Reconstruction of the initial state from the data measured on a sphere for plasma-acoustic wave equations

We establish a reconstruction formula for the initial information of a density (pressure) from the time records of data measured on a sphere for a certain class of wave-type equations. We first derive a reconstruction formula of integral form, and by applying the theory of Fourier Bessel series for this, we also obtain a formula of discrete version, which is more robust and numerically advantageous in the presence of noises. The reconstruction formulas we propose in this work are explicit and applicable to a wider class of wave-type equations including the plasma-acoustic wave equations.