Reflection and refraction of a planar acoustic wave in an anisotropic inhomogeneous layer

The problem of reflection and refraction of a planar acoustic wave by an inhomogeneous elastic layer whose material possesses general-type anisotropy is considered. The equations of motion of the elastic layer are reduced to a system of ordinary differential equations. The boundary-value problem for this system is solved by two methods: by reduction to problems with initial conditions and by the method of power series. Analytical expressions that describe acoustic fields outside the layer are obtained. Calculation results of the transmission factor for transversely isotropic layers inhomogeneous in thickness are presented. Tula State University, Tula 300600. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 5, pp. 179–184, September–October, 1999.     

Traveling Wave Fronts of Reaction-Diffusion Systems with Delay

This paper deals with the existence of traveling wave front solutions of reaction-diffusion systems with delay. A monotone iteration scheme is established for the corresponding wave system. If the reaction term satisfies the so-called quasimonotonicity condition, it is shown that the iteration converges to a solution of the wave system, provided that the initial function for the iteration is chosen to be an upper solution and is from the profile set. For systems with certain nonquasimonotone reaction terms, a convergence result is also obtained by further restricting the initial functions of the iteration and using a non-standard ordering of the profile set. Applications are made to the delayed Fishery–KPP equation with a nonmonotone delayed reaction term and to the delayed system of the Belousov–Zhabotinskii reaction model. An erratum to this article is available at .     

Interaction of a transient cylindrical acoustic wave with a multimode cylindrical piezoelectric transducer

The interaction between a transient cylindrical acoustic wave and a cylindrical piezoelectric transducer with sectioned electrodes is considered. The solution is obtained using the integral Laplace transform with respect to time (in the form of natural-mode series). In going to the space of originals, the problem is reduced to integral Volterra equations. The results of calculation are presented for various angular dimensions of the electrode sections and various distances from the wave source to the piezoelectric transducer. Translated from Prikladnaya Mekhanika, Vol. 35, No. 10, pp. 37–45, October, 1999.     

Vortex motions of liquid in a narrow channel

Quasi-linear integrodifferential equations that describe vortex flows of an ideal incomparessible liquid in a narrow curved channel in the Eulerian-Lagrangian coordinate system are considered. The necessary and sufficient conditions for hyperbolicity of the system of equations of motion are obtained for flows with a monotonic velocity depth profile. The propagation velocities of the characteristics and the characteristic form of the system are calculated. A particular solution is given in which the system of integrodifferential equations changes type with time. The solution of the Cauchy problem is given for linearized equations. An example of initial data for which the Cauchy problem is ill-posed is constructed. Novosibirsk State University, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 4, pp. 38–49, July–August, 1998.     

Constructing an analytical solution for lamb waves using the cosserat continuum approach

The problem of propagation of a Lamb elastic wave in a thin plate is considered using the Cosserat continuum model. The deformed state is characterized by independent displacement and rotation vectors. Solutions of the equations of motion are sought in the form of wave packets specified by a Fourier spectrum of an arbitrary shape for three components of the displacement vector and three components of the rotation vector which depend on time, depth, and the longitudinal coordinate. The initial system of equations is split into two systems, one of which describes a Lamb wave and the second corresponds to a transverse wave whose amplitude depends on depth. Analytical solutions in displacements are obtained for the waves of both types. Unlike the solution for Lamb waves, the solution obtained for the transverse wave has no analogs in classical elasticity theory. The solution for the transverse wave is compared with the solution for the Lamb wave. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 1, pp. 143–150, January–February, 2007. An erratum to this article is available at .     

On solving the problem of reflection of linear waves in a fluid from a porous half-space saturated by this fluid

Solutions of the problem of reflection of a stepwise pressure wave in a linearly compressed fluid from a flat boundary of a porous medium of infinite length saturated by the same fluid are obtained in the acoustic approximation. Based on analytical solutions, a numerical analysis is performed to reveal the specific features of the reflected and incident waves, depending on porosity and permeability of the porous half-space. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 16–26, September–October, 2006.     

Anti-plane dynamic hole–crack interaction in a functionally graded piezoelectric media

The anti-plane dynamic problem of a functionally graded piezoelectric plane containing a hole–crack system is treated by a non-hypersingular traction-based boundary integral equation method. The material parameters vary exponentially in the same manner in an arbitrary direction. The system is loaded by an incident SH-type wave, and impermeable boundary conditions are assumed. Using a frequency-dependent fundamental solution of the wave equation, the boundary value problem is transformed into a system of integro-differential equations along the boundary of the hole and on the crack line. Its numerical solution yields the dynamic stress intensity factors and stress concentration factors. A parametric study reveals their dependence on the hole–crack scenario and its geometry, characteristics of the dynamic load and magnitude and direction of material inhomogeneity.     

Asymptotic Stability of Combination of Viscous Contact Wave with Rarefaction Waves for One-Dimensional Compressible Navier–Stokes System

We are concerned with the large-time behavior of solutions of the Cauchy problem to the one-dimensional compressible Navier–Stokes system for ideal polytropic fluids, where the far field states are prescribed. When the corresponding Riemann problem for the compressible Euler system admits the solution consisting of contact discontinuity and rarefaction waves, it is proved that for the one-dimensional compressible Navier–Stokes system, the combination wave of a “viscous contact wave”, which corresponds to the contact discontinuity, with rarefaction waves is asymptotically stable, provided the strength of the combination wave is suitably small. This result is proved by using elementary energy methods.     

The Acoustic Analogy and the Prediction of the Noise of Rotating Blades

The acoustic analogy was introduced into acoustics by Lighthill in 1952 to understand and predict the noise generated by the jet of an aircraft turbojet engine. The idea behind the acoustic analogy is simple but powerful. The entire noise generation process is mathematically reduced to the study of wave propagation in a quiescent medium with the effect of flow replaced by quadrupole sources. In jet noise theory, Lighthill was able to obtain significant and useful qualitative results from the acoustic analogy. The acoustic analogy has influenced the theoretical and experimental research on jet noise since the early 1950s. This paper, however, focuses on another area in which the acoustic analogy has had a significant impact, namely, the prediction of the noise of rotating machinery. The governing equation for this problem was derived by Ffowcs Williams and Hawkings in 1969. This equation is a wave equation for perturbation density with three source terms, which have become known as thickness, loading, and the quadrupole source terms, respectively. The Ffowcs Williams–Hawkings (FW–H) equation has been used for the successful prediction of the noise of helicopter rotors, propellers, and fans. Several reasons account for the success and popularity of the acoustic analogy. First, the problems of acoustics and aerodynamics are separated. Second, because the FW–H equation is linear, powerful analytical methods from linear operator theory can be used to obtain closed-form solutions. Third, advances in digital computers and computational fluid dynamics algorithms have resulted in high-resolution near-field aerodynamic calculations that are suitable for noise prediction. We present some of the mathematical results for noise prediction based on the FW–H equation, including examples for helicopter rotors. In particular, we discuss the prediction of blade-vortex interaction noise and high-speed impulsive noise of helicopter rotors. For high-speed propellers, we briefly discuss the derivation of a singularity-free solution of the FW–H equation for a supersonic panel on a blade. Received 24 January 1997 and accepted 30 May 1997     

Diffraction of an arbitrary acoustic wave by a strip of finite width

The problem of the diffraction of an arbitrary acoustic wave by a strip of finite width is solved. The solution is constructed by means of a generalization of the previously obtained integral for the problem of the diffraction of acoustic waves by a half-plane [5]. The problem of the diffraction of an arbitrary acoustic wave by the Riemannian manifold corresponding to the strip of finite width is first found. After this, by substitution of the values of the polar angle a solution is obtained for the reflected wave associated with diffraction on the Riemannian manifold, and then the boundary conditions on the surface of the strip are satisfied by means of a linear combination of these solutions. The problem of the diffraction of an arbitrary acoustic wave by a slit of finite width could be constructed in exactly the same way.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 171–175, March–April, 1991.     

Construction of piecewise-analytical gas flows joined through shock waves near the axis or center of symmetry

Analytical solutions of a quasilinear system of equations with partial derivatives are constructed in the case where the initial data for different functions are specified on different surfaces and the resultant problem has singularities of the form u/x and w/x. Conditions for existence and uniqueness of a solution in the form of formal power series for the problem posed and sufficient conditions for convergence of the series are indicated. A generalization of the problem considered is given. Results of the study are used to solve the problem of the focussing of a compression wave generated by a piston moving smoothly in a quiescent gas: a solution for t=0, including determination of the piston trajectory, and a solution for t<0, including unequivocal construction of the front of a reflected shock wave, are uniquely constructed from the distribution of gas-dynamic quantities for t>0. The solution of this problem is a generalization to the case of two independent variable self-similar Sedov's solutions. Ural State Academy of Communications, Ekaterinburg 620034. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 5, pp. 25–38, September–October, 1998.     

Reflection and radiation of a wave system at the open end of a submerged elastic pipe

The reflection and radiation of a wave system at the open end of a submerged semi-infinite elastic pipe are studied. This wave system consists of a flexural wave in the pipe, an acoustic surface wave in the fluid exterior to the pipe and an acoustic wave in the pipe’s interior. Fourier transform techniques are used to formulate this semi-infinite geometry problem rigorously as a Wiener-Hopf type equation. An approximate solution is obtained by using a perturbation method in which the ratio of the massdensities of the fluid and the pipe material is regarded as a small parameter. The calculation of the reflection coefficient is emphasized, and the polar plots of the radiation coefficient are also presented.     

New expressions for the radiation force function of spherical targets in stationary and quasi-stationary waves

A new expression for the radiation force function – which is the radiation force per unit energy density and unit cross-sectional surface area – for spheres in a stationary (or standing) and quasi-stationary wave is obtained based on the far-field acoustic scattering field. The radiation force function formulation has been simplified mathematically and improved into a more general form. Numerical results are presented for rigid and elastic spheres, air bubbles in water as well as liquid drops in air to illustrate the theory. It is demonstrated that expressions for the radiation force functions obtained from the far-field derivation approach are equivalent to those obtained from the near-field-based derivation.     

Wave processes in a perfect liquid layer impacted on by a blunted solid

The problem of impact of a smooth blunt solid upon a compressible-liquid layer of finite depth is addressed. A mixed initial-boundary-value problem with an unknown moving boundary is formulated. In the general case, the problem is reduced to an infinite system of integral Volterra equations of the second kind. It is solved numerically, using quadrature formulas and truncation method. An exact analytic solution to the problem is obtained in the special case where the body moves with a constant velocity at the initial stage of submergence. This solution makes it possible to examine the effect of successive wave reflections on the pressure at the frontal point and inside the layer __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 3, pp. 37–47, March 2007.     

Internal waves induced by potential vortices

A linearized equation of the internal waves developing in an ideal stratified gas under the action of potential vortices concentrated in a vertical cylinder is obtained. The Cauchy problem for the internal wave equation with right side depending on the vortex intensity is solved by the integral transform method. In the case of a vortex filament the exact solution is found. Approximate formulas are obtained on the basis of the steady-phase method when the vorticity is exponentially stratified along the vertical. Expressions for the phase velocity and amplitude of the radial wave traveling away from the cylindrical vortex are found. Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 1, pp. 118–123, January–February, 1998. The work was carried out with the support of the Russian Foundation for Fundamental Research (project No. 96-01-04599).     

Numerical simulation of the interaction of gravity waves with elastic ice floes

The self-consistent motion of a fluid and elastically oscillating plates partially covering the fluid is simulated numerically in the linear approximation. The problem is reduced to the simultaneous solution of the Laplace equation for the fluid and the equation of elastic plate oscillations for the ice. The numerical and analytical solutions, the latter obtained from an integral equation containing the Green’s function, are compared. To solve the problem numerically, the boundary element method for the Laplace equation and the finite element method for the equation describing the elastic plate are proposed. The coefficients of transmission and reflection of surface gravity waves from the floating plates are calculated. It is shown that the solution may be quasi-periodic with characteristics determined by the initial values of the wave and ice-floe parameters. The ice floes may exert a filtering effect on the surface wave spectrum, essentially reducing its most reflectable components. Sankt-Peterburg. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 123–131, May–June, 2000.     

Dynamic method of determining aerodynamic characteristics of models from on experimental results obtained in short-duration wind tunnels

A technique is developed to recover the loads acting in model testing in short-duration wind tunnels. The model together with the load-measuring device is assumed to be a linear dynamic system. Normal responses of the system to unit loads are determined experimentally. The solution of a system of integral Volterra equations of the first kind is found in the class of piecewise-constant functions. Examples are given on solving a test problem with a known exact solution and a problem of determining the aerodynamic characteristics of the HB-2 reference model on the basis of loads obtained in wind-tunnel measurements. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 5, pp. 47–55, September–October, 2006.     

Dynamics of particles near a plane boundary in the radiation field of an acoustic wave

The problem of interaction of a rigid body and a boundary of a viscous fluid under acoustic wave propagation is investigated. An example of the motion of a cylinder in the vicinity of a flat rigid wall caused by acoustic field forces is examined. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 35, No. 10, pp. 74–79, October, 1999.     

New explicit and exact traveling wave solutions for two nonlinear evolution equations

In this paper, with the aid of computer symbolic computation system such as Maple, an algebraic method is firstly applied to two nonlinear evolution equations, namely, nonlinear Schrodinger equation and Pochhammer–Chree (PC) equation. As a consequence, some new types of exact traveling wave solutions are obtained, which include bell and kink profile solitary wave solutions, triangular periodic wave solutions, and singular solutions. The method is straightforward and concise, and it can also be applied to other nonlinear evolution equations in mathematical physics.     

Stability of shock waves in the modified quintic complex Ginzburg-Landau equation

We consider a shock-type wave solution of the modified quintic complex Ginzburg-Landau equation and make a numerical study of its spatiotemporal stability. Discussions related to the behavior of this front wave are introduced and it is shown how the velocities of the wave can be used to collect information concerning the pattern formation in the system. Published in Neliniini Kolyvannya, Vol. 10, No. 2, pp. 270–276, April–June, 2007.