A d'Alembert-type formula for longitudinal oscillations of an infinite rod consisting of two segments with different densities and elasticities

A d' Alembert-type formula has been derived, which gives a Cauchy solution describing longitudinal oscillations of an infinite rod consisting of two segments. The two segments are x ≤ 0 with linear density p1 = const and Young modulus k1 = const and the segment x≥ with linear density p2 = const and Young modulus k2 = const. The problem reduces to finding a solution to the Cauchy problem for the discontinuous wave equation. It is assumed that the function φ(x) belongs to the class 22, loc on each of the half-lines x≤0 and x≥0 and satisfy the conjugation conditions.