Layer peeling bore reconstruction

Consider a junction between two infinite tubes. If a pressure wave is incident from negative infinity and no wave is incident from positive infinity, then the ratio of the backward and forward going waves is given by the reflection coefficient of equation (2.20).

Going back to the general case of an object attached to the source tube
of a reflectometer from figure 5.1, at the first time step
(ie. when the forward going pulse arrives at the input plane), there cannot
be any backward going waves on surface 2.
The ratio of the first term in the
reflection sequence (or input impulse response) and the first term in the
input sequence (which is unity) is therefore equal to the reflection
coefficient from equation (2.20). We will
label this reflection coefficient as . The subscripts indicate that
this reflection coefficient is for reflection from the discontinuity between
surfaces 1 and 2 when waves are incident from the surface 1 side only.

where we have labelled the surface area of surface 1 as and the surface area of surface 2 as . Now follows from and by rearranging equation (5.8):

Now we know the cross-section on section 2, we will proceed to calculate the forward and backward going time sequences there. The forward going wave on section 2 is equal to the sum of the transmission of the forward going wave on section 1 and reflection of the backward going wave on section 2. Similarly, the backward going wave on section 1 is equal to the sum of the transmission of the backward going wave on section 2 and the reflection of the forward going wave on section 1. It can be expressed in matrix notation as follows:

gives the reflection coefficient from the discontinuity between surfaces 2 and 1 when waves are incident from the surface 2 side only:

(5.12) |

(5.13) |

(5.14) |

This equation is performed for all values of from to . For the current situation, the forward going sequence on surface 1 is an impulse, , and the backward going pressure sequence on surface 1 is the input impulse response, . Notice that

(5.16) |

The forward travelling pressure sequence, at the right
hand side of the cylindrical section, will be found simply by adding a
delay time of to the forward going wave on the left, to account
for the time taken to travel a distance of .

The backward travelling wave at surface 3 is found by subtracting a delay of
from the backward travelling wave at surface 2.

(5.19) |

The ratio of the backward and forward travelling waves at
surface 3, at time , is equal to the reflection coefficient
since this corresponds to a primary reflection.

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