Autonomous structural health monitoring (SHM) of aerostructures strengthens the reliability, increases the lifetime, and reduces the maintenance cost of aerovehicles such as airplanes and unmanned aerial vehicles (UAV). The continuous monitoring of aerostructures for early damage detection and identification is made possible through a wireless network of sensors deployed on the structure. Usually, the data collected by these sensors is communicated to a central unit for real-time data processing using electromagnetic waves at radio frequencies (RF). However, the emission of RF signals for autonomous SHM creates additional sources of interference to on-board RF communication systems used for aircraft control and safety-related services. To overcome this issue, we propose in this paper an acoustic data communication system for autonomous health monitoring of aerostructures which are modeled as thin plate-like structures. In the proposed system, both damage detection and wireless communication are performed using guided elastic waves. Data communication across an elastic channel is challenging because of the severe frequency-dispersive and multimodal propagation in solid media which distorts, delays, and greatly attenuates the transmitted data signals. To cope with this problem, we introduce a sensor network based on time-reversal pulse position modulation that compensates for channel dispersion and improves the signal-to-noise ratio of the communication link without relying on sophisticated channel estimation algorithms. We demonstrate the viability of the presented system by conducting experiments on an homogeneous and isotropic aluminum plate specimen using Lead Zirconate Titanate (PZT) sensor discs at a resonant frequency of 300 kHz.