Triple-wave ensembles in a thin cylindrical shell — страница 5

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in the particular case the triple-wave phase matching is reduced to the so-called resonance 2:1. This one can be proposed as the main instability mechanism explaining some experimentally observed patterns in shells subject to periodic cinematic excitations [4]. It was pointed out in the paper [5] that the resonance 2:1 is a rarely observed in shells. The so-called resonance 1:1 was proposed instead as the instability mechanism. This means that the primary axisymmetric mode (with ) can be unstable one with respect to small perturbations of the asymmetric mode (with ) possessing a natural frequency closed to that of the primary one. From the viewpoint of theory of waves this situation is treated as the degenerated four-wave resonant interaction. In turn, one more mechanism

explaining the loss of stability of axisymmetric waves in shells based on a paradigm of the so-called nonresonant interactions can be proposed [6,7,8]. By the way, it was underlined in the paper [6] that theoretical prognoses relevant to the modulation instability are extremely sensible upon the model explored. This means that the Karman-type equations and Donnell-type equations lead to different predictions related the stability properties of axisymmetric waves. Self-action The propagation of any intense bending waves in a long cylindrical shell is accompanied by the excitation of long-wave displacements related to the in-plane tensions and rotations. In turn, these long-wave fields can influence on the theoretically predicted dependence between the amplitude and frequency of

the intense bending wave. Moreover, quasi-harmonic bending waves, whose group velocities do not exceed the typical propagation velocity of shear waves, are stable against small perturbations within the lowest-order nonlinear approximation analysis. However amplitude envelopes of these waves can be unstable with respect to small long-wave perturbations in the next approximation. Amplitude-frequency curve Let us consider a stationary wave traveling along the single direction characterized by the ''companion'' coordinate . By substituting this expression into the first and second equations of the set (1)-(2), one obtains the following differential relations (15) Here while where and . Using (15) one can get the following nonlinear ordinary differential equation of the fourth order:

(16), which describes simple stationary waves in the cylindrical shell (primes denote differentiation). Here where and are the integration constants. If the small parameter , and , , satisfies the dispersion relation (4), then a periodic solution to the linearized equation (16) reads where are arbitrary constants, since . Let the parameter be small enough, then a solution to eq.(16) can be represented in the following form (17) where the amplitude depends upon the slow variables , while are small nonresonant corrections. After the substitution (17) into eq.( 16) one obtains the expression of the first-order nonresonant correction and the following modulation equation (18), where the nonlinearity coefficient is given by . Suppose that the wave vector is conserved in the nonlinear

solution. Taking into account that the following relation holds true for the stationary waves, one gets the following modulation equation instead of eq.(18): or , where the point denotes differentiation on the slow temporal scale . This equation has a simple solution for spatially uniform and time-periodic waves of constant amplitude : , which characterizes the amplitude-frequency response curve of the shell or the Stocks addition to the natural frequency of linear oscillations: (19). Spatio-temporal modulation of waves Relation (19) cannot provide information related to the modulation instability of quasi-harmonic waves. To obtain this, one should slightly modify the ansatz (17): (20) where and denote the long-wave slowly varying fields being the functions of arguments and

(these turn in constants in the linear theory); is the amplitude of the bending wave; , and are small nonresonant corrections. By substituting the expression (20) into the governing equations (1)-(2), one obtains, after some rearranging, the following modulation equations (21) where is the group velocity, and . Notice that eqs.(21) have a form of Zakharov-type equations. Consider the stationary quasi-harmonic bending wave packets. Let the propagation velocity be , then eqs.(21) can be reduced to the nonlinear Schrцdinger equation (22), where the nonlinearity coefficient is equal to , while the non-oscillatory in-plane wave fields are defined by the following relations and . The theory of modulated waves predicts that the amplitude envelope of a wavetrain governed by eq.(22) will