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The solutions to the partial differential equations (7) and
(8) are obtained using the method of characteristics.
The characteristic curves are curves on which the partial
differential equation in several independent parameters (in
our case the parameters (*x*_{0}, *t*_{0}) ) can be transformed
into a system of ordinary differential equations in a single
parameter along the characteristic curve.
A partial differential equation of the form

can be reduced to the form
if we can find a parameter such that
so that we can express the variables (*x*, *t*) in the form
As a result we have

which we can introduce in the original equation.
For equation (7) the system of associated equations is

| |
(9) |

which after integration becomes
| |
(10) |

The constants *c*_{1}, *c*_{2}, *c*_{3} are functions of the parameter
given on the initial curve, and are determined by setting in equations (10).
The initial condition is given along an ellipse, and has the form

| |
(11) |

As a result the solution for the function *u* is given
in parametric form
| |
(12) |

For equation 8 the system of associated equations is

| |
(13) |

which after integration becomes
| |
(14) |

The constants *c*_{1}, *c*_{2}, *c*_{3} are functions of the parameter
given on the initial curve, and are determined by setting in equations (14).
The initial condition is given along a curve of constant
parameter, and has the form

| |
(15) |

As a result the solution for the function *u* is given
in parametric form

| |
(16) |

The amplitudes of the function *u* along the characteristic curves
are found by calculating the integrals in
equations (9) and (14).

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Stanford Exploration Project

1/13/1998