The aim of the current study is to explore the formation of delta-shock and vacuum for the pressureless Euler system with zeroth-order terms modelling acoustic generation. For this purpose, we solve the Riemann problem for the perturbed logarithmic Euler equations and obtain four kinds of different structures composed of rarefaction wave(s) and/or shock(s) through the use of the variable substitution method. Unlike the homogeneous case and the nonhomogeneous case containing only a friction term, all waves propagate at an exponential rate. It is proved that the solution involving delta-shock of the pressureless Euler system with zeroth-order terms can be formed as limits of the solution consisting of two shocks of the perturbed logarithmic Euler equations, and the solution including vacuum of the pressureless Euler system with zeroth-order terms can be formed as limits of the solution formed by two rarefaction waves of the perturbed logarithmic Euler equations.
