On the Law of Motion of the End of a Semi-Bounded String, Which Extinguishes Reflected Waves

The problem of oscillation of a semi-bounded string at a given initial perturbation '(x), when the left end of the string moves according to a given law f(t), is considered (t time). The solution of the problem is obtained in the final form for arbitrary functions '(x) and f(t). The direct, inverse and reflected from the end of the string wave components that make up the resulting waves are constructed. The influence of the function f(t) on the magnitude of the reflected waves is investigated. For an arbitrary initial perturbation of a string, the law of motion of its end is found, at which the reflected waves disappear. The relevance of the article is determined by a wide range of practical problems related to the vibrations of extended objects (bridges, beams, antennas), when the source of motion is an initial perturbation and a given external force applied to the end of the object.