One-dimensional single-sided acoustic focusing in a Goupillaud layered model

Using the single-sided reflection response of an unknown layered medium, one-dimensional single-sided acoustic focusing is a process that constructs a normally incident plane waveform such that the sound field collapses to a point inside this medium at a specified time. A recursive method for finding the incident waveform of the discrete single-sided focusing in the Goupillaud layered model is presented. The method is formulated using the standard Z-transform and provides an intuitive way to understand how single-sided focusing works in an unknown medium. Two numerical scattering experiments, with different layered structures, are simulated to verify the proposed method. In addition, the responses to virtual sources in the examples are reconstructed.