### ASA 125th Meeting Ottawa 1993 May

## 5pUW3. Prufer transforms and numerical solutions to the Helmholtz
equation.

**Arthur B. Baggeroer
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*MIT, Rm. 5-204, Cambridge, MA 02139
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The Prufer transformation maps the second-order, linear Helmholtz
differential equation to two first-order, nonlinear differential equations. The
coupling between the two separates such that the equation for the eigenvalue is
a single nonlinear first-order equation. This transformation has long been
useful in theoretical studies of Strum--Liouville where many properties of the
eigenvalues have been derived using it. It turns out also to have some useful
numerical properties especially for problems involving waveguides where the
coupling between the boundaries is poorly conditioned. It also identifies
explicitly the both stability points in the evanescent sections of the modes
and the local oscillation frequency in the ducted section. The numerical
aspects of using the Prufer transform and some examples are discussed in the
presentation.