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4 Wave Equation

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 ([14], [5], [6], and [3]). 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.