We study the fluctuations in a one-dimensional conductor, coupled to a current source. The latter produces shot noise at the boundary of the conductor. The conductor is treated as a classical system of mobile negative charges (with a neutralizing fixed positive background). The current source is represented by considering time-dependent boundary conditions for the system. We also assume the existence of dissipation in the system, and describe the interacting particles by a Langevin equation. The problem is then transformed into an equivalent quantum-mechanical problem (in an imaginary time) of charged interacting bosons and is solved in the limit of small density fluctuations. We show how the fluctuations, induced by the source, decay along the wire. We find that whereas the boundary conditions represent uncorrelated shot noise, the power spectrum of the current fluctuations in the conductor has a 1/f tail at low frequencies.