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The acoustic wave equation describes sound waves in a liquid or gas.
Another more complicated set of equations describes elastic waves in solids.
Begin with the acoustic case.
Define

`
= mass per unit volume of the fluid
`*u* = velocity flow of fluid in the *x*-direction
*w* = velocity flow of fluid in the *z*-direction
*P* = pressure in the fluid

Newton's law of momentum conservation says that a small volume within a
gas will accelerate if there is an applied force.
The force arises from pressure differences at opposite sides of
the small volume.
Newton's law says

| |
(9) |

| (10) |

The second physical process is energy storage by compression and volume change.
If the velocity vector *u* at exceeds that at *x*, then
the flow is said to be diverging.
In other words, the
small volume between *x* and is expanding.
This expansion must lead to a pressure drop.
The amount of the pressure drop is in proportion to a property of the fluid
called its
*incompressibility*
*K*.
In one dimension the equation is

| |
(11) |

| |
(12) |

In two dimensions it is
| |
(13) |

To arrive at the one-dimensional wave equation from
(9) and (12),
first divide (9) by and
take its *x*-derivative:
| |
(14) |

Second, take the time-derivatives of (12) and (13).
In the solid-earth sciences we are fortunate
that the material in question does not change during our experiments.
This means that *K* is a constant function of time:
| |
(15) |

Inserting (14) into (15),
the one-dimensional scalar wave equation appears.
| |
(16) |

In two space dimensions, the exact, acoustic scalar wave equation is
| |
(17) |

You will often see the scalar wave equation in a simplified form,
in which it is assumed that is
not a function of *x* and *z*.
Two reasons are often given for this approximation.
First, observations are generally unable to determine density,
so density may as well be taken as constant.
Second, we will soon see that
Fourier methods of solution do not work for space variable coefficients.
Before examining the validity of this approximation,
its consequences will be examined.
It immediately reduces (17) to the usual form
of the scalar wave equation:
| |
(18) |

To see that this equation is a restatement of
the geometrical concepts of previous sections,
insert the trial solution

| |
(19) |

What is obtained is the
*
dispersion relation of the two-dimensional scalar wave equation:
*
| |
(20) |

Later
an equation like (20) will be developed by
considering only the geometrical behavior of waves.
In that development the wave velocity squared is found
where stands in equation (20).
Thus physics and geometry will be reconciled by the association
| |
(21) |

Last, let us see why Fourier methods fail when the velocity is
space variable.
Assume that , *k*_{x}, and *k*_{z} are constant functions
of space.
Substitute (19) into (18)
and you get the contradiction that
, *k*_{x}, and *k*_{z} must be space variable
if the velocity is space variable.
Try again assuming space variability,
and the resulting equation is still a differential equation,
not an algebraic equation like (20).

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Stanford Exploration Project

10/31/1997