  We consider non-stationary oscillations of an infinite string with
time-varying tension. The string lies on the Winkler foundation with a point
inhomogeneity (a concentrated spring of negative stiffness). In such a system
with constant parameters (the string tension), under certain conditions a
trapped mode of oscillation exists and is unique. Therefore, applying a
non-stationary external excitation to this system can lead to the emergence of
the string oscillations localized near the inhomogeneity. We provide an
analytical description of non-stationary localized oscillations of the string
with slowly time-varying tension using the asymptotic procedure based on
successive application of two asymptotic methods, namely the method of
stationary phase and the method of multiple scales. The obtained analytical
results were verified by independent numerical calculations based on the finite
difference method. The applicability of the analytical formulas was
demonstrated for various types of external excitation and laws governing the
varying tension. In particular, we have shown that in the case when the trapped
mode frequency approaches zero, localized low-frequency oscillations with
increasing amplitude precede the localized string buckling. The dependence of
the amplitude of such oscillations on its frequency is more complicated in
comparison with the case of a one degree of freedom system with time-varying
stiffness.
