Abstract
AbstractHere we study the motion of a vibrating string in the presence of an arbitrary obstacle. We show that if the string always rebounds on the concave parts of the obstacle, it can either rebound or roll on the convex parts. The latter is the case if the velocity of the string is null at the contact point just before contact, or if the contact point propagates at a characteristic speed. Four examples are given. The three first correspond to the same obstacle, a sinusoidal arc, but with different initial conditions. In the first case, the string rebounds on the whole of the obstacle and the motion is explicitly determined when it is periodic. In the second case, the string rolls on the convex part of the obstacle up to the inflexion point and then rebounds on the concave part and unwinds on the convex part. In the third case, the string is initially at rest on the obstacle; then it instantaneously leaves the concave part while it unwinds progressively on the convex part. The fourth case is similar to the third but with a different obstacle; the motion, which is periodic, is determined explicitly.
Highlights
Problems related to the study of the motion of a vibrating string in presence of an obstacle were first considered by Amerio and Prouse in 1975 [3]
In the case of curvilinear obstacles a fundamental difference appears between concave obstacles and convex obstacles
There are only a few papers related to the cases of curvilinear obstacles, in particular one by Schatzman [12] on concave obstacles, while Bumdge et al [4], and Ameno [2] considered convex obstacles
Summary
Problems related to the study of the motion of a vibrating string in presence of an obstacle were first considered by Amerio and Prouse in 1975 [3]. If the obstacle is concave, the string rebounds instantaneously, while if the obstacle is convex the string can either rebound or can roll up while remaining in contact with the obstacle for some time. This last behaviour appears when the string vibrates in the presence of a point-mass obstacle [ll], [S). In all cases the reflection of the string on the obstacle takes place with conservation of energy For both obstacles, periodic motions have been obtained, the string being initially at rest with an appropriate shape ($$ 5 and 8); if the initial conditions are different, the periodic character of the motion disappears. The determination of the limits of the arcs of the string in contact with the obstacle during the wrapping and during the unwrapping is a free boundary problem, as Amerio has already shown [2]
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