To develop the wave equation, Equation 1, from first principles we will consider the disturbance of a fluid-like medium. The conservation of mass and momentum provide the basis for development of the acoustic wave equation (, , , and ). The mass density is , the particle velocity is , and the fluid pressure is P. The three spatial coordinates are for the domain . Particle velocities are for each direction .
The conservation of momentum is
The conservation of mass, the continuity equation, is
The incompressible fluid flow equation can be derived from the Navier--Stokes equations. The form used here is Euler's equation when the viscosity is zero and uses D as the Substantial Derivative
The body forces are negligible, . Notice that the repeated index used in the equations indicate a tensor summation.