Periodic changes in the tension of a taut string parametrically excite transverse motion in the string when the driving frequency is close to twice the natural frequency of any transverse normal mode of the string. The literature on this phenomenon is synthesized and extended to include the effects of damping as well as nonlinearity. It is shown that it is nonlinearity rather than damping that limits the growth of a resonantly excited mode, although damping is needed for steady-state oscillations to occur. The validity of the usual approximation that the string tension depends only on time and not on space is checked by modeling a string as point masses joined by massless linear springs. It is found that although this approximation is likely to be violated in practice, the violation does not have a significant effect on the results. The source of the disagreement in the literature for the speed of longitudinal waves in a stretched string is identified.
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Equation (16) shows that if and then This condition reveals one of the features of parametric excitation, namely that a parametrically excited mode will only grow if it is “seeded” by random fluctuations in the properties of the system. This seeding is done mathematically by assuming that is a small but nonzero value (see Figs. 345).
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24.
Equation (33) with η given by Eq. (8) gives The amplitudes of both the terms in this equation are small compared to unity because in practice and the amplitude of the second term is one-quarter of the square of the amplitude of and the validity of the third-order wave equations depends on This result means that there are regimes where is no longer negligible in comparison to unity, but still is.
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These solutions are referred to as pseudo-unstable because like unstable solutions, perturbations from the equilibrium solution initially grow exponentially. However, unlike unstable solutions and more like stable solutions, they do not do so without bound and, like stable solutions, periodically return to the perturbation value. (As shown in Ref. 8, the pseudo-unstable solutions travel on closed rather than open curves in the phase plane, although these curves do not remain in the vicinity of the equilibrium solution.)
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Reference 29, Chap. 12.
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