The Wave Equation for a Moving Source and a Moving Receiver - Hrvoje Dodig (Mathematics, 2024)
The ordinary 3D wave equation for nondissipative, homogeneous, isotropic media admits solutions where the point sources are permitted to move, but as shown in this paper, it does not admit solutions where the receiver is allowed to move. To overcome this limitation, a new wave equation that permits both the receiver and the source to move is derived in this paper. This new wave equation is a generalization of the standard wave equation, and it reduces to the standard wave equation when the receiver is at rest. To derive this new wave equation, we first mathematically define a diverging spherical wave caused by a stationary point source. From this purely mathematical definition, the wave equation for a stationary source and a moving receiver is derived, together with a corresponding free-space Green function. Utilizing the derived Green function, it is shown that unlike the standard wave equation this new wave equation also permits solutions where both the receiver and the source are permitted to move. In conclusion, this paper demonstrates that, instead of an ordinary wave equation, the wave equation for a moving source and a moving receiver governs the waves emitted by moving point sources and received by moving receivers. This new wave equation has possible applications in acoustics, electrodynamics, and other physical sciences.