Next: Example: A Warm-Core Up: Normal Modes for Previous: One-way Coupled Modes

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.

Next: Example: A Warm-Core Up: Normal Modes for Previous: One-way Coupled Modes

Michael B. Porter
Tue Oct 28 13:27:38 PST 1997