On the non-linear vibration problem of the elastic string

Abstract

Fig 3. the ordinates should be labeled P/P*. ** Equation (15) indicates more precisely the meaning of t.

Hyperbolic Parabolic Equations with Nonlinearity of Kirchhoff-Carrier Type

“…We observe titat witen tite abstract results are realized in tite concret case (iii) with Al we obtain tite results proved by tite metitod of Fourier transform, by Dickey [6] and Menzala [19].…”

“…Kírcititoff [14] and Carrier [6], titat describe small vibrations of a stretcited string when we consider only vertical component for tite tension and titere is time dependence. In titís case tite temperature of tite string at time tis considered.…”

Interactions in a stretched string

“…The problem we shall study is the continuation of Zinc to the upper half plane t > 0. The solutions we obtain are approximate and are based on the assumption that the shear wave speed at Yo> namely the constant c0 defined in (15), is much smaller than the longitudinal wave speed at y0, namely the constant i \ -i 1 d*(yp) clong(To) \l Pq dy ■ A similar hypothesis was invoked by Carrier [1] and Dickey [2,3] in their analysis of the vibrations of a finite string. Since our primary interest is in the behavior of the vertical displacement field and not the detailed structure of the longitudinal field, we shall assume that f(-) behaves linearly near y0.…”

Global Existence and Decay Properties of Solutions for Coupled Degenerate Dissipative Hyperbolic Systems of Kirchhoff Type

“…The coupled hyperbolic system (1.3)-(1.4) will be useful for the research of amplitude vibrations of two kinds of elastic stretched strings (see [19] with uð0Þ ¼ u 0 , u t ð0Þ ¼ u 1 , and uj qW ¼ 0, which is called a non-degenerate equation when m > 0 and a degenerate one when m ¼ 0. The single equation (1.5) was introduced by Kirchho¤ [7] (also see [5], [6]). …”