The one-way coupled mode approach discussed in the previous
section provides a great speed-up in execution time but in many cases
still remains too time consuming. For this reason a further
approximation is often invoked in which one neglects the
cross-coupling terms which allow energy from one mode to transfer
into other modes. Instead, one assumes that in going from one range
to the next the modes will couple * adiabatically*, that is,
without any transfer of energy to other modes. This approximation
was introduced to ocean acoustics problems by Pierce [54]
based on analogous results for the Schrödinger equation. Our
derivation follows Pierce, however we note that a somewhat more
formal derivation is given by Weinberg and Burridge in Ref.
[55].

To derive this approximation, we return to the Helmholtz equation in two-dimensions:

Since the modes form a complete set, we can represent the solution at any range as a sum of local modes. We therefore seek a solution of the range-dependent problem in the form

where, are the local modes defined by

and primes denote differentiation with respect to **z**. Thus, at any
range **r**, is found by solving the depth-separated modal
equation with the environmental properties at that range.
Substituting in the Helmholtz equation yields:

where we have used Eq. () to eliminate the **z**-derivatives.
Rearranging terms leads to,

For simplicity we shall now assume that is
independent of **r**. Then we can apply
the operator

and because of the orthogonality property many of the terms in the sum will disappear. The result is:

where,

Note that , since differentiating

gives

Equation () is a statement of coupled modes written for the case of continuous variation of sound speed. It can be solved directly by, for instance, finite differences. The adiabatic approximation can now be stated simply as the assumption that the coupling matrices and are negligible. (Some authors retain the diagonal terms which alters the results slightly.) This yields the set of decoupled equations

The WKB approximation then yields the solution:

The value of **A** is found by requiring that the WKB solution must
match our normal solution, Eq. (), when the problem is
range-independent. Thus,

Substituting this results back into Eq. () we obtain the final result:

In practice, the eigenfunctions and eigenvalues are normally
calculated at a discrete set of ranges: values at intermediate
ranges are then calculated by linear interpolation. Note that the
adiabatic form is sensitive to the polarity of the modes; that is, if
you flip the sign of at some particular range, the
computed pressure is changed. Therefore, care must be taken that
the modes are polarized in a consistent fashion in range. In
* KRAKEN*
the modes are polarized so that they assume a positive value
at the turning point nearest to the surface.

Tue Oct 28 13:27:38 PST 1997