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Let *x*_{i}, where *i*=1,2,3 be three orthogonal directions and the position vector in a coordinate system associated with the three
directions. Let us define as
the mass per volume unit in the acoustic medium, as the velocity of the acoustic medium and K
as the bulk modulus of the acoustic medium. The second law of dynamics
states that mass acceleration = force = - pressure gradient:

| |
(15) |

Energy can be stored by compression and volume variation. If

| |
(16) |

we say that the flow diverges (the volume changes). This leads to a
pressure variation, proportional to the divergence of the velocity:

| |
(17) |

The wave equation in an acoustic medium can be deduced from
(15) and (17) as follows. Derivate (17)
with respect to time:

| |
(18) |

Divide (15) by and derivate it with respect to the
axis *x*_{i}:

| |
(19) |

Plug (19) in (18):

| |
(20) |

Approximation: is a constant that does not depend on the
position vector. By denoting the acoustic waves propagation velocity
through the medium by v, where

| |
(21) |

we obtain the acoustic wave equation:

| |
(22) |

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Stanford Exploration Project

6/8/2002