Consider a layer with waves **reverberating** in it
as shown in Figure 4.

8-4
Some rays corresponding
to resonance in a layer.
Figure 4 |

Let the delay operator *Z* refer to the time delay of a travel path
across the layer and back again.
The wave seen above the layer
is an **infinite sequence** that is expressible as a simple denominator
(because ).

It is no accident that the infinite
series may be summed.
We will soon see that for *n* layers the waves,
which are of infinite duration,
may be expressed as simple polynomials of degree *n*.
We consider many layers and the general problem
of determining waves given reflection coefficients.

Equation (11) relates to
Figure 5 and shows how from the waves
*U* and *D*' we extrapolate into the future
to get *U*' and *D*.

8-5
Waves incident and
reflected from an interface.
Figure 5 |

(11) |

Let us rearrange (11) to get *U*' and *D*'
on the right and *U* and *D* on the left
to get an equation
which extrapolates from the primed medium to the unprimed medium.
We get

Arrange to matrix form

Premultiply by the inverse of the left-hand matrix

getting a result we want, an equation to extrapolate the infinitesimal distance from the bottom of one layer to the top of the next.

(12) |

Now we consider a layered medium
where each layer has the same travel-time thickness [**Goupillaud** medium].
Presumably, any continuously variable medium
can be approximated this way.
Examine the definitions in the layered medium in Figure 6.

8-6
Goupillaud-type layered medium
(layers have equal travel time).
Figure 6 |

For this arrangement of layers, (12) may be written

(13) |

Inserting (13) into (12) we
get a * layer matrix*
that takes us from the top of one layer
to the top of the next deeper one.

(14) |

3/1/2001