use of singular integral equations. For example, some problems of aerodynamics [1], diffraction of electromagnetic and acoustic waves [2], elasticity [3], etc. can be reduced to the solution of such equations. The theory of singular integral equations was developed long ago [3], but related applied problems required the solution of these equations in the class of absolutely integrable functions. It was mentioned in [4] that, under specific conditions imposed on the coefficients of a singular integral equation, one can find solutions in the class of fimctions with nonintegrable singularity of the form 1/x as x -* 0. But the absence of applied problems leading to such solutions, numerical methods for their construction, and a theory of distributions developed for singular integral equations hindered the investigation of such singular solutions. However, in the framework of aerodynamics [5, 6], the mechanics of wings has recently been analyzed with the use of ejection of an external flow. It was shown in [6] that, in the case of a thin profile, this problem can be reduced to solving a singular integral equation of the first kind on a closed interval in the class of fimctions that have a nonintegrable singularity of the above-mentioned type at the ejection point. By [7], the investigation of the diffraction of an electromagnetic wave on an ideal conductor with a current thread on the surface leads to a similar mathematical problem. Such problems were numerically solved in [1, 7, 8] in sorne special cases with the use of the discrete vortex and discrete current methods. However, the integral operators constructed in these papers are continuously invertible in related pairs of spaces but have the property that their squares map the entire subspace of singular solutions to zero. In the present paper, in view of [9, 10], we give another mathematical treatment of the same applied problems, which permits one to consider them in the framework of the theory of pseudodifferential operators on a closed curve. Then we describe the construction of Sobolev type weighted spaces and develop the theory of linear operators (similar to pseudodifferential ones) in these spaces for open curves. We present examples illustrating the difference between the notions of integral in the sense of the Cauchy principal value and pseudodifferential operators on distributions that are singular solutions. Similar linear operators are constructed for singular integral equations of the second kind with Hilbert kernel and constant coefficients on a closed interval. We indicate the possibility of the generalization of the suggested construction of linear operators to some spatial problems.

Nonlinear distortion and shock formation in planar surface acoustic waves in anisotropic crystals have been modeled without [Hamilton et al., JASA (1999)] and with [Cormack et al., JASA 2022] piezoelectricity. Weak diffraction has been included in the paraxial approximation for nonlinear surface wave beams in isotropic solids [Shull et al., JASA (1995)]. Here, a procedure for including strong diffraction, i.e., without the paraxial approximation, in the model for nonlinear surface waves in crystals is presented, which incorporates the angular spectrum approach described by Kharusi and Farnell (JASA 1970). Anisotropy is defined by expressing the phase speed of a plane wave, and therefore, the magnitude of the corresponding wavenumber, as a function of the direction of propagation. Next, an explicit expression relating the two wavenumber components in the planar surface along which the wave propagates must be obtained as a function of direction, requiring iterative solution of a transcendental relation. The beam is then propagated incrementally away from the source, advancing the angular spectrum in k-space and including nonlinear interactions in the spatial domain to characterize the combined effects of diffraction and nonlinearity. Preliminary results are presented for converging nonlinear surface waves in crystals. [Work supported by IR&D at ARL:UT.]

A new method termed Directive Line Source Model (DLSM) is presented for predicting the diffracted field produced by a sound wave incident on a rigid or pressure release half plane. In the new method the edge of the half plane is modeled as an infinite set of directive point sources continuously distributed along the edge. Because DLSM is fast, simple and intuitive, it is a promising tool for the study of diffraction. It can be applied for several types of incident radiation: omnidirectional cylindrical and spherical waves, plane waves, as well as waves from directional sources. Wedges may also be treated. Finally, DLSM can handle diffraction by an arbitrarily shaped edge profile, for example, a half plane having an edge that is jagged instead of straight. Results for plane, cylindrical and spherical incident waves, as well as for arrays of line and point sources, are presented and their agreement with known analytical solutions is demonstrated. Predictions based on DLSM compare favorably with experimental data.

In the field of diffraction of acoustic, electro-magnetic and elastic waves the ray method has proven to be a powerful tool to obtain reliable results for the behaviour of short waves. In ship hydrodynamics where the generation of short waves by a slowly sailing vessel and the diffraction of short surface waves by these vessels is of importance recently the ray method is applied successfully. In this presentation we present two cases, each with its own free surface condition, treated with this method.

Using the rigorous wave-front formulation for scalar wave diffraction of Kraus [ J. Opt. Soc. Am. A6, 1196 ( 1989); J. Opt. Soc. Am. A9, 1132 ( 1992)], it is shown that the two-dimensional integral used to calculate the diffraction of spherical waves by a circular aperture may be reduced to a one-dimensional integral by choosing an appropriate coordinate frame. Both the two-dimensional integral and the one-dimensional integral must be evaluated numerically, but because each dimension must be sampled at approximately N locations to calculate accurately the integral (where N is the number of wavelengths across the aperture) the two-dimensional integration will require of the order of N2 evaluations of the integrand, whereas the one-dimensional integration will require of the order of only N evaluations, a substantial decrease in computing time for apertures that are large compared with optical wavelengths.