Nonlinear evolution equations for degenerate transverse waves in anisotropic elastic solids

Transverse elastic waves behave differently in nonlinear isotropic and anisotropic media. Quadratically nonlinear coupling in the evolution equations for wave amplitudes is not possible in isotropic solids, but such a coupling may occur for certain directions in anisotropic materials. We identify the expression responsible for the coupling and we derive coupled canonical evolution equations for transverse wave amplitudes in the case of two-fold and three-fold symmetry acoustic axes. We illustrate our considerations by examples for a cubic crystal.Comment: 4 page

Fourieruppdelning av en plan ickelinjär ljudvåg och dess övergång från Fubinis till Fays lösning till Burgers ekvation

Burgers' equation describes plane sound wave propagation through a thermoviscous fluid. If the boundary condition at the sound source is given as a pure sine wave, the exact solution is given by the Cole-Hopf transformation as a quotient between two Fourier series. Two approximate Fourier series representations of this solution are known: Fubini's (1935) solution, neglecting dissipation and valid at short distance from the sound source, and Fay's (1931) solution, valid far from the source. In the present investigation a linear system of equations is found, from which the coefficients in a series expansion of each Fourier coefficient can be derived one by one. Curves which join smoothly to Fubini's solution (valid up to slightly before shock formation) and to Fay's solution (valid for approximately three shock formation distances). Maxima for the Fourier coefficients of the higher harmonics are given. These maxima are situated in a region where neither Fubini's nor Fay's solution is valid

Fourier decomposition of a plane nonlinear sound wave and transition from Fubini´s to Fay´s solution of Burger´s equation

Burgers' equation describes plane sound wave propagation through a thermoviscous fluid. If the boundary condition at the sound source is given as a pure sine wave, the exact solution is given by the Cole-Hopf transformation as a quotient between two Fourier series. Two approximate Fourier series representations of this solution are known: Fubini's (1935) solution, neglecting dissipation and valid at short distance from the sound source, and Fay's (1931) solution, valid far from the source. In the present investigation a linear system of equations is found, from which the coefficients in a series expansion of each Fourier coefficient can be derived one by one. Curves which join smoothly to Fubini's solution (valid up to slightly before shock formation) and to Fay's solution (valid for approximately three shock formation distances). Maxima for the Fourier coefficients of the higher harmonics are given. These maxima are situated in a region where neither Fubini's nor Fay's solution is valid

Finite-Amplitude Standing Acoustic Waves in a Cubically Nonlinear Medium

The behavior of the wave field in a resonator containing a cubically nonlinear medium is studied. The field is constructed as a linear superposition of two counter-propagating and strongly distorted waves. As distinct from a quadratic nonlinear medium the waves traveling in opposite directions are connected through their averaged (over the period) intensities. Both free and forced standing waves are studied. Profiles of discontinuous vibrations containing shock fronts of both compression and rarefaction are constructed. Resonant curves depicting the dependence of mean intensity on the difference between the frequency of vibration of the boundary and the natural frequency of one of the resonator’s mode are calculated. The structure of temporal profiles of strongly distorted forced waves is analyzed. It is shown, that shocks can appear only if the difference between the mean intensity and the discrepancy takes on definite negative values. The discontinuities are studied as jumps between the different solutions of a nonlinear integro-differential equation degenerating at weak dissipation to a third order algebraic equation with an undetermined coefficient. The dependence of the intensity of shocked vibrations on the frequency of vibration of the boundary is found. Nonlinear saturation is shown to appear: the intensity of wave field inside the resonator does not depend on the amplitude of boundary vibration when the amplitude is large. If the amplitude tends to infinity, the intensity tends to its limiting value determined by the nonlinear absorption at shock fronts. This maximum can be reached by smooth increase in frequency above the linear resonance. A hysteresis area and bistability appears, in analogy with the nonlinear resonance phenomena in localized vibration systems described by ordinary differential equations

Stående och fortplantande vågor i ett kubiskt olinjärt medium

The paper has three parts. In the first part a cubically nonlinear equation is derived for a transverse finite-amplitude wave in an isotropic solid. The cubic nonlinearity is expressed in terms of elastic constants. In the second part a simplified approach for a resonator filled by a cubically nonlinear medium results in functional equations. The frequency response shows the dependence of the amplitude of the resonance on the difference between one of the resonator's eigenfrequencies and the driving frequency. The frequency response curves are plotted for different values of the dissipation and differ very much for quadratic and cubic nonlinearities. In the third part a propagating N-wave is studied, which fulfils a modified Burgers' equation with a cubic nonlinearity. Approximate solutions to this equation are found for new parts of the wave profile