Revealing the acoustic spin and beam invariants from the Dirac-like equation

Exploiting the spin of waves is of fundamental interest for wave physics, with pivotal significance for diverse applications ranging from topological matters to information devices. However, only until recently has the spin of acoustic waves been unveiled. Here we theoretically reveal the spin in acoustics from the perspective of the Dirac-like equation. By introducing a four-component wave function that contains both pressure and velocity fields, the analogy between the coupled acoustic equations and the massless Dirac equation is established. Based on this, we find that the orbital angular momentum operator does not commute with the acoustic Dirac-like Hamiltonian, necessitating an additional spin operator to satisfy the angular momentum conservation, which naturally elucidates the origin of acoustic spin. The expression of the spin operator indicates that the scalar pressure field is spin 0 and the vector velocity field is spin 1. We also demonstrate that different types of dynamical properties, such as the canonical-Minkowski and the kinetic-Abraham types, can be unified in our theory framework by applying a weighting factor to the different components of the wave function. Furthermore, we present the invariance of the expectation values of dynamical operators during wave propagation, which corresponds to the conservation of dynamical properties with rational energy normalization. Our method offers insights into acoustics beyond traditional single-variable wave equations.