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.
